text:
AM

AM or Am may refer to:

Music

 * A minor, a minor scale in music
 * A.M. (Chris Young album)
 * A.M. (Wilco album)
 * AM (Abraham Mateo album)
 * AM (Arctic Monkeys album)
 * AM (musician), American musician
 * Am, the A minor chord symbol
 * Armeemarschsammlung (Army March Collection), catalog of German military march music
 * Andrew Moore (musician), Canadian musician known as A.M.
 * DJ AM, American DJ and producer
 * Skengdo & AM, British hip hop duo

Television and radio

 * AM (radio program), Australian current affairs radio program
 * American Morning, American morning television news program
 * Am, Antes del Mediodía, Argentine current affairs television program
 * Am, a character in the anthology Star Wars: Visions
 * @fter Midnight (TV series) (logo: ), U.S. late night comedy celebrity game show

Other media

 * Allied Mastercomputer, the antagonist of the short story "I Have No Mouth, and I Must Scream"

Education

 * Active Minds, a mental health awareness charity
 * Arts et Métiers ParisTech, a French engineering school
 * Australian Museum, a museum in Australia
 * Master of Arts, an academic degree

Military

 * A US Navy hull classification symbol: Minesweeper (AM)
 * Air marshal, a senior air officer rank used in Commonwealth countries
 * Anti-materiel rifle, rifle designed for use against military equipment
 * Aviation structural mechanic, a U.S. Navy occupational rating

Science

 * AM, a complexity class related to Arthur–Merlin protocol
 * Adrenomedullin, a protein
 * Air mass (astronomy), measure of the amount of air along the line of sight in astronomical observations
 * Am, tropical monsoon climate in the Köppen climate classification
 * Americium, symbol Am, a chemical element
 * Attometre, a unit of length
 * attomolar (aM), a unit of molar concentration

Technology

 * Amplitude modulation, an electronic communication technique
 * AM broadcasting, radio broadcasting using amplitude modulation
 * Additive manufacturing, or 3-D printing, a process of making a three-dimensional solid object of virtually any shape from a digital model.
 * Agile modeling, a software engineering methodology for modeling and documenting software systems
 * Automated Mathematician, an artificial intelligence program
 * .am, Internet domain for Armenia
 * .am, a file extension associated with Automake software

Timekeeping

 * a.m., ante meridiem, the time period from midnight to noon, written e.g., 6a.m.
 * Anno Mundi, a calendar era based on the biblical creation of the world

Transportation

 * A.M. (automobile), a 1906 French car
 * Aeroméxico (IATA airline code AM), airline in Mexico
 * All-mountain, a discipline of mountain biking
 * Arkansas and Missouri Railroad

Other uses

 * First-person singular present of the copula verb to be
 * Am (cuneiform), a written syllable
 * Member of the Order of Australia, postnominal letters which can be used by a Member of the Order
 * Assembly Member (disambiguation), a political office
 * formerly a Member of the National Assembly for Wales, now Member of the Senedd
 * Member of the London Assembly
 * Amharic language (ISO 639-1 language code am)
 * Anguilla, LOC MARC code AM
 * Armenia (ISO country code AM)
 * Attacking midfielder, a position in association football
 * The Book of Amos, part of the Tanakh and Old Testament

title: AM


text:
Talk:Axiom

"True" and "universal" hopeless dream?
In the twentieth century, the grand goal of finding a "true" and "universal" set of axioms was shown to be a hopeless dream by Gödel and others.

 * That's not what Gödel showed. :-) --LMS

Well, that's a reasonable paraphrase of what he showed, which was that no set of axioms sufficiently large for ordinary mathematics could be both (1) complete, i.e., capable of proving every truth; and (2) consistent, i.e., never proving an untruth. Or to put it yet another way, there must exist some assertions that are true but unprovable. --LDC

 * Not even that... He showed that any formal system strong enough to have equality, addition, and multiplication, had those properties. (It didn't have to encompass all of ordinary mathematics.) [He also had a loophole that might have allowed non Omega consistant logics to bypass the problems, but Rosser closed that hole. Nahaj 01:57:56, 2005-09-08 (UTC)

 * That's first incompleteness theorem. There was also a second one, which is what LMS refers to — Kallikanzaridtalk 08:57, 6 March 2011 (UTC)

Hey... I was just following the style guidelines that said that I should leave something hanging! (I still stand be the statement that Incompleteness can be colloquially said to imply that there is no universal and true set of axioms. There are definitely complete and consistent systems such as real arithmetic, but they lack the power of, say, integer arithmetic and thus can be said not to be universal.  Another way to read what I was saying is that Principia was a hopeless task and not just because of a few paradoxes that might someday be weaseled around. -- TedDunning

 * What do you mean by real arithmetic? Not arithmetic of real numbers, surely, because that includes integer arithmetic as a subset and so is just as powerful.  -- Josh Grosse

Actually, real arithmetic does not include integer arithmetic as a subset. The reals include the integers, but logical systems built on the two fields are not equivalent. In particular, real arithmetic is generally taken as not including comparison while integer arithmetic has comparison. The exclusion of comparison is generally due to the complexity of the definitions of the reals. The completeness of the real system was proved (I think) by either Banach or Tarski in the middle of the twentieth century.

My own personal view is that Incompleteness is just a guise of the Halting problem. Since you can solve the Halting problem with real arithmetic where the reals are defined using bit-strings and you are allowed to look at and compare a finite prefix of any real. The trick is that the algorithm requires an initial condition that is not a computable real (TANSTAAFL!) -- TedDunning

Illegible
The Greek word in the etymology in this article is illegible on this browser (Netscape) and looks like a sequence of question marks. Contrast this:

 * αβ&gamma

and this:

 * $$\alpha\beta\gamma$$

the first is also illegible on Netscape, but you can tell what was intended; the second is perfectly legible. Michael Hardy 18:45 Mar 10, 2003 (UTC)

 * Note also that the first isn't valid HTML. Character entities *MUST* have the closing ";" to be valid. (I.E. It should be "&gamma;" instead of "&gamma") Since browser behavior is (as far as the standards are concerned) undefined if the HTML is invalid, one ought not to expect that the first case do anything reasonable.    That said, the use of the math markup is preferable anyway, in my opinion. Nahaj 01:54:53, 2005-09-08 (UTC)

Self-evident?
"As the word axiom is understood in mathematics, an axiom is not a proposition that is self-evident."

The Liddell and Scott entry for (axioma) says the exact opposite --Dwight 15:36, 12 Apr 2004 (UTC)

 * That remark seems very silly. The only "Liddell and Scott" I've been able to find is a lexicon translating ancient Greek words into English.  They would therefore be expected to write about what the word meant in Ancient Greek, not about what it means in the usage of modern mathematicians.  Liddell and Scott are probably right, and the statement you quote above about use in mathematics is also right.  They do not contradict each other; they are about two different things.  Liddell and Scott do not appear to be mathematicians and cannot be supposed to have expertise in that area.  I, on the other hand, am a mathematician, and I am quite familiar with both usages.  I suggest you read the whole Wikipedia article, and you will see that there is no contradiction between these points. Michael Hardy 22:33, 12 Apr 2004 (UTC)

Defined by Websters as a "self evident truth." It is one of those things that you think up while sitting on the can, or when when you can't sleep at 3:30 in the morning and you have some huge presentation to give the next day. You know, it just sort of hits you, but you knew it all along. Not to be confused with an epiphany. —The preceding unsigned comment was added by 172.198.219.243 (talk • contribs) 07:53, June 23, 2004 (UTC)

 * Nor with the trivial and obvious, which are theorems =) 142.177.126.230 21:19, 4 Aug 2004 (UTC)

 * ...except that there is are technical definitions of self-evident in epistemology. See self-evidence. Michael Hardy 01:41, 5 Aug 2004 (UTC)

Axiom and postulate are different things. Axioms are taken as self evident. Postulates are accepted because the theory that is derived from them is proven to be correct. Manuel, march 2008.

 * I'm a layman in logic, but from what I read in other aricles, treating axioms as self-evident is an outdated approach, they are rather considered starting points of theories. In fact, there are many different logics (classical logic, intuitionist logic etc.) so treating axioms as self-evident becomes philosophically moot. — Kallikanzaridtalk 08:54, 6 March 2011 (UTC)

The article has a logicistic attitude. It suggests non-logical axioms are not assumed to be true, but mathematical axioms are just as self-evidenct as logical axioms. — Preceding unsigned comment added by 72.238.115.40 (talk) 02:20, 11 December 2013 (UTC)

Completeness?
Gödel's completeness theorem establishes that every deductive system with a consistent set of non-logical axioms is complete

Does it? Doesn't it apply to a certain set of logical axioms and rules of deduction? Take a typical deductive system and remove a logical axiom schema or modus ponens. You still have a deductive system with a consistent set of non-logical axioms--albeit one that would not ordinarily be used (except perhaps by an intuitionist)--but it's not complete. Josh Cherry 02:07, 24 Oct 2004 (UTC)

Uniqueness of the reals?
We are fortunate enough to have that the standard model of "real analysis", described by the axioms of a complete ordered commutative field, is unique up to isomorphism.

This seems to say that there is a set of axioms that picks out the reals uniquely (up to isomorphism). What about the Löwenheim-Skolem theorem and such? I presume that although the reals are the unique complete ordered commutative field, completeness can not be expressed axiomatically, at least in systems to which the L-S theorems apply. Josh Cherry 02:58, 24 Oct 2004 (UTC)

 * L-S is about first-order theories, and this axiomization isn't one. You can come up with a first-order theory of all first-order sentences true of the real numbers using the ring functions of addition and multiplication, as well as the order relation, and then L-S would apply, and would tell us it has a countable model. We get the theory of formally real fields in this way, but not the real numbers uniquely. Gene Ward Smith 08:01, 1 Dec 2004 (UTC)

OK, I've changed the article to discuss this point. Josh Cherry 23:50, 1 Dec 2004 (UTC)

Examples
This editorial text was removed from the end of the examples page and is reproduced here:

''[OK. The later two are being presumed to actually be logical axioms, i.e. valid formulas. It would better be to say "valid formulas, as follows..." The proofs of these facts are definitely a technical issue, but interesting enough on their own.]''

Hu 20:36, 2004 Nov 22 (UTC)

Link suggestions
An automated Wikipedia link suggester has some possible wiki link suggestions for the Axiom article, and they have been placed on this page for your convenience. Tip: Some people find it helpful if these suggestions are shown on this talk page, rather than on another page. To do this, just add to this page. — LinkBot 10:39, 17 Dec 2004 (UTC)

When are axioms used?
The article now claims:

 * Basic theories, such as arithmetic, real analysis (sometimes referred to as the theory of functions of one real variable), linear algebra, and complex analysis (a.k.a. complex variables), are often introduced non-axiomatically in mostly technical studies, but any rigorous course in these subjects always begins by presenting its axioms.

It seems to me this just isn't true. Most often, what is done instead is to present definitions inside of set theory, and the set theory used is normally naive set theory. Linear algebra uses axioms when it wants to talk about vector spaces over arbitary fields, but that is quite different than using axioms to define the integers, real numbers, or complex numbers. If this sentence isn't given a good defense I'll remove it. Gene Ward Smith 19:34, 29 April 2006 (UTC)

Galois and geometry
While Galois theory was successfully applied to classic questions of geometry, the names here should be Gauss and Pierre Wantzel, not Galois. Gene Ward Smith 02:02, 2 May 2006 (UTC)

I have a complaint.
I probably shouldn't be saying this. And this maybe should be deleted. But I'm a little annoyed how every equation in wiki makes absolutely no sense. I would think there should be easy and hard ones to demonstrate how it works. —The preceding unsigned comment was added by 208.186.255.18 (talk • contribs) 05:54, June 3, 2006 (UTC)

 * Firstly, could you identify yourself by user name so we don't need to do detective work with the edit history to find out who wrote this?

 * Secondly, your complaint is horribly vague. Please explain what in the world you mean and cite examples. Michael Hardy 00:31, 5 June 2006 (UTC)

I think what this persons complaint was trying to convey is the language and explanations provided assume the average person knows as much as you do, I ended up here in the process of reading about a prescription drug which in the study cited, refers to percentages from (n=(some number), so in my search to find what specifically they were referring to I wiki'd statistics, proceeded to standard deviation, than on to algebraic symbols, than epsilon, than summation, harmonic numbers (though that was out of curiosity). I appreciate that knowledge especially mathematical is built on a chain of previous knowledge and that people take the time to share this. I did find what I was looking for as well and maybe better off for the journey, however I have been discouraged by other articles which seem more technical and maybe partly driven by the types of debates I've read. I think accessibility and relevance should be priority over precision as these topic's tend to mirror the tangent's of the contributors who may be focused on something more technical than required for a basic understanding and maybe a little less accessible for most people. However if I had a sample I could show you a graph now. Thanks again. —Preceding unsigned comment added by 64.53.203.180 (talk) 17:23, 31 August 2010 (UTC)

Lack of sources
Sorry, I forgot to post here after adding. There's not a single reference in the entire article, so I think the tag is warranted until the problem can be addressed. Simões ( talk/contribs ) 01:17, 22 October 2006 (UTC)

abstracted 'truth' out of introductory paragraph
Assuming it is not controversial to make the point that an 'axiom' does not necessarily connote a notion of "truth" or actuality, (except perhaps in the realms of epistemology, deontology, etc.) and therefore axioms are subject to whatever motivation is deemed appropriate under the circumstances, intro paragraph should reflect this. drefty.mac 07:00, 28 October 2006 (UTC)

Given
I came here looking for the goalkeeper Shay Given, and typed in "Given". Was redirected here. Obviously the disambiguation page for "axiom" was no use for me. Somebody might want to look into this. —Preceding unsigned comment added by 212.64.98.189 (talk • contribs) 22:41, March 7, 2007 (UTC)

Misleading Statement
The article states that

"...for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that no parallels through a point outside a line exist, that exactly one exists, or that infinitely many exist."

I think it might be a good idea to change part of this statement. It is true that the fifth postulate is independent of the first four, but the assumption that no parallels exist is not - it is a much stronger statement than the negation of Euclid's fifth and is inconsistent with the first four postulates. The difference between the two kinds of geometry with the first four postulates already given is that in non-euclidean geometry parallels are not unique, whereas in euclidean geometry they are. See "Euclidean and non-euclidean geometries" by Greenberg. Stephen Thompson 01:14, 29 April 2007 (UTC)

φ, ψ, χ or φ, χ, ψ?
Under "Mathematical logic" the article says: "...φ, ψ, and χ can be any formulae of the language...". Are these letters in the correct order? The article "Greek alphabet" says the alphabetical order of these letters is φ, χ, ψ. (Complex Buttons 20:03, 4 July 2007 (UTC))

acceptation?
Is this a word??? maybe Bush wrote it? —Preceding unsigned comment added by 128.232.238.250 (talk) 22:58, 24 November 2007 (UTC)

Yes, "acceptation" is a word. It means the generally recognized meaning or sense attributed to a word. It is a term frequently used in both philosophy (especially logic and epistemology) and linguistics. (I think the confusion here may be originating from a failure to maintain the distinction between words that do not exist and words that one simply does not know.) Mardiste (talk) 12:58, 28 January 2008 (UTC)
 * Snork Mardiste! :) 85.210.15.219 (talk) 23:29, 6 May 2011 (UTC)

Should we write about "bad axiom"??
formally, axiom is just anything you want to call an axiom...  you can define any system you want and call any random sequence of symbols the "axioms" of the system...

Just because every system we ACTUALLY USE are nice and useful and consistent and have nice little axioms as starting points doesn't mean we HAVE TO have axioms like that... we can just as well have an inconsistent axiom that totally screw up the entire system. Alternatively we can have an axiom that doesn't imply anything (for example in systems where there are no inference rules that can be used to derive theorems from the axiom)..

Basically it is quite possible to have "non-ideal" axioms... but the thing is, i'm not entirely sure if it'd be useful to point out that we can have these "non-ideal" axioms... most wikipedia readers are probably not going to find the comment useful... and it's probably confusing to non-specialists Philosophy.dude (talk) 01:19, 2 December 2008 (UTC)

btw, i think the comment that logical axioms are universally "true" is not entirely correct. .. there are plenty of formal axiomatic systems that does not assume truth at all... Philosophy.dude (talk) 01:26, 2 December 2008 (UTC)

 * The article already says, in the lede, "In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived." Does it say somewhere that axioms, in the contemporary sense, have to be nice in some way?

 * Regarding the use of the word "truth", I think the article seems reasonable. The contemporary definition of a "logic" involves both syntactic and semantical parts, and thus includes a notion of truth. Systems of mathematical interest with no semantics are extremely rare. — Carl (CBM · talk) 02:42, 2 December 2008 (UTC)

Scheme / schema
Both 'Axiom Schema' and 'Axiom Scheme' are used in this article. Are they both correct? Passingtramp (talk) 09:45, 21 May 2009 (UTC)

An error?
On the first line of the third paragraph there is an example given in the first set of parentheses 'e.g., A and B implies A'. This appears to be mis-written. Then again, what do I know. I will leave it up to editors of these types of pages. If I am wrong, please disregard. 220.255.1.30 (talk) 03:49, 16 January 2011 (UTC)

Axiom / Postulate
The distinction between axoims and postulates is never explicitly stated, which is regrettable since Postulate redirectes here. Anybody up for rectifying this? --93.206.135.188 (talk) 08:02, 13 September 2011 (UTC)

axioms are not assumed to be true. please correct the language used in this article.
technically, no position is taken on the truth of a set of axioms. they are merely premises that might be true. the formal process of deduction states that if the set of axioms is true than the set of deductions follows; a theorem is said to be true if the axioms that led to it's deduction are. but, the correctness of an axiom is neither discussed nor relevant. the relevant concept is consistency.

it's a subtle point, but glossing over it can have serious consequences. the rejection of absolute truth is the great insight of modern mathematics, an insight that has yet to work it's way to other fields. explicitly making this point whenever possible should be done to get the idea out and circulating, and aid in the abolition of superstition.

i may come back and do it myself, but i'm low on time and would prefer somebody else take the initiative, if they have the opportunity, please. — Preceding unsigned comment added by 70.53.24.43 (talk) 09:27, 27 January 2013 (UTC)

This seems to have been corrected in the current page. Anaxiomatic (talk) 10:58, 12 February 2013 (UTC)

Agreed. There are no absolute, universally true axioms. Sentence deleted. BlueMist (talk) 22:27, 19 December 2015 (UTC)
 * OK, look, the question is subtle and complicated. The IP contributor, 70.53.24.43, seems to be taking a formalist view, but this is by no means the only relevant modern view.  This article needs to be balanced between formalism and mathematical realism.  From a realist view, it is definitely of interest to inquire whether or not a given axiom is true.
 * I have no huge objection to User:Blue Mist 1's second change, this one; it does not make very much difference whether we say "accepted" or "accepted as true". However, as this is explicitly called out as the "classical" notion of axiom, we should probably say something about self-evidence; certainly classically self-evidence was an essential part of the notion of axiom.
 * As to the first change, this one, we have a different problem. It is true that it has been tagged with "citation needed" and I don't immediately have a citation for the exact claim.  However I can definitely give sources for workers grappling with the issue of what it means for an axiom to be true.  Here are two just from Penelope Maddy's bio:     That's just one example.
 * This is somewhat of a recurring problem; it is often easy to source important thinkers who take a certain position, but more difficult to source the meta-claim that various important thinkers take the position. But it's a problem that we ought to solve for this article; this is an important article that should present all the significant views. --Trovatore (talk) 23:06, 19 December 2015 (UTC)

Trovatore, you're an experienced editor. You know not to revert an edit agreed upon by three other editors. Please undo your revert, and discuss the issue here, as required by Wikipedia, before enforcing your personal, unsupported preferences ! BlueMist (talk) 23:40, 19 December 2015 (UTC)
 * I am discussing it. I will not revert it.  You made the bold edit, which I reverted.  There is no consensus here, just two other editors from almost three years ago. --Trovatore (talk) 23:41, 19 December 2015 (UTC)
 * By the way, not at all "unsupported" — I supplied specific citations, for one point of view. I can get others for that point of view and for others.  I don't know how easily I could find one specifically for "multitude of views"; that's the problem I mentioned above.  However there are a multitude of views that I can source individually. --Trovatore (talk) 23:47, 19 December 2015 (UTC)

 * I am deeply unhappy with the claims made here that axioms are the same thing as premises and postulates.

 * Axioms are one leg of a system of reasoning; they are statements taken to be true for the purposes of that system. The other leg is the rules of inference.

 * Premises are statements that are not taken to be true; rather, reasoning that proceeds from premises leads to conclusions that depend on those premises. For example, if the reasoning leads to a contradiction, it follows that one or more of the premises must have been false. That is not possible for axioms; there is no kind of reasoning that can lead one to conclude that an axiom is false.

 * A postulate is a different thing again. It is a statement that someone believes to be a theorem, but for which they have so far failed to construct a proof. Some fields of mathematics rely on the truth of such an unproven postulate (i.e. they treat the postulate as if it were an axiom, while fully accepting that it is no such thing). A postulate can be proven false; it then ceases to be a postulate, and becomes simply a false statement.

 * I fear to attempt to edit this page, because it is full of this belief that these terms all mean the same thing, because so many editors seem to share this 'all the same' view, and because I can't imagine what kind of source would suffice to prove they are distinct (I could cite Beginning Logic, by Lemmon; but that would presumably not satisfy those who believe that logical axioms are just one kind of axiom - a claim I do not accept).

 * If they do all mean the same thing, it becomes dramatically harder to talk about reasoning. And why would one use any of these terms, if not to talk about reasoning?

 * On the truth of axioms: if one considers an axiom in the context of the system of reasoning in which it is embedded, then it cannot be considered to be true, as if it were a fact about the world. A purported fact can be shown to be false, but it is impossible to falsify an axiom, either by logic or by reference to reality. I am unable to check the OED reference for this: "As used in modern logic, an axiom is simply a premise or starting point for reasoning", but I seriously doubt the OED gives any such vague definition for this term. My Concise OED certainly doesn't say that. MrDemeanour (talk) 16:01, 22 March 2018 (UTC)

Perhaps these references will help: — Carl (CBM · talk) 23:12, 20 December 2015 (UTC)
 * https://math.stanford.edu/~feferman/papers/newaxioms.pdf
 * http://www.scientificamerican.com/article/infinity-logic-law/
 * http://home.uchicago.edu/~wwtx/objectivity.pdf

This question is extrinsic and philosophical, not intrinsic and mathematical. There are a number of competing supported positions that need to be in that paragraph, or none at all. It's a WP:NPOV requirement. Please also see my talk page. BlueMist (talk) 00:25, 21 December 2015 (UTC)
 * Blue Mist, they are all there. Not all in detail, of course (actually, not any of them in detail).  But every single position on the question, including yours, is a position about "[w]hether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be 'true'".
 * The previous wording that you challenged said instead "[w]hat it means for an axiom, or any mathematical statement, to be 'true'", and that also included your position, though I take it your answer would be "nothing". However I can see that it is possible to read that text as implying that it does mean something (the text didn't actually imply that, but it was possible to read it that way), so I changed it to make it clear that "nothing" was one of the answers being considered.  I can't see how you can possibly object to this on NPOV grounds, since it includes all the possible POVs. --Trovatore (talk) 06:11, 21 December 2015 (UTC)

My position is that I agree with the above two editors, 70.53.24.43 and Anaxiomatic that the sentence needs to be deleted because it is biased in favor of dogmatism. I urged you to undo your revert of this deletion in compliance with NPOV. BlueMist (talk) 07:07, 21 December 2015 (UTC)
 * How is it biased, and in particular in favor of dogmatism? You have as yet not explained that. --Trovatore (talk) 07:12, 21 December 2015 (UTC)
 * BlueMist (talk) 16:09, 21 December 2015 (UTC)
 * Not responsive. I didn't ask for a definition of dogma.  I asked how the text you challenged was biased, and in particular in favor of dogmatism. --Trovatore (talk) 19:41, 21 December 2015 (UTC)

On another note, I find in the history that this language was put into place in May 2015, and there were various attempts before we settled on this. This was one attempt of mine, and in some ways I still like it better than what we have now:
 * In modern mathematics, it is controversial whether and when to consider axioms "true" or "false" outside the context of a deductive system starting with those axioms. In broad strokes, mathematical realists, sometimes called Platonists, consider some axioms to be true or false in structures that they believe to be well-specified, whereas formalists may take formal derivability from the axioms as simply the meaning of the derived statements (making axioms "true" by definition).

Blue Mist, what do you think of that? I didn't spend really very long on it at the time, and no doubt it can be improved, especially the formalism half, but I think it is at least balanced between the two.

One possible quibble is that some formalists may consider axioms "true by definition" whereas others may simply decline to consider their truth. My take on this is that this is not so much a difference in substance as it is in preferred terminology. However certainly both versions could be mentioned.

Also, I'm not sure how readable the "formal derivability from the axioms as simply the meaning" language is. I know what it means, but then I wrote it. No doubt that section can be written more clearly. --Trovatore (talk) 07:30, 21 December 2015 (UTC)

I don't see how the following sentence from the article is biased in favor of any particular position:
 * Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be "true" is a central question in the philosophy of mathematics, with modern mathematicians holding a multitude of different opinions.

The sentence does not even state a particular viewpoint on the issue, it only states that there are several of them. — Carl (CBM · talk) 11:39, 21 December 2015 (UTC)

 * Carl, I've stated my position clearly enough above. I deleted the sentence following the suggestion of two (2) other editors, in accordance with WP:Consensus, as it was judged to be in violation of NPOV.

 * As an ex-admin, you know that I am not required at Wikipedia to defend my reasons. I am not on trial here.

 * That sentence is plain nonsense in philosophy of mathematics terms, as even the idea it suggests as "controversial" is untenable according to the very citation that was added there in its defense. Just click on the reference and read the freely available first page of that article to see. Therefore, the sentence should also be removed as also a WP:NOR violation.

 * The paragraph that follows explains axioms, reality, and truth fully enough that this disputed sentence adds nothing to the content of the article.

BlueMist (talk) 20:40, 21 December 2015 (UTC)

 * I also fail to see how that sentence can be read as being "biased in favor of dogmatism". Blue Mist would you please address this specific point? How would you rewrite it so as to make it neutral? Paul August ☎ 20:45, 21 December 2015 (UTC)

 * It's the judgment of the first 3 editors. In fact. If you disagree, then the issue is closed. BlueMist (talk) 21:56, 21 December 2015 (UTC)
 * Your history is a little off here, Blue Mist. The disputed language dates to May 2015, and the discussion involved at least three editors at that time.  The last contribution from the two other editors you cite (at least, from those accounts) was in February 2013, so they definitely did not pronounce themselves on that language per se.  However, if the issue is closed for you, then that's OK with me. --Trovatore (talk) 22:05, 21 December 2015 (UTC)

Of course, the issue being "closed" for Blue Mist does not mean that there are no improvements to be made.
 * The current text (which is mostly the work of User:Sammy1339, I think) should indeed be better referenced, if we keep it.
 * I still think it might be good to call out specific realist/formalist/other positions, along the lines of my attempt reproduced above in my comment of 07:30, 21 December 2015. The sentence structure of that attempt definitely needs work, though.
 * More should be said about the classical (e.g. Euclid) notion of axiom as self-evident truth. (I think that's the Euclidean notion but I am not a scholar of that era; I am open to correction on that point.)
 * The overall flow of the lead could use work, especially the second paragraph, the one that starts "[i]n mathematics". The text that we are currently discussing is about mathematics, so in that sense perhaps it belongs in the second paragraph.  On the other hand, if we (as I think we should) restore a discussion of the classical Euclidean notion to the first paragraph, then some discussion of modern views on truth might also make sense there. --Trovatore (talk) 22:14, 21 December 2015 (UTC)

Unclear: "When an equal amount is taken from equals, an equal amount results."
This example axiom is unclear; what does taken mean?

I can see only two interpretations: 1) Subtracting equal amounts from equals produces equal amounts; then the quote should be revised to read "equal amounts result", as there are two quantities being compared 2) Subtracting equal amounts from two quantities and adding each to a third produces an amount equal to the first two; this is plainly false

So perhaps the quote should be revised.Anaxiomatic (talk) 10:57, 12 February 2013 (UTC)

Reflist
I changed it to use reflist. This is pretty standard common stuff for > 10 refs (or more), maybe 40em would be better than 30em with these ref widths but what's the objection? Widefox ; talk 08:53, 26 September 2015 (UTC)
 * WP:CITEVAR The established reference formatted should not be changed randomly; there is no "standard" that requires multiple columns. - User:CBM
 * My understanding of CITEVAR is that's more about "citation style" than use of reflist. I can only see in WP:CITEVAR WP:ASL that one may use either, so not compulsory either but improves by reducing whitespace.
 * 70% of articles use reflist and don't flip randomly. Widefox ; talk 08:53, 26 September 2015 (UTC)
 * There is no "standard" formatting that applies across all articles. Moreover, there is no benefit to the multiple columns, because there is an unlimited amount of scroll space in the browser. Personally, I don't see any extra white space in the references - perhaps your window is too wide? There is no way to get consistent wrapping, because monitors, browsers, and font sizes vary too much.

 * So, from my perspective, there was no objective benefit for the change, and it was just based on someone's personal preference to have multiple columns. But for things that are personal preferences, there is a guideline, WP:CITEVAR, which is to leave the established style in place, rather than trying to standardize different articles. Otherwise, other people could go around removing the columns from other articles, or changing the number of columns to suit their taste, etc. — Carl (CBM · talk) 14:05, 26 September 2015 (UTC)

 * What, figure of speech "pretty standard" (from an edit summary), struck through and replaced by "common" here (for clarity), and it's still not clear the exact precise intended word was "common"?
 * There's an inevitability of increasing refs. Good luck, Widefox ; talk 12:32, 27 September 2015 (UTC)

Mismatching parentheses in formula
The third formula in 3.2.1.1 Arithmetic seems to have mismatching parenteses, there is 1 unnecessary opening parentesis. Could someone who understands the formula well repair it?213.125.95.58 (talk) 14:36, 7 May 2017 (UTC)
 * ✅ --Bill Cherowitzo (talk) 14:52, 7 May 2017 (UTC)

Axioms, premises and postulates
I am deeply unhappy with the claims made here that axioms are the same thing as premises and postulates.

Axioms are one leg of a system of reasoning; they are statements taken to be true for the purposes of that system. The other leg is the rules of inference.

Premises are statements that are not taken to be true; rather, reasoning that proceeds from premises leads to conclusions that depend on those premises. For example, if the reasoning leads to a contradiction, it follows that one or more of the premises must have been false. That is not possible for axioms; there is no kind of reasoning that can lead one to conclude that an axiom is false.

A postulate is a different thing again. It is a statement that someone believes to be a theorem, but for which they have so far failed to construct a proof. Some fields of mathematics rely on the truth of such an unproven postulate (i.e. they treat the postulate as if it were an axiom, while fully accepting that it is no such thing). A postulate can be proven false; it then ceases to be a postulate, and becomes simply a false statement.

I fear to attempt to edit this page, because it is full of this belief that these terms all mean the same thing, because so many editors seem to share this 'all the same' view, and because I can't imagine what kind of source would suffice to prove they are distinct (I could cite Beginning Logic, by Lemmon; but that would presumably not satisfy those who believe that logical axioms are just one kind of axiom - a claim I do not accept).

If they do all mean the same thing, it becomes dramatically harder to talk about reasoning; we would need a new word for "axiom". And why would one use any of these terms, if not to talk about reasoning?

On the truth of axioms: if one considers an axiom in the context of the system of reasoning in which it is embedded, then it cannot be considered to be "true", as if it were a fact about the world. A purported fact can be shown to be false, but it is impossible to falsify an axiom, either by logic or by reference to reality.

I am unable to check the OED reference for this: "As used in modern logic, an axiom is simply a premise or starting point for reasoning", but I seriously doubt the OED gives any such vague definition for this term. My Concise OED certainly doesn't say that. MrDemeanour (talk) 16:01, 22 March 2018 (UTC)

 * This is an eternal topic of discussion. To start the new thread, I have copied a couple sources below, and asked a question about the source you mentioned. — Carl (CBM · talk) 18:52, 22 March 2018 (UTC)

 * Here are the four OED definitions, verbatim, for the convenience of those who cannot check them.
 * A proposition that commends itself to general acceptance; a well-established or universally-conceded principle; a maxim, rule, law.
 * b. Specially restricted by Bacon to: An empirical law, a generalization from experience.
 * Logic. A proposition (whether true or false).
 * Logic and Math. ‘A self-evident proposition, requiring no formal demonstration to prove its truth, but received and assented to as soon as mentioned’ (Hutton).
 * — Carl (CBM · talk) 18:39, 22 March 2018 (UTC)

 * The (Springer) Encyclopedia of Mathematics includes:
 * "Axiomatic method: A way of arriving at a scientific theory in which certain primitive assumptions, the so-called axioms (cf. Axiom), are postulated as the basis of the theory, while the remaining propositions of the theory are obtained as logical consequences of these axioms."
 * This seems to conflate postulates with axioms. — Carl (CBM · talk) 18:42, 22 March 2018 (UTC)

 * Which part of Lemmon's "Beginning Logic" has an explanation of the difference he maintains between axioms, postulates, and premises? — Carl (CBM · talk) 18:50, 22 March 2018 (UTC)

 * Good question! I seem to have lost my old copy; I've had a pretty good hunt for it. Also, I'm sure I haven't read it for thirty years, so I guess my recollection of what he did and didn't say is pretty suspect.
 * Update: I found my copy of Lemon, and to my astonishment he doesn't discuss axioms at all.
 * Well, if I'm wrong, that's a big deal for me. There's a lot of stuff I'd have to review, if it turns out that an axiom is the same as a premise. MrDemeanour (talk) 19:28, 23 March 2018 (UTC)
 * No one is saying you're "wrong", exactly. You're just not "right" in quite the universal sense you might think.  Different people use these terms different ways.  You seem to be taking a certain POV, one of many possible. --Trovatore (talk) 21:33, 23 March 2018 (UTC)

 * I don't think anyone is wrong - I am simply curious to find out about another usage of terminology that I was not familiar with. As a logician I have always identified "axiom", "postulate", "assumption", etc. - in first-order logic there is no difference between them. There is at least one sensible way to distinguish "premise" as a hypothesis used by an inference rule in a proof, even if that premise is not itself one of the assumptions of the proof. But the terminology can become more interesting when we start looking at more "traditional" logic texts, especially those written before the dominance of first-order logic or those that are interested in reasoning that is not captured in systems like first-order logic. So I would genuinely like to know if there is some position that keeps a distinction between "axiom" and "postulate", even if that position is not common in modern texts. — Carl (CBM · talk) 13:07, 24 March 2018 (UTC)
 * MrDemeanour seems to be using "postulate" in the sense that mathematicians more commonly use "conjecture". I have the notion that this terminology is common among physicists, though I don't have anything to point you to to back that up. --Trovatore (talk) 20:11, 24 March 2018 (UTC)
 * Interesting - so that would be more like the "Riemann hypothesis" for mathematicians. As you know "postulate" is not used much in math anymore - it makes me think of Euclid. — Carl (CBM · talk) 20:25, 24 March 2018 (UTC)

Flaws in The Elements
One of the discoveries since the ancient Greeks is that there are incorrect proofs in Euclid's Elements and that, in fact, some of the stated theorems do not follow from the stated axioms and postulates. Should this not be discussed when mentioning the Elements?. In that context, a reference to, e.g., Hilbert's Grundlagen der Geometrie, may be appropriate, Shmuel (Seymour J.) Metz Username:Chatul (talk) 02:07, 8 July 2020 (UTC)
 * It seems to me more about the Elements than it is about axioms, so I would tend to think it does not really belong in this article. However I wouldn't dig in my heels to prevent a passing mention, if you can word it so it doesn't sound out of place. --Trovatore (talk) 05:55, 8 July 2020 (UTC)
 * To clarify, the article already mentions the Elements in Axiom; I'm concerned that the existing text promises more than the Elements can deliver. Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:26, 8 July 2020 (UTC)
 * Of course we should mention the Elements, as an early (maybe the earliest?) explicit example of the axiomatic method. That's about axioms.  When you start talking about whether all the specific results in the Elements follow from the axioms in the Elements, then it seems to me that maybe you've left off talking about axioms, and started talking about the Elements.  --Trovatore (talk) 15:02, 8 July 2020 (UTC)
 * Well, would you object to a short footnote, e.g., well-illustrated, and to a brief mention in #Modern development of adding axioms? Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:30, 8 July 2020 (UTC)

Aproach Edit
Hello Community🙂❤

Here below my edit aproach:

"In historical sciences/context you can see axiomata perhaps as more inclusive as it, for example the third reich here, is satisfying enough if it fully encovers the empathic aspekt abeling one with permission to go on with further assumptions. Inclusive in this way, that it is possible to work with prezise statements regarding empathic ideas without necessarly having great knowledge of formally required information normally required in this topic. So for example: "If ones an ex nationalsocialist it is appropriate for him as a way of caesura in his further life to go on with maybe dramatically social ideas maybe in a slightly risky kontext at least in generating this ideas mentally". So this axiomata is more inclusive as through fully satisfying every aspekt (at that point not yet formally) regarding empathy and through this at least this specific phrase in this specific context, if satisfying the empathic site enough automatically creates valid statements to work with. It is also inclusive in that way that in its origin, the messenger is able to collect information and knowledge for this specific case without explicit formal reputation/education so it enables the working with this discipline to a broader number of people."

So first my provided sources The autor of this article says that the ability to work with statistical math requires the same way of thinking as with connecting to someones feelings. The autor of this article works with our core sozial values as mathematical axioma and how society changes if changing this values. This (also a bit more Professional) source says that with an AI, working with empathy and with this training it in social complex situations outperformes other AIs in methodological learning.

Also I tried writing down some context below, first in case of reposting my edit, but because I love wikipedia too much to get blocked, I switched to the Talk section so Here it is:

"First, I want to say, I love Wikipedia and sometimes I Love editing. Also english is not my native language and I know I make grammatical and orthographically mistakes. Second, I provided some Sources which stated that empathy have significantly Impact on mathematics. But thats just a Part of what I wanted to say. As said before I Love wikipedia and this article. Im not an expert, but first, I tried Just writing about things, that, as far as I guess am able to understand. So second I tried to say that when you, in the process of working with axiomatas, switch the things referred to in this process, to themes you are more familiar with, maybe for this parts of the process Its easier to work with making this Part of mathematics maybe more inclusive. So with this said maybe I can work on this edit to Point this Out more clearly. Second, I guess that, If just regarding the improved performance when work with themes, one is more familiar with, and, regarding my explicit example of working with the third reich in this context, which, is from a science aspect one of the few themes, that, are finished in some way completely and almost every statement is true working With an anti fascist View, maybe this theme gives somehow an extra boost. To be honest, surely, I don't know if the parts I describe in the process with axiomatas, even when performance is better, wether it is possible to get outcomes which are satisfying enough for this mathematical aspect even with, as it is with me also, little lackings of full knowledge of this aspect. To the critiques regarding non dictionary language I want to say that the people I wanted to reach are nonexperts explicitly so I wanted to decrease the limitations for gaining more safety in performing this theme a bit, mostly just as I do in thinking of this theme relatively correct, so it isn't even my aproach to reach people working, regarding this theme, in expressing their workings down to sheets of paper or professionals."

So if you read all this, I would love to hear your critiques, improving ideas, and, if you think all this is pure trash, let me hear your opinion, don't worry Im not THAT emotional unstable😁❤ Materie34 (talk) 03:00, 20 November 2023 (UTC)

 * Wikipedia is not a forum for discussion and not a platform for publishing your own ideas. If any of the material above is somehow being proposed as something to include in our article at Axiom, I have to observe that it is not proper English, even in a single clause of it, and it appears to desire to dwell on Nazism, which would be very undue. If the term "axiom" has something to do with the Nazi doctrine, that could perhaps be covered somewhere in the article, but only if there are reliable sources that show it to be central to that doctrine, and an important historical aspect of the meaning of axiom. The material you've provided has no clear purpose, direction, or theme, and seems to be a bunch of quasi-philosophical rambling, veering from Nazis, to empathy, to statement precision, to messengers of some kind, to reputation, to AI, and it makes no sense at all, much less to the average encyclopedia reader. There's nothing for us to work with here. Your "sources" are a self-published opinion piece at a social media site (which doesn't mention axioms); psychologic.science, which is obviously just a bunch of self-published material on pseudo-scientific nonsense like "alpha males"; and a paper published by MDPI which is listed at WP:Reliable_sources/Perennial_sources#MDPI as "questionable" to "predatory", and the paper has no mention of "axiom" at all any way, so it is not any kind of source for this topic.  Please do not waste Wikipedia editors' time with more material of this sort.  If you are actually interested in working on an encyclopedia, which means working with  sources and summarizing what they say without engaging in any form of original research of your own on any topic, then you would do well to focus on whichever edition of Wikipedia is in your native language.  — SMcCandlish ☏ ¢ 😼  04:36, 20 November 2023 (UTC)

Might be controversial: Is addtive commutative an axiom?
Hello community,I think additive commutative law are not considered as an axiom but an theory derived from the Peano Axiom? I did found some of the people call it an "axiom" in arithmetic. However, in early undergraduate analysis courses, it's often used as an example of basic reasoning to derive some laws in natural numbers from Peano Axiom. I doubt if it's a good example here. Alexliyihao (talk) 02:03, 30 January 2024 (UTC)
 * The only appearance of commutative is in #Non-logical axioms, where the context is group theory. The Peano postulates are not relevant in that context. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:43, 30 January 2024 (UTC) theory.
 * Commutativity of addition is a theorem in Peano theory, see these course notes. Paradoctor (talk) 12:38, 21 April 2024 (UTC)
 * How is that relevant to group theory? The article is about axioms, not about theorems.
 * As I said,
 * As an illustration of the importance of context, arithmetic discusses two approaches, in one of which the Peano Postulates are themselves theorems. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:13, 22 April 2024 (UTC)
 * Are we reading the same encyclopedia? It says that the Peano axiom are theorems in some set theory!
 * You also appear to misunderstand the point Alexliyihao made: the lead uses commutativity of addition as an example of a non-logical axiom, which is misleading. Arithemic is normally axiomatized using the Peano axioms, or some set-theoretic model thereof. There, commutativity of addition is a theorem. Surely we ought to do better? Paradoctor (talk) 13:37, 22 April 2024 (UTC)
 * Yes, and I never claimed that the wiki article said that. Many elementary course in Set theory derive the Peano postulate from the definition of $$\mathbb{N}$$ from the construction mentioned in arithmetic.
 * is bogus, because it ignored the context and the wording. The text is clearly talking about Group theory, not about the Peano postulates. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:21, 22 April 2024 (UTC)
 * Seriously, what are you reading?!? Alexlihiyao did not talk about group theory, never mentioned it. That's something you imported here. All he did is criticize the use of commutativity of addition as an example of a non-logical axiom, where it is connected to . is the context we're talking about. Paradoctor (talk) 16:38, 22 April 2024 (UTC)
 * Arithmetic is not the same as Peano arithmetic. It's perfectly possible to consider commutativity of addition as an axiom of arithmetic; it would be a different set of axioms, but the same subject matter.  I think you're over-focusing on a particular axiomatization, which isn't mentioned at the point in question in the text. --Trovatore (talk) 19:02, 22 April 2024 (UTC)
 * I focus on the most likey interpretation, given that we're not a specialty encyclopedia. Anyway, I finally found out what tripped up Chatul: The lead does not properly represent the article. I took the liberty of fixing that. Unless someone has a better idea, that should conclude this discussion. — Preceding unsigned comment added by Paradoctor (talk • contribs) 19:35, 22 April 2024 (UTC)
 * "The most likely interpretation" of arithmetic is hardly the Peano axioms; that's a much more "specialty" notion than arithmetic per se.
 * In any case, your new text is problematic because commutativity is not an axiom of group theory, and also because we shouldn't be assuming that people know about group theory at this point in the article. --Trovatore (talk) 19:42, 22 April 2024 (UTC)
 * We could replace it with a + 0 = a, maybe? --Trovatore (talk) 19:47, 22 April 2024 (UTC)
 * MOS:LEAD: The lead should [...] summarize the body of the article
 * Commutativity for groups is the first example mentioned in § Non-logical axioms, and the only one mentioned in its introduction. Neutral element of addition is not mentioned at all.
 * not an axiom of group theory It is in the theory of commutative groups. So we add a word. Paradoctor (talk) 20:50, 22 April 2024 (UTC)
 * OK, the lead should generally summarize the body, but we don't have to be ultra-rigid about it. If it's useful to put an example in the lead that doesn't appear in the body, I think that's fine.  And I don't think we should mention groups in the lead; too specific for the general article about axioms. --Trovatore (talk) 20:59, 22 April 2024 (UTC)
 * 🤦 Paradoctor (talk) 21:14, 22 April 2024 (UTC)
 * Obviously, the comments relating to Axiom, NOT TO THE LEAD.
 * . The text in contention, Which part of  did you not understand? And, yes, there were subsequent updates to the lead, but that has nothing to do with the validity of comments posted before them.
 * . Patently false: the lead, AT THAT TIME, did not contain any such text. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:16, 25 April 2024 (UTC)
 * when Alexliyihao . Or you could just look at the passage they quoted.
 * the comments relating to Axiom, NOT TO THE LEAD Then you're missing the point, because the discussion is about a passage from the lead. Paradoctor (talk) 12:54, 25 April 2024 (UTC)
 * Repeating a bogus claim won't make it magically come true. The word commutative does not appear in the lead of permalink/1213388710, nor does it appear in the lead of permalink/1197258286. The first appearance is in Axiom. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:01, 25 April 2024 (UTC)
 * paraphrase 🤦 Paradoctor (talk) 20:52, 25 April 2024 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Axiom


text:
Talk:Automated Alice

Untitled
''Er, quick...is this copyrighted? I looked on the web and found no evidence whatever that it is public domain. Therefore, you should entirely remove this from Wikipedia, correct? Curiously, LMS''

Errrrrr nooooooo! I read Automated Alice and just finished it yesterday, what I've wrote so far are my (Neeklamy's) own words, I didn't lift them from anywhere!

I find it a bit disconcerting to be accused of plagiarism - was it not obvious that this text is just a summary? That I wrote it (in good heart) in the style of the book? The book itself came out in 1996 and has been published a number of times over since - it's out there for anyone to track down ISBN: 0552999059 and read and it runs to a bit more than a few mere paragraphs of poorly revised text (more like 252 pages of some very witty dialogue).

I'm sure you didn't mean offence, what with a number of indivduals stealing wholesale from various resources, but seriously, idle accusations of copyright infringement aren't appreciated at all - personally I would never plagiarise and would never do anything to harm this new forum.

(half inch = pinch, Cockney rhyming slang)

I assume that's why Larry left the text here and merely asked the question, rather than simply deleting it as should be done for a clear infringement (NOTE: copyright infringement and plagiarism are two entirely different things--don't confuse them). It is the nature of Wikipedia that authors here must be thick-skinned in the face of such questions, and peer edits, and commentary about articles, and many other things. It is important that such potentially offensive things get said and replied to rather than being supressed out of a misguided and counterproductive sense of "politeness". --LDC

Hate to repeat myself - but read what I wrote, hardly a novel is it? And if it were I'd ask for my money back :-)

Common sense - it's cheap, limitless and enviro-friendly - use it. Thank you

The third book of the adventures of Alice by Charles Dodgson, er, that isn't right, Lewis Carroll, no, no, by Zenith O'Clock...

by Jeff Noon...?

yes! that's it!

Steam Punk
In the genre box at the top of the article this is described as cyberpunk. Surely it is in fact steam punk. e.g. The computer in the book is hardly high tech Schizocarp (talk) 15:29, 4 November 2012 (UTC)

title: Talk:Automated Alice


text:
Antigua and Barbuda

Antigua and Barbuda is a sovereign island country in the Caribbean. It lies at the conjuncture of the Caribbean Sea and the Atlantic Ocean in the Leeward Islands part of the Lesser Antilles.

The country consists of two major islands, Antigua and Barbuda, which are approximately 40 km apart, and several smaller islands, including Great Bird, Green, Guiana, Long, Maiden, Prickly Pear, York, and Redonda. The permanent population is approximately 97,120 ( estimates), with 97% residing in Antigua. St. John's, Antigua, is the country's capital, major city, and largest port. Codrington is Barbuda's largest town.

In 1493, Christopher Columbus surveyed the island of Antigua, which he named for the Church of Santa María La Antigua. Great Britain colonized Antigua in 1632 and Barbuda in 1678. A part of the Federal Colony of the Leeward Islands from 1871, Antigua and Barbuda joined the West Indies Federation in 1958. With the breakup of the federation in 1962, it became one of the West Indies Associated States in 1967. Following a period of internal self-governance, it gained full independence from the United Kingdom on 1 November 1981. Antigua and Barbuda is a member of the Commonwealth and a Commonwealth realm; it is a constitutional monarchy with Charles III as its head of state.

The economy of Antigua and Barbuda is largely dependent on tourism, which accounts for 80% of its GDP. Like other island nations, Antigua and Barbuda is vulnerable to the effects of climate change, such as sea level rise, and increased intensity of extreme weather like hurricanes. These cause coastal erosion, water scarcity, and other challenges.

Antigua and Barbuda offers a citizenship by investment program. The country levies no personal income tax.

Etymology
Antigua is Spanish for 'ancient' and barbuda is Spanish for 'bearded'. The island of Antigua was originally called Wadadli by the Arawaks and is locally known by that name today; the Caribs possibly called Barbuda Wa'omoni. Christopher Columbus, while sailing by in 1493, may have named it Santa Maria la Antigua, after an icon in the Spanish Seville Cathedral. The "bearded" of Barbuda is thought to refer either to the male inhabitants of the island, or the bearded fig trees present there.

Pre-colonial period
Antigua was first settled by archaic age hunter-gatherer Native Americans called the Ciboney. Carbon dating has established the earliest settlements started around 3100 BC. They were succeeded by the ceramic age pre-Columbian Arawak-speaking Saladoid people who migrated from the lower Orinoco River. They introduced agriculture, raising, among other crops, the famous Antigua Black Pineapple (Ananas comosus), corn, sweet potatoes, chiles, guava, tobacco, and cotton. Later on the more bellicose Caribs also settled the island, possibly by force.

European arrival and slavery
Christopher Columbus was the first European to sight the islands in 1493. The Spanish did not colonise Antigua until after a combination of European and African diseases, malnutrition, and slavery eventually extirpated most of the native population; smallpox was probably the greatest killer.

The English settled on Antigua in 1632; Christopher Codrington settled on Barbuda in 1685. Tobacco and then sugar was grown, worked by a large population of slaves transported from West Africa, who soon came to vastly outnumber the European settlers.

Colonial era
The English maintained control of the islands, repulsing an attempted French attack in 1666. The brutal conditions endured by the slaves led to revolts in 1701 and 1729 and a planned revolt in 1736, the last led by Prince Klaas, though it was discovered before it began and the ringleaders were executed. Slavery was abolished in the British Empire in 1833, affecting the economy. This was exacerbated by natural disasters such as the 1843 earthquake and the 1847 hurricane. Mining occurred on the isle of Redonda, however, this ceased in 1929 and the island has since remained uninhabited.

Part of the Leeward Islands colony, Antigua and Barbuda became part of the short-lived West Indies Federation from 1958 to 1962. Antigua and Barbuda subsequently became an associated state of the United Kingdom with full internal autonomy on 27 February 1967. The 1970s were dominated by discussions as to the islands' future and the rivalry between Vere Bird of the Antigua and Barbuda Labour Party (ABLP) (Premier from 1967 to 1971 and 1976 to 1981) and the Progressive Labour Movement (PLM) of George Walter (Premier 1971–1976). Eventually, Antigua and Barbuda gained full independence on 1 November 1981; Vere Bird became prime minister of the new country. The country opted to remain within the Commonwealth, retaining Queen Elizabeth as head of state, with the first governor, Sir Wilfred Jacobs, as governor-general. Succeeding Sir Wilfred Jacobs were Sir James Carlisle (June 10, 1993 – June 30, 2007), Dame Louise Lake-Tack (July 17, 2007 – August 14, 2014.), and the present governor, Sir Rodney Williams: (August 14, 2014 – present).

Independence era
The first two decades of Antigua's independence were dominated politically by the Bird family and the ABLP, with Vere Bird ruling from 1981 to 1994, followed by his son Lester Bird from 1994 to 2004. Though providing a degree of political stability, and boosting tourism to the country, the Bird governments were frequently accused of corruption, cronyism and financial malfeasance. Vere Bird Jr., the elder son, was forced to leave the cabinet in 1990 following a scandal in which he was accused of smuggling Israeli weapons to Colombian drug-traffickers. Another son, Ivor Bird, was convicted of selling cocaine in 1995.

In 1995, Hurricane Luis caused severe damage on Barbuda.

The ABLP's dominance of Antiguan politics ended with the 2004 Antiguan general election, which was won by Winston Baldwin Spencer's United Progressive Party (UPP). Winston Baldwin Spencer was Prime Minister of Antigua and Barbuda from 2004 to 2014. However the UPP lost the 2014 Antiguan general election, with the ABLP returning to power under Gaston Browne. ABLP won 15 of the 17 seats in the 2018 snap election under the leadership of incumbent Prime Minister Gaston Browne.

In 2016, Nelson's Dockyard was designated as a UNESCO World Heritage Site.

Most of Barbuda was devastated in early September 2017 by Hurricane Irma, which brought winds with speeds reaching 295 km/h (185 mph). The storm damaged or destroyed 95% of the island's buildings and infrastructure, leaving Barbuda "barely habitable" according to Prime Minister Gaston Browne. Nearly everyone on the island was evacuated to Antigua. Amidst the following rebuilding efforts on Barbuda that were estimated to cost at least $100 million, the government announced plans to revoke a century-old law of communal land ownership by allowing residents to buy land; a move that has been criticised as promoting "disaster capitalism".

Geography
Limestone formations, rather than volcanic activity, have had the most impact on the topography of both Antigua and Barbuda, which are both relatively low-lying islands. Boggy Peak, also known as Mt. Obama from 2008 to 2016, is the highest point on both Antigua and Barbuda. It is the remnant of a volcanic crater and rises a total of 402 meters. Boggy Peak is located in the southwest of Antigua (1,319 feet).

Both of these islands have very irregularly shaped coastlines that are dotted with beaches, lagoons, and natural harbors. There are reefs and shoals that surround the islands on all sides. Because of the low amount of rainfall, there are not many streams. On neither of these islands can sufficient quantities of fresh groundwater be found.

Redonda is a small, uninhabited island located about 40 kilometers (25 miles) to the south-west of Antigua. Redonda is a rocky island.

Cities and villages
The most populous cities in Antigua and Barbuda are mostly on Antigua, being Saint John's, All Saints, Piggotts, and Liberta. The most populous city on Barbuda is Codrington. It is estimated that 25% of the population lives in an urban area, which is much lower than the international average of 55%.

Islands
Antigua and Barbuda consists mostly of its two namesake islands, Antigua, and Barbuda. Other than that, Antigua and Barbuda's biggest islands are Guiana Island and Long Island off the coast of Antigua, and Redonda island, which is far from both of the main islands.

Climate
Rainfall averages 990 mm per year, with the amount varying widely from season to season. In general the wettest period is between September and November. The islands generally experience low humidity and recurrent droughts. Temperatures average 27 °C, with a range from 23 °C to 29 °C in the winter to from 25 °C to 30 °C in the summer and autumn. The coolest period is between December and February.

Hurricanes strike on an average of once a year, including the powerful Category 5 Hurricane Irma, on 6 September 2017, which damaged 95% of the structures on Barbuda. Some 1,800 people were evacuated to Antigua.

Officials quoted by Time indicated that over $100 million would be required to rebuild homes and infrastructure. Philmore Mullin, Director of Barbuda's National Office of Disaster Services, said that "all critical infrastructure and utilities are non-existent – food supply, medicine, shelter, electricity, water, communications, waste management". He summarised the situation as follows: "Public utilities need to be rebuilt in their entirety... It is optimistic to think anything can be rebuilt in six months ... In my 25 years in disaster management, I have never seen something like this."

Ethnic groups
Antigua has a population of, mostly made up of people of West African, British, and Portuguese descent. The ethnic distribution consists of 91% Black, 4.4% mixed race, 1.7% White, and 2.9% other (primarily East Indian). Most Whites are of British descent. Christian Levantine Arabs and a small number of East Asians and Sephardic Jews make up the remainder of the population.

An increasingly large percentage of the population lives abroad, most notably in the United Kingdom (Antiguan Britons), the United States and Canada. A minority of Antiguan residents are immigrants from other countries, particularly from Dominica, Guyana and Jamaica, and, increasingly, from the Dominican Republic, St. Vincent and the Grenadines and Nigeria. An estimated 4,500 American citizens also make their home in Antigua and Barbuda, making their numbers one of the largest American populations in the English-speaking Eastern Caribbean. 68.47% of the population was born in Antigua and Barbuda.

Languages
The language most commonly used in business is English. There is a noticeable distinction between the Antiguan accent and the Barbudan one.

When compared to Antiguan Creole, Standard English was the language of choice in the years leading up to Antigua and Barbuda's attainment of their independence. The Antiguan Creole language is looked down upon by the upper and middle classes in general. The Antiguan Creole language is discouraged from use in the educational system, and instruction is carried out in Standard (British) English instead.

A significant number of the words that are used in the Antiguan dialect are derived from both the British and African languages. This is readily apparent in phrases such as "Innit?" which literally translates to "Isn't it?" Many common island proverbs can be traced back to Africa, such as the pidgin language.

Approximately 10,000 people are able to speak in Spanish.

Religion
A majority (77%) of Antiguans are Christians, with the Anglicans (17.6%) being the largest single denomination. Other Christian denominations present are Seventh-day Adventist Church (12.4%), Pentecostalism (12.2%), Moravian Church (8.3%), Roman Catholics (8.2%), Methodist Church (5.6%), Wesleyan Holiness Church (4.5%), Church of God (4.1%), Baptists (3.6%), Mormonism (<1.0%), as well as Jehovah's Witnesses.

Government
Antigua and Barbuda is a unitary parliamentary democracy under a constitutional monarchy. The current Constitution of Antigua and Barbuda was adopted upon independence on 1 November 1981. This replaced the pre-independence constitution of the Associated State of Antigua, which did not thoroughly define the relationship between the two islands. The island of Barbuda maintains much autonomy, while the island of Antigua is directly governed by the national government.

The executive branch has two primary leaders. The Governor-General, currently Rodney Williams, exercises the functions of the Monarch of Antigua and Barbuda, in whom executive power is vested in. The Governor-General serves at the pleasure of the Monarch, and usually serves a similar term to that of the Prime Minister. The Prime Minister, currently, is the head of government, and is appointed by the Governor-General. The Prime Minister must be a member of the House of Representatives, and must be the member of the House of Representatives who is most likely to command the support of the majority of members. The Governor-General has the ability to dissolve Parliament on the advice of the Prime Minister, or when the majority of the members of the House of Representatives pass a motion of no confidence, and the Prime Minister does not within seven days resign or advise the Governor-General to dissolve Parliament.

The legislative power of Antigua and Barbuda is vested in Parliament, which is composed of the Monarch, the Senate, and the House of Representatives. The Senate is composed of seventeen members, who are appointed by the Governor-General. Ten of the members are appointed on the advice of the Prime Minister, these members being known as government senators. An eleventh government senator is also appointed on the advice of the Prime Minister, who must be an inhabitant of Barbuda. Four of the members are appointed on the advice of the Leader of the Opposition, these senators being known as opposition senators. One of the members is appointed on the advice of the Barbuda Council, and an independent senator is appointed under the discretion of the Governor-General himself. The House of Representatives is currently composed of seventeen elected members, as well as the Speaker of the House, who is elected by the members of the House itself. The Attorney General, while currently an elected member of Parliament, Steadroy Benjamin, may also be appointed to the House of Representatives as an ex officio member. The Attorney-General also attends sittings of the Senate. Any bill except money bills may be introduced in either chamber: money bills may only be introduced in the House. Parliament may not amend the Barbuda Local Government Act without the consent of the Barbuda Council.

The judiciary of Antigua and Barbuda is composed of the magistrates' courts, the Supreme Court including the High Court and the Court of Appeal, and finally the Judicial Committee of the Privy Council, the country's final court. Antiguan and Barbudan voters rejected a proposal to make the Caribbean Court of Justice the final court in 2018. Antigua and Barbuda is composed of three magistrates' courts districts, and is part of the Eastern Caribbean Supreme Court system. The acting chief justice of the Supreme Court is Mario Michel, serving since 5 May 2024.

Since the 1990s, the two major parties in Antigua have been the centre-right (formerly left-wing) Antigua and Barbuda Labour Party, and the left-wing social democratic United Progressive Party. The Labour Party and its predecessors have traditionally been the dominant party on the national level since the 1946 general elections, with brief pauses during the Progressive Labour Movement government (predecessor of the UPP) from 1971 to 1976, and the United Progressive Party government from 2004 until 2014. On Barbuda, dominant party is traditionally the Barbuda People's Movement, being the only political grouping in the Barbuda Council since 2023.

Administrative divisions
Antigua and Barbuda is composed of six parishes and two dependencies. Saint John is the most populous parish, home to well over half of Antigua and Barbuda's population. During colonial times, the parishes were governed by parish vestries, however, the parishes now lack any sort of government. Since the 2023 general elections, various proposals have been made to establish parish councils, however, as of June 2024, none have been established. The dependency of Redonda is part of the parish of Saint John under the Redonda Annexation Act, in Magistrates' District "A". Local government in Antigua and Barbuda is completely inactive, except for the Barbuda Council which is enshrined in the Constitution. Antigua historically had a system of village councils in the 1940s (although the legislation was never repealed), however, the Gaston Browne administration has expressed opposition to all forms of local governance. St. John's also historically had a city council during the late 1800s and early 1900s, however the St. John's Development Corporation has since consumed most of its functions.

Foreign relations
The Minister of Foreign Affairs, Agriculture, Trade & Barbuda Affairs is responsible for overseeing the foreign relations of Antigua and Barbuda. The current minister is Paul Chet Greene. Antigua and Barbuda is a founding member of the Organisation of Eastern Caribbean States, as well as a member of the United Nations, the Caribbean Community, the Alliance of Small Island States, and the World Trade Organization.

Antigua and Barbuda's foreign policy has been described by Gaston Browne as a system of "we are friends of all; enemies of none". Antigua and Barbuda has also rejected the notion that it is in any country's "backyard". Antigua and Barbuda usually maintains close relations with other Small Island Developing States, and has hosted various summits on that subject. The United Nations has also praised Antigua and Barbuda for its "United Nations-based multilateralism" efforts. Antigua and Barbuda also has close relations with many Caribbean countries and territories, especially Montserrat, which Antigua and Barbuda accepted 3,000 refugees from in 1997 after the Soufrière Hills eruption. Many policies adopted by the Antiguan and Barbudan government have also often had an impact on Montserrat, due to Antigua and Barbuda hosting the only air and transportation links into the territory.

Defence and national security
The Minister of Finance, Corporate Governance & Public Private Partnerships is responsible for the Antigua and Barbuda Defence Force, the country's military. The Minister of Legal Affairs, Public Safety, Immigration and Labour is responsible for the national security of Antigua and Barbuda. The Defence Force consists of the Regiment (army), the Air Wing, the Coast Guard, and the Service and Support Battalion. The Defence Force is led by the Chief of Defence Staff, who is subject to the orders of the Governor-General. The Defence Force is headquartered at Camp Blizzard.

The National Security Council is responsible for the coordination of Antigua and Barbuda's national security. The National Security Adviser is a member of the council and is responsible for the gathering of intelligence and information on national security matters.

The Royal Police Force of Antigua and Barbuda is the national police force of Antigua and Barbuda. The Special Service Unit is Antigua and Barbuda's police tactical unit. The Police Force is composed of four lettered regional divisions, and subordinated service districts.

Human rights
As of July 2022, Same-sex sexual activity is legal in Antigua and Barbuda.

Economy
Tourism dominates the economy, accounting for more than half of the gross domestic product (GDP). As a destination for the most affluent travelers, Antigua is well known for its extensive collection of five-star resorts. However, weaker tourist activity in lower and middle market segments since the beginning of the year 2000 has slowed the economy and put the government into a tight fiscal corner. Antigua and Barbuda has enacted policies to attract high-net-worth citizens and residents, such as enacting a 0% personal income tax rate in 2019.

The provision of investment banking and financial services also constitutes a significant portion of the economy. Major international financial institutions such as the Royal Bank of Canada (RBC) and Scotiabank both maintain offices in Antigua. PriceWaterhouseCoopers, Pannell Kerr Forster, and KPMG are some of the other companies in the financial services industry that have offices in Antigua. The United States Securities and Exchange Commission has leveled allegations against the Antigua-based Stanford International Bank, which is owned by the Texas billionaire Allen Stanford, of orchestrating a massive fraud that may have resulted in the theft of approximately $8 billion from investors.

The nation, which consists of two islands, directs the majority of its agricultural production toward the markets that are found within the nation. This is done despite the fact that the nation has a limited water supply and a shortage of laborers as a result of the higher wages offered in the tourism and construction industries.

Manufacturing comprises 2% of GDP and is made up of enclave-type assembly for export, the major products being bedding, handicrafts, and electronic components. Prospects for economic growth in the medium term will continue to depend on income growth in the industrialised world, especially in the United States, from which about one-third to one-half of all tourists come.

Access to biocapacity is lower than world average. In 2016, Antigua and Barbuda had 0.8 global hectares of biocapacity per person within its territory, much less than the world average of 1.6 global hectares per person. In 2016, Antigua and Barbuda used 4.3 global hectares of biocapacity per person – their ecological footprint of consumption. This means they use more biocapacity than Antigua and Barbuda contains. As a result, Antigua and Barbuda are running a biocapacity deficit.

The Citizenship by Investment Unit (CIU) is the government authority responsible for processing all applications for Agent's Licenses as well as all applications for Citizenship by Investment made by applicants and their family members. This unit was established by the Prime Minister and is known as the Citizenship by Investment Unit.

Culture
The music of Antigua and Barbuda has a primarily African character, with minimal influence from European music. The first known records of music in Antigua and Barbuda dates back to Christopher Columbus' discovery of the island nation in 1493, when it was still home to Arawak and Carib people. Still, very little research has been done on early music from the islands. African labourers are documented in history to have danced outside in the 1780s to the toombah (later tum tum), a drum adorned with tin and shell jingles, and the banjar (later bangoe, maybe related to the banjo). Antigua's indigenous music, known as Benna, came into being after slavery was abolished. Benna uses a call-and-response format, and its audience is typically interested in obscene gossip and rumours. Benna was widely utilised as a popular communication tool by the beginning of the 20th century, disseminating information around the island.

The art of Antigua and Barbuda began with the Arawak people. Their artwork included pictographs and petroglyphs. These geometric shapes, animals, and plant artworks are said to have been used for ceremonial or religious purposes. Painting, sculpture, and ceramics were among the artistic traditions that European settlers brought to Antigua and Barbuda. Local painters used European art forms to produce Antiguan and Barbudan art in their own unique styles. Social issues, nature, and Caribbean identity were the subjects of this artwork. Traditional crafts from Antigua and Barbuda include scrimshaw, pottery, sculptures, ethnic dolls, and photography.

Every year, on the island of Antigua, people celebrate their freedom from slavery with the Antigua Carnival. Over thirteen days, there are brightly coloured costumes, talent events, beauty pageants and music. The celebration runs from late July to Carnival Tuesday, the first Tuesday in August. On the island, Carnival Tuesday and Monday are both observed as public holidays. In an effort to boost travel to Antigua and Barbuda, the Old Time Christmas Festival was replaced in 1957 by the Antiguan Carnival. Another annual festival held in Antigua is Antigua Sailing Week. Sailing Week is a week-long yacht regatta held in the waters of English Harbour. Sailing Week was founded in 1967 and is known for being one of the top regattas in the world. The main festival held in Barbuda is Caribana. Caribana takes place every year during Whit Monday weekend and features various pageants, calypso competitions, and weekend beach parties.

Antigua and Barbuda has eleven public holidays. On the advice of the Cabinet, the Governor-General may also proclaim other holidays. Historically, about three weeks before Christmas Day, carol singers would roam the various villages, carrying carol trees and lanterns. "John Bulls" are replicas of "masked African witch doctors", that often dominated the country's Christmas festivities. Jazz bands were also common sights, dressed in red and green clown costumes.

Cuisine
Fungee (pronounced "foon-jee") and pepperpot are the national dishes. Fungee is a cornmeal-based dish that resembles Italian polenta. Other regional cuisines include saltfish, lobster (from Barbuda), ducana, and seasoned rice. Additionally, there are regional confections such peanut brittle, sugar cake, fudge, and raspberry and tamarind stew. The Antigua black pineapple is prized for its juicy, sweet flesh, which is claimed to taste different from other pineapple kinds. It is a well-liked fruit in the area and is included into many regional specialties and sweets. It is said to be the sweetest variety of pineapple.

An important part of the Antiguan and Barbudan breakfast is Antigua Sunday bread. It is sold in many bakeries on both islands, and instead of being made with butter, it is made with lard. There are often decorative twists on the crust of the bread. Antiguan raisin buns, often called "bun and cheese", is another traditional bread, which is sweet and most popular during Easter. It is sometimes made with spices such as nutmeg.

Sport
Cricket is the most popular sport within the islands. With Sir Isaac Vivian Alexander Richards who represented the West Indies cricket team between 1974 and 1991, Antigua had one of the world's most famous batsmen ever. The Antigua and Barbuda national cricket team represented the country at the 1998 Commonwealth Games, but Antiguan cricketers otherwise play for the Leeward Islands cricket team in domestic matches and the West Indies cricket team internationally. Teams from the various villages and parishes compete in the Parish League.

Association football is the second most popular sport in the country, with the Antigua and Barbuda national football team being founded in 1928.

title: Antigua and Barbuda


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Azincourt

Azincourt is a commune in the Pas-de-Calais department in northern France. It is situated 12 mi north-west of Saint-Pol-sur-Ternoise on the D71 road between Hesdin and Fruges. It is often erroneously known as Agincourt in English, as the village had never been called as such throughout its history, however, there is a village that is named "Agincourt", located in the Meurthe-et-Moselle department in Eastern France.

The Late Medieval Battle of Azincourt between the English and the French took place in the commune in 1415.

Toponym
The name is attested as Aisincurt in 1175, derived from a Germanic masculine name Aizo, Aizino and the early Northern French word curt (which meant a farm with a courtyard; derived from the Late Latin cortem). The name has no etymological link with Agincourt, Meurthe-et-Moselle (attested as Egincourt 875), which is derived separately from another Germanic male name *Ingin-.

History
Azincourt is known for being near the site of the battle fought on 25 October 1415 in which the army led by King Henry V of England defeated the forces led by Charles d'Albret on behalf of Charles VI of France, which has gone down in history as the Battle of Agincourt. According to M. Forrest, the French knights were so encumbered by their armour that they were exhausted even before the start of the battle.

After he became king in 1509, Henry VIII is purported to have commissioned an English translation of a Life of Henry V so that he could emulate him, on the grounds that he thought that launching a campaign against France would help him to impose himself on the European stage. In 1513, Henry VIII crossed the English Channel, stopping by at Azincourt.

The battle, as was the tradition, was named after a nearby castle called Azincourt. The castle has since disappeared and the settlement now known as Azincourt adopted the name in the seventeenth century.

John Cassell wrote in 1857 that "the village of Azincourt itself is now a group of dirty farmhouses and wretched cottages, but where the hottest of the battle raged, between that village and the commune of Tramecourt, there still remains a wood precisely corresponding with the one in which Henry placed his ambush; and there are yet existing the foundations of the castle of Azincourt, from which the king named the field."

Sights
The original battlefield museum in the village featured model knights made out of Action Man figures. This has now been replaced by the Centre historique médiéval d'Azincourt (CHM)—a more professional museum, conference centre and exhibition space incorporating laser, video, slide shows, audio commentaries, and some interactive elements. The museum building is shaped like a longbow similar to those used at the battle by archers under King Henry.

Since 2004 a large medieval festival organised by the local community, the CHM, The Azincourt Alliance, and various other UK societies commemorating the battle, local history and medieval life, arts and crafts has been held in the village. Prior to this date the festival was held in October, but due to the inclement weather and local heavy clay soil (like the battle) making the festival difficult, it was moved to the last Sunday in July.

International relations
Azincourt is twinned with Middleham, United Kingdom.

title: Azincourt