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Talk:Cantor's diagonal argument

Set of all sets
''Moved to arguments page. --Trovatore (talk) 06:30, 16 July 2021 (UTC)''

diagonal counterexample
''Moved to arguments page. --Trovatore (talk) 06:30, 16 July 2021 (UTC)''

In his 1891 article...
"In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one):"

No, he didn't. In his 1891 article, Cantor considered a bicoloring by m and w. It is standard in wikipedia for modified proofs to be indicated as such and it is amateurish not to do so when the article itself is referred to by date and linked to in the footnotes. The proof should either conform to the reference or (if we're so endeared to the modification that we prefer it to the original) be indicated as an updated or simplified restatement thereof. — Preceding unsigned comment added by 2603:7000:8E03:3E7E:6979:D94F:B78E:B1DA (talk) 18:56, 15 November 2021 (UTC)

Cantor's proof is unrelated to binary sequences. Binary sequences are related to Cantor's proof. — Preceding unsigned comment added by 2603:7000:8E01:2B47:F8AA:FA7F:B5CA:53D8 (talk) 19:59, 19 April 2022 (UTC)

diagonal = contradiction
Duplicate of talk:Cantor's diagonal argument/Arguments

Clarifying subtle points
Most people learn CDA as a compact, one-step proof. If you assume that R (or sometimes the range [0,1]) can be enumerated, then you can prove that R has not been enumerated. The contradiction "disproves" the assumption.

Ignoring the fact that Cantor (explicitly) did not apply diagonalization to real numbers, this is not valid as a proof by contradiction. The supposed proof never uses the assumption that all members of R (or whatever set) are enumerated, only that the enumeration contains members of R. And the statement that is contradicted is the part of the assumption that isn't used. That is closer to a direct proof.

The previous Wiki did get all this right, but was subject to intentional misinterpretation. Many Cantor Doubters claim to "disprove' CDA based on this self-contradictory assumption. And use the previous text of this Wiki as evidence of what Cantor "did wrong." Since he did no such thing, the Wiki should not allow people to believe he did.

So I feel that several points needed to be made more explicit. First, that the diagonalization procedure is only ever applied to a subset of T that is known, not assumed, to be countable. Literally (in the notation used here), ""If s1, s2, …, sn, … is any simply infinite series of elements of T ..."

Technically, we should demonstrate that such subsets do exist, but examples are trivial and Cantor did not. So it is acceptable to not do so. Cantor proved that any countable subset of T necessarily omits an element s that is in T. This is the first part of a two-part proof.

But the second part is the important one: IfT is countable, then we know that the proven result applies to it without actually diagonalizing it. And the contradiction Cantor uses is not that T both contains, and does not contain, every sequence. It is that the sequence s both is, and is not, in T.

For reference, I use the translation at http://www.logicmuseum.com/cantor/diagarg.htm JeffJor (talk) 15:38, 23 March 2023 (UTC)

 * If you have a point in your above arguments, then I didn't get it. (Maybe you tried to elaborate on a typical fringe disproof, and make explicit where it is wrong?)
 * The proof in the article's current version is pretty clear. Introducing (as you did in your recent edit) a set S = { s1, ..., sn, ... } doesn't add clarity, imo; this is whhy I reverted you. - Jochen Burghardt (talk) 16:43, 23 March 2023 (UTC)

Notion orthogonal to theorems
With respect to this edit, what does it mean to say that a notion is orthogonal to a theorem? It doesn't mean anything to a lay reader; does it mean anything to mathematically-trained readers? MartinPoulter (talk) 16:37, 22 April 2023 (UTC)

 * The word seems to have survived a lot of rewriting, particularly since about March this year, but has been there for longer in a slightly different sentence and place in the paragraph with an explicit concrete example, through which it may have been clearer at the time. I think those last two sentences from the section 'Ordering of cardinals' back then might make it clearer. I don't think 'orthogonal' is used in a particularly technical mathematical way here though. It seems it just means to say that they don't particularly have much to do with each other, that subcountability is irrelevant to the statement about the cardinality, countability and uncountability and is independent in that it doesn't really immediately have anything to do with that. I think really just to say that, since it is defined in terms of surjection, it has no bearing on theorems about injections, so in that way they are independent. It is just a certain broader meaning of orthogonal.
 * The newer wording appears to simply be more general, I am not certain the wording about orthogonality of a notion is the best and clearest, perhaps it should be expanded a bit with explanation, but I think what is meant is as simple as that subcountability is defined in terms of surjection, so anything being subcountable has to do with surjections, and theorems about injections are about injections, which is something else than surjections. So one could prove theorems or assert things about the one without that necessarily having bearing upon stuff to do with the other. MathsWolf (talk) 12:15, 15 June 2023 (UTC)
 * Thanks User:MathsWolf for the explanation. Glad to hear from someone much more familiar with the topic than I. Would you like to have a go at rephrasing that sentence so the word "orthogonal" isn't used. It's an unfamiliar term to a lot of readers and it seems excessive precision to use that when the intended meaning is something like "irrelevant" or "different". Cheers, MartinPoulter (talk) 15:05, 15 June 2023 (UTC)

not written for the average user
like so many math/logic/science articles, this is not written for the average user why are you math people so bad at describing stuff in terms an ordinary person can understand ? that is mean; that is just reality — Preceding unsigned comment added by 2601:197:d00:3ca0:adb1:4bc0:d097:b9a6 (talk) 04:12, 21 February 2024 (UTC)
 * You are more than welcome to suggest improvements to the article, including ways it can be made more understandable to more readers. --Trovatore (talk) 20:30, 23 February 2024 (UTC)
 * Nice paradox! He says he can't understand it so you suggest he rephrases it! :) BioImages2000 (talk) 11:49, 21 May 2024 (UTC)
 * There was no actionable complaint in the post. Comments that just whine about the article being too hard to understand are of minimal value.  If they give specifics as to where they're getting stuck, there might be something to fix.  --Trovatore (talk) 21:10, 21 May 2024 (UTC)

Cantor proved this for reals in [0,1]--for a reason
The page needs to clearly state that these strings of digits represent binary fractions between 0 and 1 inclusive.

This version of the proof differs from Cantor's original in two ways: (1) it uses base 2 numbers instead of base 10 numbers; and (2) it uses arbitrary strings of digits--which include leading zeros-- rather than place notation. The proof as given holds for arbitrary strings of digits, but not if the digits are interpreted as numbers.

When digits are used to express whole numbers, leading zeros are not significant. 1 = 01 = 001 = 0001 = 000001 etc. So too with rational and real numbers greater than 1: 1.1 = 01.1 = 001.1 = 0001.1, etc. That is why Cantor used decimal fractions in [0,1]: 0.1 is different from 0.01. etc.

So in Wiki's form of the proof, to prove that the number 7 is missing from say a 6 row x 6 column table, one must show that 1 1 1 0 0 0, 0 1 1 1 0 0, 0 0 1 1 1 0, and 0 0 0 1 1 1 are missing--since all these numerals represent the same number. That is not shown--and indeed is not true--making the proof (which is otherwise air-tight) invalid. Mgryan (talk) 01:50, 16 May 2024 (UTC)

 * The proof (text between "Cantor considered the set T of all infinite sequences of binary digits" and "Therefore, T is uncountable.") uses infinite sequences of binary digits. These are mathematical objects by their own, and are not interpreted as natural or real numbers. Lateron, in subsection Cantor's_diagonal_argument, the uncountability of the set of real numbers between 0 and 1 is shown as a corollary. - Jochen Burghardt (talk) 05:20, 16 May 2024 (UTC)

Proposed disproof
Consider the set of all numbers to N elements, there are 2 power N possible numbers. S only traverses N numbers so it can never keep up and lags further and further behind as N increases. Especially when N reaches infinity. So S is not-equal to only a small subset of the numbers in the set and may well equal one of the others. BioImages2000 (talk) 11:45, 21 May 2024 (UTC)

 * This belongs to the page Talk:Cantor's_diagonal_argument/Arguments. I'd recommend to try to make your argument "waterproof" before. - Jochen Burghardt (talk) 08:06, 22 May 2024 (UTC)
 * ...or even "bulletproof" :) CiaPan (talk) 10:31, 22 May 2024 (UTC)
 * I am the only decent antiset theorist editor on Wikipedia.
 * The disproof is valid except is missing several parts. For one it should not be applied first to the general set theory diagonal argument but to the powerset proof that exponential of cardinal infinity does not equal the same infinity but rather its exponential only, second should point out certain tree proofs prove exponential of cardinal aleph0 does equal aleph0 .  And a few other things missing.
 * the author is apparently not a mathematician so I don’t see that the arguments page would be appropriate, unless you people want to argue it amongst yourselves. Even then the arguments page doesn’t properly in its current form link, or post changes in Victor Kosko (talk) 23:32, 22 May 2024 (UTC)
 * Or post changes in watchlist Victor Kosko (talk) 23:37, 22 May 2024 (UTC)
 * You just have to add it separately to your personal watchlist. - Jochen Burghardt (talk) 04:13, 23 May 2024 (UTC)

title: Talk:Cantor's diagonal argument


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Hyperreal number

In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number $$x$$ is said to be finite if, and only if, $$|x|<n$$ for some integer $$n$$. $$x$$ is said to be infinitesimal if, and only if, $$|x|<1/n$$ for all positive integers $$n$$. The term "hyper-real" was introduced by Edwin Hewitt in 1948.

The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since $$\sin({\pi n})=0$$ for all integers n, one also has $$\sin({\pi H})=0$$ for all hyperintegers $$H$$. The transfer principle for ultrapowers is a consequence of Łoś's theorem of 1955.

Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated.

The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis. One immediate application is the definition of the basic concepts of analysis such as the derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes $$f'(x) = \operatorname{st}\left( \frac{f(x + \Delta x) - f(x)}{\Delta x} \right)$$ for an infinitesimal $$\Delta x$$, where st(⋅) denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.

Transfer principle
The idea of the hyperreal system is to extend the real numbers R to form a system *R that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form "for any number x ..." that is true for the reals is also true for the hyperreals. For example, the axiom that states "for any number x, x + 0 = x" still applies. The same is true for quantification over several numbers, e.g., "for any numbers x and y, xy = yx." This ability to carry over statements from the reals to the hyperreals is called the transfer principle. However, statements of the form "for any set of numbers S ..." may not carry over. The only properties that differ between the reals and the hyperreals are those that rely on quantification over sets, or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties. The kinds of logical sentences that obey this restriction on quantification are referred to as statements in first-order logic.

The transfer principle, however, does not mean that R and *R have identical behavior. For instance, in *R there exists an element ω such that
 * $$ 1<\omega, \quad 1+1<\omega, \quad 1+1+1<\omega, \quad 1+1+1+1<\omega, \ldots. $$

but there is no such number in R. (In other words, *R is not Archimedean.) This is possible because the nonexistence of ω cannot be expressed as a first-order statement.

Use in analysis
Informal notations for non-real quantities have historically appeared in calculus in two contexts: as infinitesimals, like dx, and as the symbol ∞, used, for example, in limits of integration of improper integrals.

As an example of the transfer principle, the statement that for any nonzero number x, 2x ≠ x, is true for the real numbers, and it is in the form required by the transfer principle, so it is also true for the hyperreal numbers. This shows that it is not possible to use a generic symbol such as ∞ for all the infinite quantities in the hyperreal system; infinite quantities differ in magnitude from other infinite quantities, and infinitesimals from other infinitesimals.

Similarly, the casual use of 1/0 = ∞ is invalid, since the transfer principle applies to the statement that zero has no multiplicative inverse. The rigorous counterpart of such a calculation would be that if ε is a non-zero infinitesimal, then 1/ε is infinite.

For any finite hyperreal number x, the standard part, st(x), is defined as the unique closest real number to x; it necessarily differs from x only infinitesimally. The standard part function can also be defined for infinite hyperreal numbers as follows: If x is a positive infinite hyperreal number, set st(x) to be the extended real number $$+\infty$$, and likewise, if x is a negative infinite hyperreal number, set st(x) to be $$-\infty$$ (the idea is that an infinite hyperreal number should be smaller than the "true" absolute infinity but closer to it than any real number is).

Differentiation
One of the key uses of the hyperreal number system is to give a precise meaning to the differential operator d as used by Leibniz to define the derivative and the integral.

For any real-valued function $$f,$$ the differential $$df$$ is defined as a map which sends every ordered pair $$(x,dx)$$ (where $$x$$ is real and $$dx$$ is nonzero infinitesimal) to an infinitesimal
 * $$df(x,dx) := \operatorname{st}\left(\frac{f(x + dx) - f(x)}{dx}\right) \ dx.$$

Note that the very notation "$$dx$$" used to denote any infinitesimal is consistent with the above definition of the operator $$d,$$ for if one interprets $$x$$ (as is commonly done) to be the function $$f(x)=x,$$ then for every $$(x,dx)$$ the differential $$d(x)$$ will equal the infinitesimal $$dx$$.

A real-valued function $$f$$ is said to be differentiable at a point $$x$$ if the quotient
 * $$\frac{df(x,dx)}{dx}=\operatorname{st}\left(\frac{f(x + dx) - f(x)}{dx}\right)$$

is the same for all nonzero infinitesimals $$dx.$$ If so, this quotient is called the derivative of $$f$$ at $$x$$.

For example, to find the derivative of the function $$f(x)=x^2$$, let $$dx$$ be a non-zero infinitesimal. Then,

 * $$\frac{df(x,dx)}{dx}$$
 * $$=\operatorname{st}\left(\frac{f(x + dx) - f(x)}{dx}\right)$$
 * $$=\operatorname{st}\left(\frac{x^2 + 2x \cdot dx + (dx)^2 -x^2}{dx}\right)$$
 * $$=\operatorname{st}\left(\frac{2x \cdot dx + (dx)^2}{dx}\right)$$
 * $$=\operatorname{st}\left(\frac{2x \cdot dx}{dx} + \frac{(dx)^2}{dx}\right)$$
 * $$=\operatorname{st}\left(2x + dx\right)$$
 * $$=2x$$
 * }
 * $$=\operatorname{st}\left(\frac{2x \cdot dx + (dx)^2}{dx}\right)$$
 * $$=\operatorname{st}\left(\frac{2x \cdot dx}{dx} + \frac{(dx)^2}{dx}\right)$$
 * $$=\operatorname{st}\left(2x + dx\right)$$
 * $$=2x$$
 * }
 * $$=\operatorname{st}\left(2x + dx\right)$$
 * $$=2x$$
 * }
 * $$=2x$$
 * }
 * $$=2x$$
 * }

The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity. Dual numbers are a number system based on this idea. After the third line of the differentiation above, the typical method from Newton through the 19th century would have been simply to discard the dx2 term. In the hyperreal system, dx2 ≠ 0, since dx is nonzero, and the transfer principle can be applied to the statement that the square of any nonzero number is nonzero. However, the quantity dx2 is infinitesimally small compared to dx; that is, the hyperreal system contains a hierarchy of infinitesimal quantities.

Using hyperreal numbers for differentiation allows for a more algebraically manipulable approach to derivatives. In standard differentiation, partial differentials and higher-order differentials are not independently manipulable through algebraic techniques. However, using the hyperreals, a system can be established for doing so, though resulting in a slightly different notation.

Integration
Another key use of the hyperreal number system is to give a precise meaning to the integral sign ∫ used by Leibniz to define the definite integral.

For any infinitesimal function$$ \ \varepsilon(x), \ $$one may define the integral $$\int(\varepsilon) \ $$as a map sending any ordered triple $$(a,b,dx)$$ (where$$ \ a \ $$and$$ \ b \ $$are real, and$$ \ dx \ $$is infinitesimal of the same sign as $$ \, b-a$$) to the value
 * $$\int_a^b(\varepsilon,dx):=\operatorname{st}\left(\sum_{n=0}^N\varepsilon(a+n \ dx)\right),$$

where$$ \ N \ $$is any hyperinteger number satisfying$$ \ \operatorname{st}(N \ dx) = b-a.$$

A real-valued function $$f$$ is then said to be integrable over a closed interval$$ \ [a,b] \ $$if for any nonzero infinitesimal$$ \ dx, \ $$the integral
 * $$\int_a^b(f \ dx,dx)$$

is independent of the choice of$$ \ dx.$$ If so, this integral is called the definite integral (or antiderivative) of $$f$$ on$$ \ [a,b].$$

This shows that using hyperreal numbers, Leibniz's notation for the definite integral can actually be interpreted as a meaningful algebraic expression (just as the derivative can be interpreted as a meaningful quotient).

Properties
The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology.

The use of the definite article the in the phrase the hyperreal numbers is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by Vladimir Kanovei and Saharon Shelah shows that there is a definable, countably saturated (meaning ω-saturated but not countable) elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers. Furthermore, the field obtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if one assumes the continuum hypothesis.

The condition of being a hyperreal field is a stronger one than that of being a real closed field strictly containing R. It is also stronger than that of being a superreal field in the sense of Dales and Woodin.

Development
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed.

From Leibniz to Robinson
When Newton and (more explicitly) Leibniz introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as Euler and Cauchy. Nonetheless these concepts were from the beginning seen as suspect, notably by George Berkeley. Berkeley's criticism centered on a perceived shift in hypothesis in the definition of the derivative in terms of infinitesimals (or fluxions), where dx is assumed to be nonzero at the beginning of the calculation, and to vanish at its conclusion (see Ghosts of departed quantities for details). When in the 1800s calculus was put on a firm footing through the development of the (ε, δ)-definition of limit by Bolzano, Cauchy, Weierstrass, and others, infinitesimals were largely abandoned, though research in non-Archimedean fields continued (Ehrlich 2006).

However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. Robinson developed his theory nonconstructively, using model theory; however it is possible to proceed using only algebra and topology, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers per se, aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic, although they were discovered by the application of model theoretic techniques from logic. Hyper-real fields were in fact originally introduced by Hewitt (1948) by purely algebraic techniques, using an ultrapower construction.

Ultrapower construction
We are going to construct a hyperreal field via sequences of reals. In fact we can add and multiply sequences componentwise; for example:
 * $$ (a_0, a_1, a_2, \ldots) + (b_0, b_1, b_2, \ldots) = (a_0 +b_0, a_1+b_1, a_2+b_2, \ldots) $$

and analogously for multiplication. This turns the set of such sequences into a commutative ring, which is in fact a real algebra A. We have a natural embedding of R in A by identifying the real number r with the sequence (r, r, r, …) and this identification preserves the corresponding algebraic operations of the reals. The intuitive motivation is, for example, to represent an infinitesimal number using a sequence that approaches zero. The inverse of such a sequence would represent an infinite number. As we will see below, the difficulties arise because of the need to define rules for comparing such sequences in a manner that, although inevitably somewhat arbitrary, must be self-consistent and well defined. For example, we may have two sequences that differ in their first n members, but are equal after that; such sequences should clearly be considered as representing the same hyperreal number. Similarly, most sequences oscillate randomly forever, and we must find some way of taking such a sequence and interpreting it as, say, $$7+\epsilon$$, where $$\epsilon$$ is a certain infinitesimal number.

Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion:
 * $$ (a_0, a_1, a_2, \ldots) \leq (b_0, b_1, b_2, \ldots) \iff (a_0 \leq b_0) \wedge (a_1 \leq b_1) \wedge (a_2 \leq b_2) \ldots $$

but here we run into trouble, since some entries of the first sequence may be bigger than the corresponding entries of the second sequence, and some others may be smaller. It follows that the relation defined in this way is only a partial order. To get around this, we have to specify which positions matter. Since there are infinitely many indices, we don't want finite sets of indices to matter. A consistent choice of index sets that matter is given by any free ultrafilter U on the natural numbers; these can be characterized as ultrafilters that do not contain any finite sets. (The good news is that Zorn's lemma guarantees the existence of many such U; the bad news is that they cannot be explicitly constructed.) We think of U as singling out those sets of indices that "matter": We write (a0, a1, a2, ...) ≤ (b0, b1, b2, ...) if and only if the set of natural numbers { n : an ≤ bn } is in U.

This is a total preorder and it turns into a total order if we agree not to distinguish between two sequences a and b if a ≤ b and b ≤ a. With this identification, the ordered field *R of hyperreals is constructed. From an algebraic point of view, U allows us to define a corresponding maximal ideal I in the commutative ring A (namely, the set of the sequences that vanish in some element of U), and then to define *R as A/I; as the quotient of a commutative ring by a maximal ideal, *R is a field. This is also notated A/U, directly in terms of the free ultrafilter U; the two are equivalent. The maximality of I follows from the possibility of, given a sequence a, constructing a sequence b inverting the non-null elements of a and not altering its null entries. If the set on which a vanishes is not in U, the product ab is identified with the number 1, and any ideal containing 1 must be A. In the resulting field, these a and b are inverses.

The field A/U is an ultrapower of R. Since this field contains R it has cardinality at least that of the continuum. Since A has cardinality
 * $$(2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0^2} =2^{\aleph_0},$$

it is also no larger than $$2^{\aleph_0}$$, and hence has the same cardinality as R.

One question we might ask is whether, if we had chosen a different free ultrafilter V, the quotient field A/U would be isomorphic as an ordered field to A/V. This question turns out to be equivalent to the continuum hypothesis; in ZFC with the continuum hypothesis we can prove this field is unique up to order isomorphism, and in ZFC with the negation of continuum hypothesis we can prove that there are non-order-isomorphic pairs of fields that are both countably indexed ultrapowers of the reals.

For more information about this method of construction, see ultraproduct.

An intuitive approach to the ultrapower construction
The following is an intuitive way of understanding the hyperreal numbers. The approach taken here is very close to the one in the book by Goldblatt. Recall that the sequences converging to zero are sometimes called infinitely small. These are almost the infinitesimals in a sense; the true infinitesimals include certain classes of sequences that contain a sequence converging to zero.

Let us see where these classes come from. Consider first the sequences of real numbers. They form a ring, that is, one can multiply, add and subtract them, but not necessarily divide by a non-zero element. The real numbers are considered as the constant sequences, the sequence is zero if it is identically zero, that is, an = 0 for all n.

In our ring of sequences one can get ab = 0 with neither a = 0 nor b = 0. Thus, if for two sequences $$a, b$$ one has ab = 0, at least one of them should be declared zero. Surprisingly enough, there is a consistent way to do it. As a result, the equivalence classes of sequences that differ by some sequence declared zero will form a field, which is called a hyperreal field. It will contain the infinitesimals in addition to the ordinary real numbers, as well as infinitely large numbers (the reciprocals of infinitesimals, including those represented by sequences diverging to infinity). Also every hyperreal that is not infinitely large will be infinitely close to an ordinary real, in other words, it will be the sum of an ordinary real and an infinitesimal.

This construction is parallel to the construction of the reals from the rationals given by Cantor. He started with the ring of the Cauchy sequences of rationals and declared all the sequences that converge to zero to be zero. The result is the reals. To continue the construction of hyperreals, consider the zero sets of our sequences, that is, the $$z(a)=\{i: a_i=0\}$$, that is, $$z(a)$$ is the set of indexes $$i$$ for which $$a_i=0$$. It is clear that if $$ab=0$$, then the union of $$z(a)$$ and $$z(b)$$ is N (the set of all natural numbers), so: Now the idea is to single out a bunch U of subsets X of N and to declare that $$a=0$$ if and only if $$z(a)$$ belongs to U. From the above conditions one can see that:
 * 1) One of the sequences that vanish on two complementary sets should be declared zero.
 * 2) If $$a$$ is declared zero, $$ab$$ should be declared zero too, no matter what $$b$$ is.
 * 3) If both $$a$$ and $$b$$ are declared zero, then $$a+b$$ should also be declared zero.
 * 1) From two complementary sets one belongs to U.
 * 2) Any set having a subset that belongs to U, also belongs to U.
 * 3) An intersection of any two sets belonging to U belongs to U.
 * 4) Finally, we do not want the empty set to belong to U because then everything would belong to U, as every set has the empty set as a subset.

Any family of sets that satisfies (2–4) is called a filter (an example: the complements to the finite sets, it is called the Fréchet filter and it is used in the usual limit theory). If (1) also holds, U is called an ultrafilter (because you can add no more sets to it without breaking it). The only explicitly known example of an ultrafilter is the family of sets containing a given element (in our case, say, the number 10). Such ultrafilters are called trivial, and if we use it in our construction, we come back to the ordinary real numbers. Any ultrafilter containing a finite set is trivial. It is known that any filter can be extended to an ultrafilter, but the proof uses the axiom of choice. The existence of a nontrivial ultrafilter (the ultrafilter lemma) can be added as an extra axiom, as it is weaker than the axiom of choice.

Now if we take a nontrivial ultrafilter (which is an extension of the Fréchet filter) and do our construction, we get the hyperreal numbers as a result.

If $$f$$ is a real function of a real variable $$x$$ then $$f$$ naturally extends to a hyperreal function of a hyperreal variable by composition:
 * $$f(\{x_n\})=\{f(x_n)\}$$

where $$\{ \dots\}$$ means "the equivalence class of the sequence $$\dots$$ relative to our ultrafilter", two sequences being in the same class if and only if the zero set of their difference belongs to our ultrafilter.

All the arithmetical expressions and formulas make sense for hyperreals and hold true if they are true for the ordinary reals. It turns out that any finite (that is, such that $$|x| < a$$ for some ordinary real $$a$$) hyperreal $$x$$ will be of the form $$y+d$$ where $$y$$ is an ordinary (called standard) real and $$d$$ is an infinitesimal. It can be proven by bisection method used in proving the Bolzano-Weierstrass theorem, the property (1) of ultrafilters turns out to be crucial.

Properties of infinitesimal and infinite numbers
The finite elements F of *R form a local ring, and in fact a valuation ring, with the unique maximal ideal S being the infinitesimals; the quotient F/S is isomorphic to the reals. Hence we have a homomorphic mapping, st(x), from F to R whose kernel consists of the infinitesimals and which sends every element x of F to a unique real number whose difference from x is in S; which is to say, is infinitesimal. Put another way, every finite nonstandard real number is "very close" to a unique real number, in the sense that if x is a finite nonstandard real, then there exists one and only one real number st(x) such that x – st(x) is infinitesimal. This number st(x) is called the standard part of x, conceptually the same as x to the nearest real number. This operation is an order-preserving homomorphism and hence is well-behaved both algebraically and order theoretically. It is order-preserving though not isotonic; i.e. $$ x \le y$$ implies $$\operatorname{st}(x) \le \operatorname{st}(y)$$, but $$x < y $$ does not imply $$\operatorname{st}(x) < \operatorname{st}(y)$$.

The map st is continuous with respect to the order topology on the finite hyperreals; in fact it is locally constant.
 * We have, if both x and y are finite, $$ \operatorname{st}(x + y) = \operatorname{st}(x) + \operatorname{st}(y) $$ $$ \operatorname{st}(x y) = \operatorname{st}(x) \operatorname{st}(y) $$
 * If x is finite and not infinitesimal. $$ \operatorname{st}(1/x) = 1 / \operatorname{st}(x) $$
 * x is real if and only if $$ \operatorname{st}(x) = x $$

Hyperreal fields
Suppose X is a Tychonoff space, also called a T3.5 space, and C(X) is the algebra of continuous real-valued functions on X. Suppose M is a maximal ideal in C(X). Then the factor algebra A = C(X)/M is a totally ordered field F containing the reals. If F strictly contains R then M is called a hyperreal ideal (terminology due to Hewitt (1948)) and F a hyperreal field. Note that no assumption is being made that the cardinality of F is greater than R; it can in fact have the same cardinality.

An important special case is where the topology on X is the discrete topology; in this case X can be identified with a cardinal number κ and C(X) with the real algebra Rκ of functions from κ to R. The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory.

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title: Hyperreal number


text:
Talk:Hyperreal number

oversimplification
in the subsection on the "intuitive approach", the following claim is made: "the true infinitesimals are the classes of sequences that contain a sequence converging to zero." This is an oversimplification that will not always be true. Depending on foundational assumptions, certain infinitesimal classes may not be representable by sequences tending to zero. Tkuvho (talk) 13:01, 23 June 2010 (UTC)

The sentence "The infinitesimals can be represented by the non-vanishing sequences converging to zero in the usual sense, that is with respect to the Fréchet filter" is just plain wrong. Tkuvho (talk) 13:08, 23 June 2010 (UTC)

Connection to P-point filter
The phrase "However, there may be infinitesimals not represented by null sequences; see P-point" was deleted in a recent edit. Why was it deleted? Tkuvho (talk) 11:51, 7 May 2013 (UTC)
 * What does P-point have to do with it. An infinitesimal in an ultra-power of R or Q (or any separable space) must, considered as a sequence, have zero as a limit point.  It doesn't necessarily have to be "null" (converge to 0).  On the other hand, a sequence converging to 0 does have to be infinitesimal (or 0).  — Arthur Rubin  (talk) 00:11, 8 May 2013 (UTC)
 * I see you said something else above in 2010. Could you explain?  If U is a non-principal ultrafilter over N, then an infinitesimal in RU must have (in RN) a subsequence (with coordinate set in U) converging to 0; conversely if a sequence in RN converges to 0, and has no zero components, then it is an infintesimal in RU.  — Arthur Rubin  (talk) 00:17, 8 May 2013 (UTC)
 * This is assuming classical logic; I don't know how ultrafilters work in intuitionistic logic. — Arthur Rubin  (talk) 00:19, 8 May 2013 (UTC)
 * The background logic is classical. The question is whether each infinitesimal is representable by a null sequence.  In other words, whether a subsequence can be chosen which is supported on a member of the ultrafilter.  For this to be true requires a special type of ultrafilter namely P-point (whose existence cannot be proved in ZFC). Tkuvho (talk) 11:19, 8 May 2013 (UTC)
 * Never mind. You're right.  The sequence (an) corresponds to an infinitesimal iff
 * $$(\forall \epsilon > 0)(\left\{ n \mid| |a_n| < \epsilon \right\} \in U).$$
 * It does follow that (an) has a null subsequence, but it doesn't follow that the subsequence is in U. I don't see that the topological definition of P-point corresponds to the necessary property of an ultrafilter so that all infinitesimals correspond to a null sequence;  It appears to be that:
 * $$(\forall n)(U_n \in U) \rightarrow (\exists V \in U)(\forall n)((V \smallsetminus U_n) \text{ is finite}).$$
 * — Arthur Rubin (talk) 20:16, 8 May 2013 (UTC)
 * This would mean that U, in $$ 2^{\mathbf N} / \text {finite}$$, is closed under countable intersections, which might correspond to a P-point filter, although not exactly a P-point. — Arthur Rubin  (talk) 20:21, 8 May 2013 (UTC)
 * Found it. If we use P-point, rather than the current redirect at P-point , then the statement as you wrote it makes sense, although could use a a more detailed argument and a source.  — Arthur Rubin  (talk) 21:58, 8 May 2013 (UTC)
 * The source is Cutland, Nigel; Kessler, Christoph; Kopp, Ekkehard; Ross, David, On Cauchy's notion of infinitesimal. British J. Philos. Sci. 39 (1988), no. 3, 375–378. Tkuvho (talk) 12:01, 9 May 2013 (UTC)

Properties of infinite numbers missing.
The section headed "Properties of infinitesimals and infinite numbers" does not mention any properties of infinite numbers. Shame, because that's what I wanted to know about. Tesspub (talk) 10:28, 29 August 2014 (UTC)

"The derivative of a function y(x) is defined not as dy/dx but as the standard part of dy/dx"
This is incorrect; using Keisler's treatment $$ \mathrm{d}x $$ and $$ \mathrm{d}y $$ are infinitesimal increments along the tangent line while $$ \Delta x $$ and $$ \Delta y $$ are infinitesimal increments along the curve. So $$ y'(x) = \frac{\mathrm{d}y }{\mathrm{d}x} = \mathrm{st} \frac{\Delta y}{\Delta x} $$. 58.169.240.244 (talk) 15:17, 4 May 2015 (UTC)

identical behavior.
This sentence "The transfer principle, however, doesn't mean that R and *R have identical behavior" is misleading. R and *R do have identical behavior as long as you don't write down statements that involve both standard and non-standard numbers. In the example, $$\omega$$ is a non-standard real whereas the dots ... are interpreted in the standard way (with the set of standard integers) (in other words, with a set that is undefinable in *R). Mixing standard with non-standard is really the only way that "non-identical" behavior can occur. If you stick with "all standard" or "all non-standard" then the behavior is identical. MvH (talk) 21:58, 7 February 2020 (UTC)
 * Well, that's if you restrict yourself to first-order logic. The models don't have all the same properties in higher-order logic.  I'm not sure whether "behavior" is the right word to explain this; maybe you can offer an improvement? --Trovatore (talk) 22:02, 7 February 2020 (UTC)
 * There is no test that could distinguish (N, R) from (*N, *R). Everything proved for (N, R) holds for (*N, *R) and vice versa. Only if you compare (N, R) with say (N, *R) will there be different properties. Higher-order logic hides, but doesn't solve, the issue by moving it to first order set theory. The point is, the behavior is identical, as long as whenever you replace R by *R you also replace N by *N,  Z by *Z,  Q by *Q etc. MvH (talk) 15:46, 21 February 2020 (UTC)

title: Talk:Hyperreal number


text:
Surreal number

In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.

The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field.

History of the concept
Research on the Go endgame by John Horton Conway led to the original definition and construction of the surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness. In his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book On Numbers and Games.

A separate route to defining the surreals began in 1907, when Hans Hahn introduced Hahn series as a generalization of formal power series, and Felix Hausdorff introduced certain ordered sets called ηα-sets for ordinals α and asked if it was possible to find a compatible ordered group or field structure. In 1962, Norman Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α and, in 1987, he showed that taking α to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.

If the surreals are considered as 'just' a proper-class-sized real closed field, Alling's 1962 paper handles the case of strongly inaccessible cardinals which can naturally be considered as proper classes by cutting off the cumulative hierarchy of the universe one stage above the cardinal, and Alling accordingly deserves much credit for the discovery/invention of the surreals in this sense. There is an important additional field structure on the surreals that isn't visible through this lens however, namely the notion of a 'birthday' and the corresponding natural description of the surreals as the result of a cut-filling process along their birthdays given by Conway. This additional structure has become fundamental to a modern understanding of the surreal numbers, and Conway is thus given credit for discovering the surreals as we know them today—Alling himself gives Conway full credit in a 1985 paper preceding his book on the subject.

Description
In the Conway construction, the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers a and b, a ≤ b or b ≤ a. (Both may hold, in which case a and b are equivalent and denote the same number.)  Each number is formed from an ordered pair of subsets of numbers already constructed:  given subsets L and R of numbers such that all the members of L are strictly less than all the members of R, then the pair { L | R } represents a number intermediate in value between all the members of L and all the members of R.

Different subsets may end up defining the same number: { L | R } and { L′ | R′ } may define the same number even if L ≠ L′ and R ≠ R′. (A similar phenomenon occurs when rational numbers are defined as quotients of integers: $1⁄2$ and $2⁄4$ are different representations of the same rational number.)  So strictly speaking, the surreal numbers are equivalence classes of representations of the form { L | R } that designate the same number.

In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: { | }. This representation, where L and R are both empty, is called 0. Subsequent stages yield forms like

 * { 0 | } = 1

 * { 1 | } = 2

 * { 2 | } = 3

and

 * { | 0 } = −1

 * { | −1 } = −2

 * { | −2 } = −3

The integers are thus contained within the surreal numbers. (The above identities are definitions, in the sense that the right-hand side is a name for the left-hand side. That the names are actually appropriate will be evident when the arithmetic operations on surreal numbers are defined, as in the section below). Similarly, representations such as

 * { 0 | 1 } = $1⁄2$

 * { 0 | $1⁄2$ } = $1⁄4$

 * { $1⁄2$ | 1 } = $3⁄4$

arise, so that the dyadic rationals (rational numbers whose denominators are powers of 2) are contained within the surreal numbers.

After an infinite number of stages, infinite subsets become available, so that any real number a can be represented by { La | Ra }, where La is the set of all dyadic rationals less than a and Ra is the set of all dyadic rationals greater than a (reminiscent of a Dedekind cut). Thus the real numbers are also embedded within the surreals.

There are also representations like

 * { 0, 1, 2, 3, ... | } = ω

 * { 0 | 1, $1⁄2$, $1⁄4$, $1⁄8$, ... } = ε

where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about 2ω or ω − 1 and so forth.

Construction
Surreal numbers are constructed inductively as equivalence classes of pairs of sets of surreal numbers, restricted by the condition that each element of the first set is smaller than each element of the second set. The construction consists of three interdependent parts: the construction rule, the comparison rule and the equivalence rule.

Forms
A form is a pair of sets of surreal numbers, called its left set and its right set. A form with left set L and right set R is written { L | R }. When L and R are given as lists of elements, the braces around them are omitted.

Either or both of the left and right set of a form may be the empty set. The form { { } | { } } with both left and right set empty is also written { | }.

Numeric forms and their equivalence classes
Construction rule
 * A form { L | R } is numeric if the intersection of L and R is the empty set and each element of R is greater than every element of L, according to the order relation ≤ given by the comparison rule below.

The numeric forms are placed in equivalence classes; each such equivalence class is a surreal number. The elements of the left and right sets of a form are drawn from the universe of the surreal numbers (not of forms, but of their equivalence classes).

Equivalence rule
 * Two numeric forms x and y are forms of the same number (lie in the same equivalence class) if and only if both x ≤ y and y ≤ x.

An ordering relationship must be antisymmetric, i.e., it must have the property that x = y (i. e., x ≤ y and y ≤ x are both true) only when x and y are the same object. This is not the case for surreal number forms, but is true by construction for surreal numbers (equivalence classes).

The equivalence class containing { | } is labeled 0; in other words, { | } is a form of the surreal number 0.

Order
The recursive definition of surreal numbers is completed by defining comparison:

Given numeric forms x = { XL | XR } and y = { YL | YR }, x ≤ y if and only if both: Surreal numbers can be compared to each other (or to numeric forms) by choosing a numeric form from its equivalence class to represent each surreal number.
 * There is no xL ∈ XL such that y ≤ xL. That is, every element in the left part of x is strictly smaller than y.
 * There is no yR ∈ YR such that yR ≤ x. That is, every element in the right part of y is strictly larger than x.

Induction
This group of definitions is recursive, and requires some form of mathematical induction to define the universe of objects (forms and numbers) that occur in them. The only surreal numbers reachable via finite induction are the dyadic fractions; a wider universe is reachable given some form of transfinite induction.

Induction rule

 * There is a generation S0 = { 0 }, in which 0 consists of the single form { | }.
 * Given any ordinal number n, the generation Sn is the set of all surreal numbers that are generated by the construction rule from subsets of $\bigcup_{i < n} S_i$.

The base case is actually a special case of the induction rule, with 0 taken as a label for the "least ordinal". Since there exists no Si with i < 0, the expression $\bigcup_{i < 0} S_i$ is the empty set; the only subset of the empty set is the empty set, and therefore S0 consists of a single surreal form { | } lying in a single equivalence class 0.

For every finite ordinal number n, Sn is well-ordered by the ordering induced by the comparison rule on the surreal numbers.

The first iteration of the induction rule produces the three numeric forms { | 0 } < { | } < { 0 | } (the form { 0 | 0 } is non-numeric because 0 ≤ 0). The equivalence class containing { 0 | } is labeled 1 and the equivalence class containing { | 0 } is labeled −1. These three labels have a special significance in the axioms that define a ring; they are the additive identity (0), the multiplicative identity (1), and the additive inverse of 1 (−1). The arithmetic operations defined below are consistent with these labels.

For every i < n, since every valid form in Si is also a valid form in Sn, all of the numbers in Si also appear in Sn (as supersets of their representation in Si). (The set union expression appears in our construction rule, rather than the simpler form Sn−1, so that the definition also makes sense when n is a limit ordinal.) Numbers in Sn that are a superset of some number in Si are said to have been inherited from generation i. The smallest value of α for which a given surreal number appears in Sα is called its birthday. For example, the birthday of 0 is 0, and the birthday of −1 is 1.

A second iteration of the construction rule yields the following ordering of equivalence classes:

 * { | −1 } = { | −1, 0 } = { | −1, 1 } = { | −1, 0, 1 }
 * < { | 0 } = { | 0, 1 }
 * < { −1 | 0 } = { −1 | 0, 1 }
 * < { | } = { −1 | } = { | 1 } = { −1 | 1 }
 * < { 0 | 1 } = { −1, 0 | 1 }
 * < { 0 | } = { −1, 0 | }
 * < { 1 | } = { 0, 1 | } = { −1, 1 | } = { −1, 0, 1 | }

Comparison of these equivalence classes is consistent, irrespective of the choice of form. Three observations follow:
 * 1) S2 contains four new surreal numbers.  Two contain extremal forms: { | −1, 0, 1 } contains all numbers from previous generations in its right set, and { −1, 0, 1 | } contains all numbers from previous generations in its left set.  The others have a form that partitions all numbers from previous generations into two non-empty sets.
 * 2) Every surreal number x that existed in the previous "generation" exists also in this generation, and includes at least one new form: a partition of all numbers other than x from previous generations into a left set (all numbers less than x) and a right set (all numbers greater than x).
 * 3) The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set.

The informal interpretations of { 1 | } and { | −1 } are "the number just after 1" and "the number just before −1" respectively; their equivalence classes are labeled 2 and −2. The informal interpretations of { 0 | 1 } and { −1 | 0 } are "the number halfway between 0 and 1" and "the number halfway between −1 and 0" respectively; their equivalence classes are labeled $1⁄2$ and −$1⁄2$. These labels will also be justified by the rules for surreal addition and multiplication below.

The equivalence classes at each stage n of induction may be characterized by their n-complete forms (each containing as many elements as possible of previous generations in its left and right sets). Either this complete form contains every number from previous generations in its left or right set, in which case this is the first generation in which this number occurs; or it contains all numbers from previous generations but one, in which case it is a new form of this one number. We retain the labels from the previous generation for these "old" numbers, and write the ordering above using the old and new labels:
 * −2 < −1 < −$1⁄2$ < 0 < $1⁄2$ < 1 < 2.

The third observation extends to all surreal numbers with finite left and right sets. (For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element.) The number { 1, 2 | 5, 8 } is therefore equivalent to { 2 | 5 }; one can establish that these are forms of 3 by using the birthday property, which is a consequence of the rules above.

Birthday property
A form x = { L | R } occurring in generation n represents a number inherited from an earlier generation i < n if and only if there is some number in Si that is greater than all elements of L and less than all elements of the R. (In other words, if L and R are already separated by a number created at an earlier stage, then x does not represent a new number but one already constructed.) If x represents a number from any generation earlier than n, there is a least such generation i, and exactly one number c with this least i as its birthday that lies between L and R;  x is a form of this c. In other words, it lies in the equivalence class in Sn that is a superset of the representation of c in generation i.

Arithmetic
The addition, negation (additive inverse), and multiplication of surreal number forms x = { XL | XR } and y = { YL | YR } are defined by three recursive formulas.

Negation
Negation of a given number x = { XL | XR } is defined by
 * $$-x = - \{ X_L \mid X_R \} = \{ -X_R \mid -X_L \},$$

where the negation of a set S of numbers is given by the set of the negated elements of S:
 * $$-S = \{ -s: s \in S \}.$$

This formula involves the negation of the surreal numbers appearing in the left and right sets of x, which is to be understood as the result of choosing a form of the number, evaluating the negation of this form, and taking the equivalence class of the resulting form. This only makes sense if the result is the same, irrespective of the choice of form of the operand. This can be proved inductively using the fact that the numbers occurring in XL and XR are drawn from generations earlier than that in which the form x first occurs, and observing the special case:
 * $$-0 = - \{ {}\mid{} \} = \{ {}\mid{} \} = 0.$$

Addition
The definition of addition is also a recursive formula:
 * $$x + y = \{ X_L \mid X_R \} + \{ Y_L \mid Y_R \} = \{ X_L + y, x + Y_L \mid X_R + y, x + Y_R \}, $$

where

 * $$X + y = \{ x + y: x \in X \}, \quad x + Y = \{ x + y: y \in Y \}$$.

This formula involves sums of one of the original operands and a surreal number drawn from the left or right set of the other. It can be proved inductively with the special cases:
 * $$0 + 0 = \{ {}\mid{} \} + \{ {}\mid{} \} = \{ {}\mid{} \} = 0$$
 * $$x + 0 = x + \{ {}\mid{} \} = \{ X_L + 0 \mid X_R + 0 \} = \{ X_L \mid X_R \} = x $$
 * $$0 + y = \{ {}\mid{} \} + y = \{ 0 + Y_L \mid 0 + Y_R \} = \{ Y_L \mid Y_R \} = y$$

For example:
 * $1⁄2$ + $1⁄2$ = { 0 | 1 } + { 0 | 1 } = { $1⁄2$ | $3⁄2$ },

which by the birthday property is a form of 1. This justifies the label used in the previous section.

Multiplication
Multiplication can be defined recursively as well, beginning from the special cases involving 0, the multiplicative identity 1, and its additive inverse −1:
 * $$\begin{align}

xy & = \{ X_L \mid X_R \} \{ Y_L \mid Y_R \} \\ & = \left\{ X_L y + x Y_L - X_L Y_L, X_R y + x Y_R - X_R Y_R \mid X_L y + x Y_R - X_L Y_R, x Y_L + X_R y - X_R Y_L \right\} \\ \end{align}$$ The formula contains arithmetic expressions involving the operands and their left and right sets, such as the expression $$X_R y + x Y_R - X_R Y_R$$ that appears in the left set of the product of x and y. This is understood as the set of numbers generated by picking all possible combinations of members of $$X_R$$ and $$Y_R$$, and substituting them into the expression.

For example, to show that the square of $1⁄2$ is $1⁄4$:
 * $1⁄2$ ⋅ $1⁄2$ = { 0 | 1 } ⋅ { 0 | 1 } = { 0 | $1⁄2$ } = $1⁄4$.

Division
The definition of division is done in terms of the reciprocal and multiplication:

 * $$\frac xy = x \cdot \frac 1y $$

where

 * $$\frac 1y = \left\{\left.0, \frac{1+(y_R-y)\left(\frac1y\right)_L}{y_R}, \frac{1+\left(y_L-y\right)\left(\frac1y\right)_R}{y_L} \,\,\right|\,\, \frac{1+(y_L-y)\left(\frac1y\right)_L}{y_L}, \frac{1+(y_R-y)\left(\frac1y\right)_R}{y_R} \right\}$$

for positive y. Only positive yL are permitted in the formula, with any nonpositive terms being ignored (and yR are always positive). This formula involves not only recursion in terms of being able to divide by numbers from the left and right sets of y, but also recursion in that the members of the left and right sets of $1⁄y$ itself. 0 is always a member of the left set of $1⁄y$, and that can be used to find more terms in a recursive fashion. For example, if y = 3 = { 2 | }, then we know a left term of $1⁄3$ will be 0. This in turn means $1 + (2 − 3)0⁄2$ = $1⁄2$ is a right term. This means
 * $$\frac{1+(2-3)\left(\frac12\right)}2=\frac14$$

is a left term. This means
 * $$\frac{1+(2-3)\left(\frac14\right)}2 = \frac 38$$

will be a right term. Continuing, this gives
 * $$\frac13 = \left\{\left. 0, \frac14, \frac5{16}, \ldots \,\right|\, \frac12, \frac38, \ldots\right\}$$

For negative y, $1⁄y$ is given by
 * $$\frac1y=-\left(\frac1{-y}\right)$$

If y = 0, then $1⁄y$ is undefined.

Consistency
It can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that:
 * Addition and negation are defined recursively in terms of "simpler" addition and negation steps, so that operations on numbers with birthday n will eventually be expressed entirely in terms of operations on numbers with birthdays less than n;
 * Multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday n will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than n;
 * As long as the operands are well-defined surreal number forms (each element of the left set is less than each element of the right set), the results are again well-defined surreal number forms;
 * The operations can be extended to numbers (equivalence classes of forms): the result of negating x or adding or multiplying x and y will represent the same number regardless of the choice of form of x and y; and
 * These operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a field, with additive identity 0 = { | } and multiplicative identity 1 = { 0 | }.

With these rules one can now verify that the numbers found in the first few generations were properly labeled. The construction rule is repeated to obtain more generations of surreals:

 * S0 = { 0 }
 * S1 = { −1 < 0 < 1 }
 * S2 = { −2 < −1 < −$1⁄2$ < 0 < $1⁄2$ < 1 < 2}
 * S3 = { −3 < −2 < −$3⁄2$ < −1 < −$3⁄4$ < −$1⁄2$ < −$1⁄4$ < 0 < $1⁄4$ < $1⁄2$ < $3⁄4$ < 1 < $3⁄2$ < 2 < 3 }
 * S4 = { −4 < −3 < ... < −$1⁄8$ < 0 < $1⁄8$ < $1⁄4$ < $3⁄8$ < $1⁄2$ < $5⁄8$ < $3⁄4$ < $7⁄8$ < 1 < $5⁄4$ < $3⁄2$ < $7⁄4$ < 2 < $5⁄2$ < 3 < 4 }

Arithmetic closure
For each natural number (finite ordinal) n, all numbers generated in Sn are dyadic fractions, i.e., can be written as an irreducible fraction $a⁄2^{b}$, where a and b are integers and 0 ≤ b < n.

The set of all surreal numbers that are generated in some Sn for finite n may be denoted as $S_* = \bigcup_{n \in N} S_n$. One may form the three classes
 * $$\begin{align}

S_{0} &= \{ 0 \} \\ S_{+} &= \{ x \in S_*: x > 0 \} \\ S_{-} &= \{ x \in S_*: x < 0 \} \end{align}$$ of which S∗ is the union. No individual Sn is closed under addition and multiplication (except S0), but S∗ is; it is the subring of the rationals consisting of all dyadic fractions.

There are infinite ordinal numbers β for which the set of surreal numbers with birthday less than β is closed under the different arithmetic operations. For any ordinal α, the set of surreal numbers with birthday less than β = ωα (using powers of ω) is closed under addition and forms a group; for birthday less than ωω α it is closed under multiplication and forms a ring; and for birthday less than an (ordinal) epsilon number εα it is closed under multiplicative inverse and forms a field. The latter sets are also closed under the exponential function as defined by Kruskal and Gonshor.

However, it is always possible to construct a surreal number that is greater than any member of a set of surreals (by including the set on the left side of the constructor) and thus the collection of surreal numbers is a proper class. With their ordering and algebraic operations they constitute an ordered field, with the caveat that they do not form a set. In fact it is the biggest ordered field, in that every ordered field is a subfield of the surreal numbers. The class of all surreal numbers is denoted by the symbol $$\mathbb{No}$$.

Infinity
Define Sω as the set of all surreal numbers generated by the construction rule from subsets of S∗. (This is the same inductive step as before, since the ordinal number ω is the smallest ordinal that is larger than all natural numbers; however, the set union appearing in the inductive step is now an infinite union of finite sets, and so this step can only be performed in a set theory that allows such a union.) A unique infinitely large positive number occurs in Sω:
 * $$ \omega = \{ S_* \mid{} \} = \{ 1, 2, 3, 4, \ldots \mid{} \}. $$

Sω also contains objects that can be identified as the rational numbers. For example, the ω-complete form of the fraction $1⁄3$ is given by:
 * $$ \tfrac{1} {3} = \{ y \in S_*: 3 y < 1 \mid y \in S_*: 3 y > 1 \}.$$

The product of this form of $1⁄3$ with any form of 3 is a form whose left set contains only numbers less than 1 and whose right set contains only numbers greater than 1; the birthday property implies that this product is a form of 1.

Not only do all the rest of the rational numbers appear in Sω; the remaining finite real numbers do too. For example,
 * $$ \pi = \left\{ 3, \tfrac{25}{8},\tfrac{201}{64}, \ldots \mid 4, \tfrac{7}{2}, \tfrac{13}{4}, \tfrac{51}{16},\ldots \right\}. $$

The only infinities in Sω are ω and −ω; but there are other non-real numbers in Sω among the reals. Consider the smallest positive number in Sω:
 * $$ \varepsilon = \{ S_- \cup S_0 \mid S_+ \} = \left\{ 0 \mid 1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \ldots \right\} = \{ 0 \mid y \in S_* : y > 0 \}$$.

This number is larger than zero but less than all positive dyadic fractions. It is therefore an infinitesimal number, often labeled ε. The ω-complete form of ε (respectively −ε) is the same as the ω-complete form of 0, except that 0 is included in the left (respectively right) set. The only "pure" infinitesimals in Sω are ε and its additive inverse −ε; adding them to any dyadic fraction y produces the numbers y ± ε, which also lie in Sω.

One can determine the relationship between ω and ε by multiplying particular forms of them to obtain:
 * ω · ε = { ε · S+ | ω · S+ + S∗ + ε · S∗ }.

This expression is only well-defined in a set theory which permits transfinite induction up to Sω2. In such a system, one can demonstrate that all the elements of the left set of ωSω·Sωε are positive infinitesimals and all the elements of the right set are positive infinities, and therefore ωSω·Sωε is the oldest positive finite number, 1. Consequently, $1⁄ε$ = ω. Some authors systematically use ω−1 in place of the symbol ε.

Contents of Sω
Given any x = { L | R } in Sω, exactly one of the following is true:
 * L and R are both empty, in which case x = 0;
 * R is empty and some integer n ≥ 0 is greater than every element of L, in which case x equals the smallest such integer n;
 * R is empty and no integer n is greater than every element of L, in which case x equals +ω;
 * L is empty and some integer n ≤ 0 is less than every element of R, in which case x equals the largest such integer n;
 * L is empty and no integer n is less than every element of R, in which case x equals −ω;
 * L and R are both non-empty, and:
 * Some dyadic fraction y is "strictly between" L and R (greater than all elements of L and less than all elements of R), in which case x equals the oldest such dyadic fraction y;
 * No dyadic fraction y lies strictly between L and R, but some dyadic fraction $$ y \in L$$ is greater than or equal to all elements of L and less than all elements of R, in which case x equals y + ε;
 * No dyadic fraction y lies strictly between L and R, but some dyadic fraction $$ y \in R$$ is greater than all elements of L and less than or equal to all elements of R, in which case x equals y − ε;
 * Every dyadic fraction is either greater than some element of R or less than some element of L, in which case x is some real number that has no representation as a dyadic fraction.

Sω is not an algebraic field, because it is not closed under arithmetic operations; consider ω+1, whose form
 * $$\omega + 1 = \{ 1, 2, 3, 4, ... \mid {} \} + \{ 0 \mid{} \} = \{ 1, 2, 3, 4, \ldots, \omega \mid {} \}$$

does not lie in any number in Sω. The maximal subset of Sω that is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities ±ω, the infinitesimals ±ε, and the infinitesimal neighbors y ± ε of each nonzero dyadic fraction y.

This construction of the real numbers differs from the Dedekind cuts of standard analysis in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in Sω with its forms in previous generations. (The ω-complete forms of real elements of Sω are in one-to-one correspondence with the reals obtained by Dedekind cuts, under the proviso that Dedekind reals corresponding to rational numbers are represented by the form in which the cut point is omitted from both left and right sets.) The rationals are not an identifiable stage in the surreal construction; they are merely the subset Q of Sω containing all elements x such that x b = a for some a and some nonzero b, both drawn from S∗. By demonstrating that Q is closed under individual repetitions of the surreal arithmetic operations, one can show that it is a field; and by showing that every element of Q is reachable from S∗ by a finite series (no longer than two, actually) of arithmetic operations including multiplicative inversion, one can show that Q is strictly smaller than the subset of Sω identified with the reals.

The set Sω has the same cardinality as the real numbers R. This can be demonstrated by exhibiting surjective mappings from Sω to the closed unit interval I of R and vice versa. Mapping Sω onto I is routine; map numbers less than or equal to ε (including −ω) to 0, numbers greater than or equal to 1 − ε (including ω) to 1, and numbers between ε and 1 − ε to their equivalent in I (mapping the infinitesimal neighbors y±ε of each dyadic fraction y, along with y itself, to y). To map I onto Sω, map the (open) central third ($1⁄3$, $2⁄3$) of I onto { | } = 0; the central third ($7⁄9$, $8⁄9$) of the upper third to { 0 | } = 1; and so forth. This maps a nonempty open interval of I onto each element of S∗, monotonically. The residue of I consists of the Cantor set 2ω, each point of which is uniquely identified by a partition of the central-third intervals into left and right sets, corresponding precisely to a form { L in Sω. This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthday ω.

Transfinite induction
Continuing to perform transfinite induction beyond Sω produces more ordinal numbers α, each represented as the largest surreal number having birthday α. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is ω+1 = { ω | }. There is another positive infinite number in generation ω+1:
 * ω − 1 = { 1, 2, 3, 4, ... | ω }.

The surreal number ω − 1 is not an ordinal; the ordinal ω is not the successor of any ordinal. This is a surreal number with birthday ω+1, which is labeled ω − 1 on the basis that it coincides with the sum of } and −1 = {. Similarly, there are two new infinitesimal numbers in generation ω + 1:
 * 2ε = ε + ε = { ε | 1 + ε, $1⁄2$ + ε, $1⁄4$ + ε, $1⁄8$ + ε, ... } and
 * $ε⁄2$ = ε · $1⁄2$ = { 0 | ε }.

At a later stage of transfinite induction, there is a number larger than ω + k for all natural numbers k:
 * 2ω = ω + ω = { ω+1, ω+2, ω+3, ω+4, ... | }

This number may be labeled ω + ω both because its birthday is ω + ω (the first ordinal number not reachable from ω by the successor operation) and because it coincides with the surreal sum of ω and ω; it may also be labeled 2ω because it coincides with the product of } and }. It is the second limit ordinal; reaching it from ω via the construction step requires a transfinite induction on
 * $$\bigcup_{k < \omega} S_{\omega + k}$$

This involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required.

Note that the conventional addition and multiplication of ordinals does not always coincide with these operations on their surreal representations. The sum of ordinals 1 + ω equals ω, but the surreal sum is commutative and produces 1 + ω = ω + 1 > ω. The addition and multiplication of the surreal numbers associated with ordinals coincides with the natural sum and natural product of ordinals.

Just as 2ω is bigger than ω + n for any natural number n, there is a surreal number $ω⁄2$ that is infinite but smaller than ω − n for any natural number n. That is, $ω⁄2$ is defined by
 * $ω⁄2$ = { S∗ | ω − S∗ }

where on the right hand side the notation x − Y is used to mean { x − y : y ∈ Y }. It can be identified as the product of ω and the form { 0 | 1 } of $1⁄2$. The birthday of $ω⁄2$ is the limit ordinal ω2.

Powers of ω and the Conway normal form
To classify the "orders" of infinite and infinitesimal surreal numbers, also known as archimedean classes, Conway associated to each surreal number x the surreal number where r and s range over the positive real numbers. If x < y then ωy is "infinitely greater" than ωx, in that it is greater than r ωx for all real numbers r. Powers of ω also satisfy the conditions so they behave the way one would expect powers to behave.
 * ωx = { 0, r ωxL | s ωxR },
 * ωx ωy = ωx+y,
 * ω−x = $1⁄ω^{x}$,

Each power of ω also has the redeeming feature of being the simplest surreal number in its archimedean class; conversely, every archimedean class within the surreal numbers contains a unique simplest member. Thus, for every positive surreal number x there will always exist some positive real number r and some surreal number y so that x − rωy is "infinitely smaller" than x. The exponent y is the "base ω logarithm" of x, defined on the positive surreals; it can be demonstrated that logω maps the positive surreals onto the surreals and that
 * logω(xy) = logω(x) + logω(y).

This gets extended by transfinite induction so that every surreal number has a "normal form" analogous to the Cantor normal form for ordinal numbers. This is the Conway normal form: Every surreal number x may be uniquely written as
 * x = r0ωy0 + r1ωy1 + ...,

where every rα is a nonzero real number and the yαs form a strictly decreasing sequence of surreal numbers. This "sum", however, may have infinitely many terms, and in general has the length of an arbitrary ordinal number. (Zero corresponds of course to the case of an empty sequence, and is the only surreal number with no leading exponent.)

Looked at in this manner, the surreal numbers resemble a power series field, except that the decreasing sequences of exponents must be bounded in length by an ordinal and are not allowed to be as long as the class of ordinals. This is the basis for the formulation of the surreal numbers as a Hahn series.

Gaps and continuity
In contrast to the real numbers, a (proper) subset of the surreal numbers does not have a least upper (or lower) bound unless it has a maximal (minimal) element. Conway defines a gap as { L | R } such that every element of L is less than every element of R, and $$ L \cup R = \mathbb{No}$$; this is not a number because at least one of the sides is a proper class. Though similar, gaps are not quite the same as Dedekind cuts, but we can still talk about a completion $$\mathbb{No}_\mathfrak{D}$$ of the surreal numbers with the natural ordering which is a (proper class-sized) linear continuum.

For instance there is no least positive infinite surreal, but the gap

 * $$ \{ x : \exists n \in \mathbb N : x < n\mid x : \forall n\in \mathbb N : x > n \} $$

is greater than all real numbers and less than all positive infinite surreals, and is thus the least upper bound of the reals in $$\mathbb{No}_\mathfrak{D}$$. Similarly the gap $$\mathbb{On} = \{ \mathbb{No} \mid{} \} $$ is larger than all surreal numbers. (This is an esoteric pun: In the general construction of ordinals, α "is" the set of ordinals smaller than α, and we can use this equivalence to write } in the surreals; $$\mathbb{On}$$ denotes the class of ordinal numbers, and because $$\mathbb{On}$$ is cofinal in $$\mathbb{No}$$ we have $$ \{ \mathbb{No} \mid {} \} = \{ \mathbb{On} \mid {} \} = \mathbb{On}$$ by extension.)

With a bit of set-theoretic care, $$\mathbb{No}$$ can be equipped with a topology where the open sets are unions of open intervals (indexed by proper sets) and continuous functions can be defined. An equivalent of Cauchy sequences can be defined as well, although they have to be indexed by the class of ordinals; these will always converge, but the limit may be either a number or a gap that can be expressed as
 * $$\sum_{\alpha\in\mathbb{No}} r_\alpha \omega^{a_\alpha}$$

with aα decreasing and having no lower bound in $$\mathbb{No}$$. (All such gaps can be understood as Cauchy sequences themselves, but there are other types of gap that are not limits, such as ∞ and $$\mathbb{On}$$).

Exponential function
Based on unpublished work by Kruskal, a construction (by transfinite induction) that extends the real exponential function exp(x) (with base e) to the surreals was carried through by Gonshor.

Other exponentials
The powers of ω function is also an exponential function, but does not have the properties desired for an extension of the function on the reals. It will, however, be needed in the development of the base-e exponential, and it is this function that is meant whenever the notation ωx is used in the following.

When y is a dyadic fraction, the power function x ∈ $\mathbb{No}$, x ↦ xy may be composed from multiplication, multiplicative inverse and square root, all of which can be defined inductively. Its values are completely determined by the basic relation xy+z = xy · xz, and where defined it necessarily agrees with any other exponentiation that can exist.

Basic induction
The induction steps for the surreal exponential are based on the series expansion for the real exponential,
 * $$\exp x = \sum_{n\ge 0} \frac{x^n}{n!}$$

more specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones. For x positive these are denoted [x]n and include all partial sums; for x negative but finite, [x]2n+1 denotes the odd steps in the series starting from the first one with a positive real part (which always exists). For x negative infinite the odd-numbered partial sums are strictly decreasing and the [x]2n+1 notation denotes the empty set, but it turns out that the corresponding elements are not needed in the induction.

The relations that hold for real x < y are then
 * exp x · [y – x]n < exp y

and
 * exp y · [x – y]2n + 1 < exp x,

and this can be extended to the surreals with the definition

 * $$ \exp z = \{0, \exp z_L \cdot [z-z_L]_n, \exp z_R\cdot[z-z_R]_{2n+1} \mid \exp z_R/[z_R-z]_n, \exp z_L/[z_L-z]_{2n+1} \}. $$

This is well-defined for all surreal arguments (the value exists and does not depend on the choice of zL and zR).

Results
Using this definition, the following hold:
 * exp is a strictly increasing positive function, x < y ⇒ 0 < exp x < exp y
 * exp satisfies exp(x+y) = exp x · exp y
 * exp is a surjection (onto $$\mathbb{No}_+$$) and has a well-defined inverse, log = exp–1
 * exp coincides with the usual exponential function on the reals (and thus exp 0 = 1, exp 1 = e)
 * For x infinitesimal, the value of the formal power series (Taylor expansion) of exp is well defined and coincides with the inductive definition
 * When x is given in Conway normal form, the set of exponents in the result is well-ordered and the coefficients are finite sums, directly giving the normal form of the result (which has a leading 1)
 * Similarly, for x infinitesimally close to 1, log x is given by power series expansion of x – 1
 * For positive infinite x, exp x is infinite as well
 * If x has the form ωα (α > 0), exp x has the form ωω β where β is a strictly increasing function of α. In fact there is an inductively defined bijection g: $\mathbb{No}_+$ → $\mathbb{No}$ : α ↦ β whose inverse can also be defined inductively
 * If x is "pure infinite" with normal form x = Σα<βrαωaα where all aα > 0, then exp x = ωΣα<βrαω g(aα)
 * Similarly, for x = ωΣα<βrαω bα, the inverse is given by log x = Σα<βrαωg –1(bα)
 * Any surreal number can be written as the sum of a pure infinite, a real and an infinitesimal part, and the exponential is the product of the partial results given above
 * The normal form can be written out by multiplying the infinite part (a single power of ω) and the real exponential into the power series resulting from the infinitesimal
 * Conversely, dividing out the leading term of the normal form will bring any surreal number into the form (ωΣγ<δtγω bγ )·r·(1 + Σα<βsαωaα), for aα < 0, where each factor has a form for which a way of calculating the logarithm has been given above; the sum is then the general logarithm
 * While there is no general inductive definition of log (unlike for exp), the partial results are given in terms of such definitions. In this way, the logarithm can be calculated explicitly, without reference to the fact that it's the inverse of the exponential.
 * The exponential function is much greater than any finite power
 * For any positive infinite x and any finite n, exp(x)/xn is infinite
 * For any integer n and surreal x > n2, exp(x) > xn. This stronger constraint is one of the Ressayre axioms for the real exponential field
 * exp satisfies all the Ressayre axioms for the real exponential field
 * The surreals with exponential is an elementary extension of the real exponential field
 * For εβ an ordinal epsilon number, the set of surreal numbers with birthday less than εβ constitute a field that is closed under exponentials, and is likewise an elementary extension of the real exponential field

Examples
The surreal exponential is essentially given by its behaviour on positive powers of ω, i.e., the function g(a), combined with well-known behaviour on finite numbers. Only examples of the former will be given. In addition, g(a) = a holds for a large part of its range, for instance for any finite number with positive real part and any infinite number that is less than some iterated power of ω (ωω · · ω for some number of levels).
 * exp ω = ωω
 * exp ω1/ω = ω and log ω = ω1/ω
 * exp (ω · log ω) = exp (ω · ω1/ω) = ωω (1 + 1/ω)
 * This shows that the "power of ω" function is not compatible with exp, since compatibility would demand a value of ωω here
 * exp ε0 = ωω ε0 + 1
 * log ε0 = ε0 / ω

Exponentiation
A general exponentiation can be defined as xy = exp(y · log x), giving an interpretation to expressions like 2ω = exp(ω · log 2) = ωlog 2 · ω. Again it is essential to distinguish this definition from the "powers of ω" function, especially if ω may occur as the base.

Surcomplex numbers
A surcomplex number is a number of the form $a + bi$, where a and b are surreal numbers and $i$ is the square root of $−1$. The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements. Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.

Games
The definition of surreal numbers contained one restriction: each element of L must be strictly less than each element of R. If this restriction is dropped we can generate a more general class known as games. All games are constructed according to this rule:

 * Construction rule : If L and R are two sets of games then { L | R } is a game.

Addition, negation, and comparison are all defined the same way for both surreal numbers and games.

Every surreal number is a game, but not all games are surreal numbers, e.g. the game { 0 | 0 } is not a surreal number. The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers. The class of surreal numbers forms a field, but the class of games does not. The surreals have a total order: given any two surreals, they are either equal, or one is greater than the other. The games have only a partial order: there exist pairs of games that are neither equal, greater than, nor less than each other. Each surreal number is either positive, negative, or zero. Each game is either positive, negative, zero, or fuzzy (incomparable with zero, such as {1 | −1}).

A move in a game involves the player whose move it is choosing a game from those available in L (for the left player) or R (for the right player) and then passing this chosen game to the other player. A player who cannot move because the choice is from the empty set has lost. A positive game represents a win for the left player, a negative game for the right player, a zero game for the second player to move, and a fuzzy game for the first player to move.

If x, y, and z are surreals, and x = y, then x z = y z. However, if x, y, and z are games, and x = y, then it is not always true that x z = y z. Note that "=" here means equality, not identity.

Application to combinatorial game theory
The surreal numbers were originally motivated by studies of the game Go, and there are numerous connections between popular games and the surreals. In this section, we will use a capitalized Game for the mathematical object {L | R}, and the lowercase game for recreational games like Chess or Go.

We consider games with these properties:
 * Two players (named Left and Right)
 * Deterministic (the game at each step will completely depend on the choices the players make, rather than a random factor)
 * No hidden information (such as cards or tiles that a player hides)
 * Players alternate taking turns (the game may or may not allow multiple moves in a turn)
 * Every game must end in a finite number of moves
 * As soon as there are no legal moves left for a player, the game ends, and that player loses

For most games, the initial board position gives no great advantage to either player. As the game progresses and one player starts to win, board positions will occur in which that player has a clear advantage. For analyzing games, it is useful to associate a Game with every board position. The value of a given position will be the Game {L|R}, where L is the set of values of all the positions that can be reached in a single move by Left. Similarly, R is the set of values of all the positions that can be reached in a single move by Right.

The zero Game (called 0) is the Game where L and R are both empty, so the player to move next (L or R) immediately loses. The sum of two Games G = { L1 | R1 } and H = { L2 | R2 } is defined as the Game G + H = { L1 + H, G + L2 | R1 + H, G + R2 } where the player to move chooses which of the Games to play in at each stage, and the loser is still the player who ends up with no legal move. One can imagine two chess boards between two players, with players making moves alternately, but with complete freedom as to which board to play on. If G is the Game {L | R}, −G is the Game {−R | −L}, i.e. with the role of the two players reversed. It is easy to show G – G = 0 for all Games G (where G – H is defined as G + (–H)).

This simple way to associate Games with games yields a very interesting result. Suppose two perfect players play a game starting with a given position whose associated Game is x. We can classify all Games into four classes as follows:

 * If x > 0 then Left will win, regardless of who plays first.
 * If x < 0 then Right will win, regardless of who plays first.
 * If x = 0 then the player who goes second will win.
 * If x || 0 then the player who goes first will win.

More generally, we can define G > H as G – H > 0, and similarly for <, = and ||.

The notation G || H means that G and H are incomparable. G || H is equivalent to G − H || 0, i.e. that G > H, G < H and G = H are all false. Incomparable games are sometimes said to be confused with each other, because one or the other may be preferred by a player depending on what is added to it. A game confused with zero is said to be fuzzy, as opposed to positive, negative, or zero. An example of a fuzzy game is star (*).

Sometimes when a game nears the end, it will decompose into several smaller games that do not interact, except in that each player's turn allows moving in only one of them. For example, in Go, the board will slowly fill up with pieces until there are just a few small islands of empty space where a player can move. Each island is like a separate game of Go, played on a very small board. It would be useful if each subgame could be analyzed separately, and then the results combined to give an analysis of the entire game. This doesn't appear to be easy to do. For example, there might be two subgames where whoever moves first wins, but when they are combined into one big game, it is no longer the first player who wins. Fortunately, there is a way to do this analysis. The following theorem can be applied:

 * If a big game decomposes into two smaller games, and the small games have associated Games of x and y, then the big game will have an associated Game of x + y.

A game composed of smaller games is called the disjunctive sum of those smaller games, and the theorem states that the method of addition we defined is equivalent to taking the disjunctive sum of the addends.

Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing Go endgames, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames into an analysis of their disjunctive sum. From this he invented the concept of a Game and the addition operator for it. From there he moved on to developing a definition of negation and comparison. Then he noticed that a certain class of Games had interesting properties; this class became the surreal numbers. Finally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals.

Alternative realizations
Alternative approaches to the surreal numbers complement Conway's exposition in terms of games.

Definitions
In what is now called the sign-expansion or sign-sequence of a surreal number, a surreal number is a function whose domain is an ordinal and whose codomain is { −1, +1 }. This is equivalent to Conway's L-R sequences.

Define the binary predicate "simpler than" on numbers by x is simpler than y if x is a proper subset of y, i.e. if dom(x) < dom(y) and x(α) = y(α) for all α < dom(x).

For surreal numbers define the binary relation < to be lexicographic order (with the convention that "undefined values" are greater than −1 and less than 1). So x < y if one of the following holds:
 * x is simpler than y and y(dom(x)) = +1;
 * y is simpler than x and x(dom(y)) = −1;
 * there exists a number z such that z is simpler than x, z is simpler than y, x(dom(z)) = − 1 and y(dom(z)) = +1.

Equivalently, let δ(x,y) = min({ dom(x), dom(y)} ∪ { α : α < dom(x) ∧ α < dom(y) ∧ x(α) ≠ y(α) }), so that x = y if and only if δ(x,y) = dom(x) = dom(y). Then, for numbers x and y, x < y if and only if one of the following holds:
 * δ(x,y) = dom(x) ∧ δ(x,y) < dom(y) ∧ y(δ(x,y)) = +1;
 * δ(x,y) < dom(x) ∧ δ(x,y) = dom(y) ∧ x(δ(x,y)) = −1;
 * δ(x,y) < dom(x) ∧ δ(x,y) < dom(y) ∧ x(δ(x,y)) = −1 ∧ y(δ(x,y)) = +1.

For numbers x and y, x ≤ y if and only if x < y ∨ x = y, and x > y if and only if y < x. Also x ≥ y if and only if y ≤ x.

The relation < is transitive, and for all numbers x and y, exactly one of x < y, x = y, x > y, holds (law of trichotomy). This means that < is a linear order (except that < is a proper class).

For sets of numbers, L and R such that ∀x ∈ L ∀y ∈ R (x < y), there exists a unique number z such that
 * ∀x ∈ L (x < z) ∧ ∀y ∈ R (z < y),
 * For any number w such that ∀x ∈ L (x < w) ∧ ∀y ∈ R (w < y), w = z or z is simpler than w.

Furthermore, z is constructible from L and R by transfinite induction. z is the simplest number between L and R. Let the unique number z be denoted by σ(L,R).

For a number x, define its left set L(x) and right set R(x) by
 * L(x) = { x|α : α < dom(x) ∧ x(α) = +1 };
 * R(x) = { x|α : α < dom(x) ∧ x(α) = −1 },

then σ(L(x),R(x)) = x.

One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike Conway's realization of the surreal numbers, however, the sign-expansion requires a prior construction of the ordinals, while in Conway's realization, the ordinals are constructed as particular cases of surreals.

However, similar definitions can be made that eliminate the need for prior construction of the ordinals. For instance, we could let the surreals be the (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule ∀g ∈ dom f (∀h ∈ dom g (h ∈ dom f )) and whose range is { −, + }. "Simpler than" is very simply defined now—x is simpler than y if x ∈ dom y. The total ordering is defined by considering x and y as sets of ordered pairs (as a function is normally defined): Either x = y,  or else the surreal number z = x ∩ y  is in the domain of x  or the domain of y  (or both, but in this case the signs must disagree). We then have x < y if x(z) = − or y(z) = + (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements of dom f in order of simplicity (i.e., inclusion), and then write down the signs that f assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is { + }.

Addition and multiplication
The sum x + y of two numbers, x and y, is defined by induction on dom(x) and dom(y) by x + y = σ(L,R), where
 * L = { u + y : u ∈ L(x) } ∪{ x + v : v ∈ L(y) },
 * R = { u + y : u ∈ R(x) } ∪{ x + v : v ∈ R(y) }.

The additive identity is given by the number 0 = { }, i.e. the number 0 is the unique function whose domain is the ordinal 0, and the additive inverse of the number x is the number −x, given by dom(−x) = dom(x), and, for α < dom(x), (−x)(α) = −1 if x(α) = +1, and (−x)(α) = +1 if x(α) = −1.

It follows that a number x is positive if and only if 0 < dom(x) and x(0) = +1, and x is negative if and only if 0 < dom(x) and x(0) = −1.

The product xy of two numbers, x and y, is defined by induction on dom(x) and dom(y) by xy = σ(L,R), where
 * L = { uy + xv − uv : u ∈ L(x), v ∈ L(y) } ∪ { uy + xv − uv : u ∈ R(x), v ∈ R(y) },
 * R = { uy + xv − uv : u ∈ L(x), v ∈ R(y) } ∪ { uy + xv − uv : u ∈ R(x), v ∈ L(y) }.

The multiplicative identity is given by the number 1 = { (0,+1) }, i.e. the number 1 has domain equal to the ordinal 1, and 1(0) = +1.

Correspondence with Conway's realization
The map from Conway's realization to sign expansions is given by f({ L | R }) = σ(M,S), where M = { f(x) : x ∈ L } and S = { f(x) : x ∈ R }.

The inverse map from the alternative realization to Conway's realization is given by g(x) = { L | R }, where L = { g(y) : y ∈ L(x) } and R = { g(y) : y ∈ R(x) }.

Axiomatic approach
In another approach to the surreals, given by Alling, explicit construction is bypassed altogether. Instead, a set of axioms is given that any particular approach to the surreals must satisfy. Much like the axiomatic approach to the reals, these axioms guarantee uniqueness up to isomorphism.

A triple $$\langle \mathbb{No}, \mathrm{<}, b \rangle$$ is a surreal number system if and only if the following hold:

 * < is a total order over $$\mathbb{No}$$
 * b is a function from $$\mathbb{No}$$ onto the class of all ordinals (b is called the "birthday function" on $$\mathbb{No}$$).
 * Let A and B be subsets of $$\mathbb{No}$$ such that for all x ∈ A and y ∈ B, x < y (using Alling's terminology, 〈 A,B 〉 is a "Conway cut" of $$\mathbb{No}$$). Then there exists a unique z ∈ $$\mathbb{No}$$ such that b(z) is minimal and for all x ∈ A and all y ∈ B, x < z < y. (This axiom is often referred to as "Conway's Simplicity Theorem".)
 * Furthermore, if an ordinal α is greater than b(x) for all x ∈ A, B, then b(z) ≤ α. (Alling calls a system that satisfies this axiom a "full surreal number system".)

Both Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.

Given these axioms, Alling derives Conway's original definition of ≤ and develops surreal arithmetic.

Simplicity hierarchy
A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich, The difference from the usual definition of a tree is that the set of ancestors of a vertex is well-ordered, but may not have a maximal element (immediate predecessor); in other words the order type of that set is a general ordinal number, not just a natural number. This construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation.

Hahn series
Alling also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of Hahn series with real coefficients on the value group of surreal numbers themselves (the series representation corresponding to the normal form of a surreal number, as defined above). This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.

This isomorphism makes the surreal numbers into a valued field where the valuation is the additive inverse of the exponent of the leading term in the Conway normal form, e.g., ν(ω) = −1. The valuation ring then consists of the finite surreal numbers (numbers with a real and/or an infinitesimal part). The reason for the sign inversion is that the exponents in the Conway normal form constitute a reverse well-ordered set, whereas Hahn series are formulated in terms of (non-reversed) well-ordered subsets of the value group.

Relation to hyperreals
Philip Ehrlich has constructed an isomorphism between Conway's maximal surreal number field and the maximal hyperreals in von Neumann–Bernays–Gödel set theory.

title: Surreal number