text:
Skull

The skull is a bone protective cavity for the brain. The skull is composed of three types of bone: cranial bones, facial bones, and ear ossicles. Two parts are more prominent: the cranium (: craniums or crania) and the mandible. In humans, these two parts are the neurocranium (braincase) and the viscerocranium (facial skeleton) that includes the mandible as its largest bone. The skull forms the anterior-most portion of the skeleton and is a product of cephalisation—housing the brain, and several sensory structures such as the eyes, ears, nose, and mouth. In humans, these sensory structures are part of the facial skeleton.

Functions of the skull include protection of the brain, fixing the distance between the eyes to allow stereoscopic vision, and fixing the position of the ears to enable sound localisation of the direction and distance of sounds. In some animals, such as horned ungulates (mammals with hooves), the skull also has a defensive function by providing the mount (on the frontal bone) for the horns.

The English word skull is probably derived from Old Norse skulle, while the Latin word cranium comes from the Greek root κρανίον (kranion). The human skull fully develops two years after birth. The junctions of the skull bones are joined by structures called sutures.

The skull is made up of a number of fused flat bones, and contains many foramina, fossae, processes, and several cavities or sinuses. In zoology, there are openings in the skull called fenestrae.

Humans
The human skull is the bone structure that forms the head in the human skeleton. It supports the structures of the face and forms a cavity for the brain. Like the skulls of other vertebrates, it protects the brain from injury.

The skull consists of three parts, of different embryological origin—the neurocranium, the sutures, and the facial skeleton. The neurocranium (or braincase) forms the protective cranial cavity that surrounds and houses the brain and brainstem. The upper areas of the cranial bones form the calvaria (skullcap). The membranous viscerocranium includes the mandible.

The sutures are fairly rigid joints between bones of the neurocranium.

The facial skeleton is formed by the bones supporting the face.

Bones
Except for the mandible, all of the bones of the skull are joined by sutures—synarthrodial (immovable) joints formed by bony ossification, with Sharpey's fibres permitting some flexibility. Sometimes there can be extra bone pieces within the suture known as Wormian bones or sutural bones. Most commonly these are found in the course of the lambdoid suture.

The human skull is generally considered to consist of 22 bones—eight cranial bones and fourteen facial skeleton bones. In the neurocranium these are the occipital bone, two temporal bones, two parietal bones, the sphenoid, ethmoid and frontal bones.

The bones of the facial skeleton (14) are the vomer, two inferior nasal conchae, two nasal bones, two maxilla, the mandible, two palatine bones, two zygomatic bones, and two lacrimal bones. Some sources count a paired bone as one, or the maxilla as having two bones (as its parts); some sources include the hyoid bone or the three ossicles of the middle ear, the malleus, incus, and stapes, but the overall general consensus of the number of bones in the human skull is the stated twenty-two.

Some of these bones—the occipital, parietal, frontal, in the neurocranium, and the nasal, lacrimal, and vomer, in the facial skeleton are flat bones.

Cavities and foramina
The skull also contains sinuses, air-filled cavities known as paranasal sinuses, and numerous foramina. The sinuses are lined with respiratory epithelium. Their known functions are the lessening of the weight of the skull, the aiding of resonance to the voice and the warming and moistening of the air drawn into the nasal cavity.

The foramina are openings in the skull. The largest of these is the foramen magnum, of the occipital bone, that allows the passage of the spinal cord as well as nerves and blood vessels.

Processes
The many processes of the skull include the mastoid process and the zygomatic processes.

Bones
The jugal is a skull bone found in most reptiles, amphibians, and birds. In mammals, the jugal is often called the zygomatic bone or malar bone.

The prefrontal bone is a bone separating the lacrimal and frontal bones in many tetrapod skulls.

Fish

The skull of fishes is formed from a series of only loosely connected bones. Lampreys and sharks only possess a cartilaginous endocranium, with both the upper and lower jaws being separate elements. Bony fishes have additional dermal bone, forming a more or less coherent skull roof in lungfish and holost fish. The lower jaw defines a chin.

The simpler structure is found in jawless fish, in which the cranium is normally represented by a trough-like basket of cartilaginous elements only partially enclosing the brain, and associated with the capsules for the inner ears and the single nostril. Distinctively, these fish have no jaws.

Cartilaginous fish, such as sharks and rays, have also simple, and presumably primitive, skull structures. The cranium is a single structure forming a case around the brain, enclosing the lower surface and the sides, but always at least partially open at the top as a large fontanelle. The most anterior part of the cranium includes a forward plate of cartilage, the rostrum, and capsules to enclose the olfactory organs. Behind these are the orbits, and then an additional pair of capsules enclosing the structure of the inner ear. Finally, the skull tapers towards the rear, where the foramen magnum lies immediately above a single condyle, articulating with the first vertebra. There are, in addition, at various points throughout the cranium, smaller foramina for the cranial nerves. The jaws consist of separate hoops of cartilage, almost always distinct from the cranium proper. In ray-finned fish, there has also been considerable modification from the primitive pattern. The roof of the skull is generally well formed, and although the exact relationship of its bones to those of tetrapods is unclear, they are usually given similar names for convenience. Other elements of the skull, however, may be reduced; there is little cheek region behind the enlarged orbits, and little, if any bone in between them. The upper jaw is often formed largely from the premaxilla, with the maxilla itself located further back, and an additional bone, the symplectic, linking the jaw to the rest of the cranium.

Although the skulls of fossil lobe-finned fish resemble those of the early tetrapods, the same cannot be said of those of the living lungfishes. The skull roof is not fully formed, and consists of multiple, somewhat irregularly shaped bones with no direct relationship to those of tetrapods. The upper jaw is formed from the pterygoids and vomers alone, all of which bear teeth. Much of the skull is formed from cartilage, and its overall structure is reduced.

Tetrapods
The skulls of the earliest tetrapods closely resembled those of their ancestors amongst the lobe-finned fishes. The skull roof is formed of a series of plate-like bones, including the maxilla, frontals, parietals, and lacrimals, among others. It is overlaying the endocranium, corresponding to the cartilaginous skull in sharks and rays. The various separate bones that compose the temporal bone of humans are also part of the skull roof series. A further plate composed of four pairs of bones forms the roof of the mouth; these include the vomer and palatine bones. The base of the cranium is formed from a ring of bones surrounding the foramen magnum and a median bone lying further forward; these are homologous with the occipital bone and parts of the sphenoid in mammals. Finally, the lower jaw is composed of multiple bones, only the most anterior of which (the dentary) is homologous with the mammalian mandible.

In living tetrapods, a great many of the original bones have either disappeared or fused into one another in various arrangements.

Birds
Birds have a diapsid skull, as in reptiles, with a prelacrimal fossa (present in some reptiles). The skull has a single occipital condyle. The skull consists of five major bones: the frontal (top of head), parietal (back of head), premaxillary and nasal (top beak), and the mandible (bottom beak). The skull of a normal bird usually weighs about 1% of the bird's total bodyweight. The eye occupies a considerable amount of the skull and is surrounded by a sclerotic eye-ring, a ring of tiny bones. This characteristic is also seen in reptiles.

Amphibians
Living amphibians typically have greatly reduced skulls, with many of the bones either absent or wholly or partly replaced by cartilage. In mammals and birds, in particular, modifications of the skull occurred to allow for the expansion of the brain. The fusion between the various bones is especially notable in birds, in which the individual structures may be difficult to identify.

Development
The skull is a complex structure; its bones are formed both by intramembranous and endochondral ossification. The skull roof bones, comprising the bones of the facial skeleton and the sides and roof of the neurocranium, are dermal bones formed by intramembranous ossification, though the temporal bones are formed by endochondral ossification. The endocranium, the bones supporting the brain (the occipital, sphenoid, and ethmoid) are largely formed by endochondral ossification. Thus frontal and parietal bones are purely membranous. The geometry of the skull base and its fossae, the anterior, middle and posterior cranial fossae changes rapidly. The anterior cranial fossa changes especially during the first trimester of pregnancy and skull defects can often develop during this time.

At birth, the human skull is made up of 44 separate bony elements. During development, many of these bony elements gradually fuse together into solid bone (for example, the frontal bone). The bones of the roof of the skull are initially separated by regions of dense connective tissue called fontanelles. There are six fontanelles: one anterior (or frontal), one posterior (or occipital), two sphenoid (or anterolateral), and two mastoid (or posterolateral). At birth, these regions are fibrous and moveable, necessary for birth and later growth. This growth can put a large amount of tension on the "obstetrical hinge", which is where the squamous and lateral parts of the occipital bone meet. A possible complication of this tension is rupture of the great cerebral vein. As growth and ossification progress, the connective tissue of the fontanelles is invaded and replaced by bone creating sutures. The five sutures are the two squamous sutures, one coronal, one lambdoid, and one sagittal suture. The posterior fontanelle usually closes by eight weeks, but the anterior fontanel can remain open up to eighteen months. The anterior fontanelle is located at the junction of the frontal and parietal bones; it is a "soft spot" on a baby's forehead. Careful observation will show that you can count a baby's heart rate by observing the pulse pulsing softly through the anterior fontanelle.

The skull in the neonate is large in proportion to other parts of the body. The facial skeleton is one seventh of the size of the calvaria. (In the adult it is half the size). The base of the skull is short and narrow, though the inner ear is almost adult size.

Clinical significance
Craniosynostosis is a condition in which one or more of the fibrous sutures in an infant skull prematurely fuses, and changes the growth pattern of the skull. Because the skull cannot expand perpendicular to the fused suture, it grows more in the parallel direction. Sometimes the resulting growth pattern provides the necessary space for the growing brain, but results in an abnormal head shape and abnormal facial features. In cases in which the compensation does not effectively provide enough space for the growing brain, craniosynostosis results in increased intracranial pressure leading possibly to visual impairment, sleeping impairment, eating difficulties, or an impairment of mental development.

A copper beaten skull is a phenomenon wherein intense intracranial pressure disfigures the internal surface of the skull. The name comes from the fact that the inner skull has the appearance of having been beaten with a ball-peen hammer, such as is often used by coppersmiths. The condition is most common in children.

Injuries and treatment
Injuries to the brain can be life-threatening. Normally the skull protects the brain from damage through its high resistance to deformation; the skull is one of the least deformable structures found in nature, needing the force of about 1 ton to reduce its diameter by 1 cm. In some cases of head injury, however, there can be raised intracranial pressure through mechanisms such as a subdural haematoma. In these cases, the raised intracranial pressure can cause herniation of the brain out of the foramen magnum ("coning") because there is no space for the brain to expand; this can result in significant brain damage or death unless an urgent operation is performed to relieve the pressure. This is why patients with concussion must be watched extremely carefully. Repeated concussions can activate the structure of skull bones as the brain's protective covering.

Dating back to Neolithic times, a skull operation called trepanning was sometimes performed. This involved drilling a burr hole in the cranium. Examination of skulls from this period reveals that the patients sometimes survived for many years afterward. It seems likely that trepanning was also performed purely for ritualistic or religious reasons. Nowadays this procedure is still used but is normally called a craniectomy.

In March 2013, for the first time in the U.S., researchers replaced a large percentage of a patient's skull with a precision, 3D-printed polymer implant. About 9 months later, the first complete cranium replacement with a 3D-printed plastic insert was performed on a Dutch woman. She had been suffering from hyperostosis, which increased the thickness of her skull and compressed her brain.

A study conducted in 2018 by the researchers of Harvard Medical School in Boston, funded by National Institutes of Health (NIH), suggested that instead of travelling via blood, there are "tiny channels" in the skull through which the immune cells combined with the bone marrow reach the areas of inflammation after an injury to the brain tissues.

Transgender procedures
Surgical alteration of sexually dimorphic skull features may be carried out as a part of facial feminization surgery or facial masculinization surgery, these reconstructive surgical procedures that can alter sexually dimorphic facial features to bring them closer in shape and size to facial features of the desired sex. These procedures can be an important part of the treatment of transgender people for gender dysphoria.

Society and culture

Artificial cranial deformation is a largely historical practice of some cultures. Cords and wooden boards would be used to apply pressure to an infant's skull and alter its shape, sometimes quite significantly. This procedure would begin just after birth and would be carried on for several years.

Osteology
Like the face, the skull and teeth can also indicate a person's life history and origin. Forensic scientists and archaeologists use quantitative and qualitative traits to estimate what the bearer of the skull looked like. When a significant amount of bones are found, such as at Spitalfields in the UK and Jōmon shell mounds in Japan, osteologists can use traits, such as the proportions of length, height and width, to know the relationships of the population of the study with other living or extinct populations.

The German physician Franz Joseph Gall in around 1800 formulated the theory of phrenology, which attempted to show that specific features of the skull are associated with certain personality traits or intellectual capabilities of its owner. His theory is now considered to be pseudoscientific.

Sexual dimorphism
In the mid-nineteenth century, anthropologists found it crucial to distinguish between male and female skulls. An anthropologist of the time, James McGrigor Allan, argued that the female brain was similar to that of an animal. This allowed anthropologists to declare that women were in fact more emotional and less rational than men. McGrigor then concluded that women's brains were more analogous to infants, thus deeming them inferior at the time. To further these claims of female inferiority and silence the feminists of the time, other anthropologists joined in on the studies of the female skull. These cranial measurements are the basis of what is known as craniology. These cranial measurements were also used to draw a connection between women and black people.

Research has shown that while in early life there is little difference between male and female skulls, in adulthood male skulls tend to be larger and more robust than female skulls, which are lighter and smaller, with a cranial capacity about 10 percent less than that of the male. However, later studies show that women's skulls are slightly thicker and thus men may be more susceptible to head injury than women. Other Sources:
 * name="Men May Be More Susceptible To Head Injury Than Women, Study Suggests">
 * However, other studies shows that men's skulls are slightly thicker in certain areas. As well as some studies showing that females are more susceptible to head injury (concussion) than males. Men's skulls have also been shown to maintain density with age, which may aid in preventing head injury, while women's skull density slightly decreases with age.
 * However, other studies shows that men's skulls are slightly thicker in certain areas. As well as some studies showing that females are more susceptible to head injury (concussion) than males. Men's skulls have also been shown to maintain density with age, which may aid in preventing head injury, while women's skull density slightly decreases with age.
 * However, other studies shows that men's skulls are slightly thicker in certain areas. As well as some studies showing that females are more susceptible to head injury (concussion) than males. Men's skulls have also been shown to maintain density with age, which may aid in preventing head injury, while women's skull density slightly decreases with age.
 * However, other studies shows that men's skulls are slightly thicker in certain areas. As well as some studies showing that females are more susceptible to head injury (concussion) than males. Men's skulls have also been shown to maintain density with age, which may aid in preventing head injury, while women's skull density slightly decreases with age.
 * However, other studies shows that men's skulls are slightly thicker in certain areas. As well as some studies showing that females are more susceptible to head injury (concussion) than males. Men's skulls have also been shown to maintain density with age, which may aid in preventing head injury, while women's skull density slightly decreases with age.
 * However, other studies shows that men's skulls are slightly thicker in certain areas. As well as some studies showing that females are more susceptible to head injury (concussion) than males. Men's skulls have also been shown to maintain density with age, which may aid in preventing head injury, while women's skull density slightly decreases with age.

Male skulls can all have more prominent supraorbital ridges, glabella, and temporal lines. Female skulls generally have rounder orbits and narrower jaws. Male skulls on average have larger, broader palates, squarer orbits, larger mastoid processes, larger sinuses, and larger occipital condyles than those of females. Male mandibles typically have squarer chins and thicker, rougher muscle attachments than female mandibles.

Craniometry
The cephalic index is the ratio of the width of the head, multiplied by 100 and divided by its length (front to back). The index is also used to categorize animals, especially dogs and cats. The width is usually measured just below the parietal eminence, and the length from the glabella to the occipital point.

Humans may be: The vertical cephalic index refers to the ratio between the height of the head multiplied by 100 and divided by the length of the head.
 * Dolichocephalic — long-headed
 * Mesaticephalic — medium-headed
 * Brachycephalic — short-headed

Humans may be:
 * Chamaecranic — low-skulled
 * Orthocranic — medium high-skulled
 * Hypsicranic — high-skulled

Terminology

 * Chondrocranium, a primitive cartilaginous skeletal structure
 * Endocranium
 * Epicranium
 * Pericranium, a membrane that lines the outer surface of the cranium

History
Trepanning, a practice in which a hole is created in the skull, has been described as the oldest surgical procedure for which there is archaeological evidence, found in the forms of cave paintings and human remains. At one burial site in France dated to 6500 BCE, 40 out of 120 prehistoric skulls found had trepanation holes.

title: Skull


text:
Well-ordering principle

In mathematics, the well-ordering principle states that every non-empty subset of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which $$x$$ precedes $$y$$ if and only if $$y$$ is either $$x$$ or the sum of $$x$$ and some positive integer (other orderings include the ordering $$2, 4, 6, ...$$; and $$1, 3, 5, ...$$).

The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem". On other occasions it is understood to be the proposition that the set of integers $$\{\ldots, -2, -1, 0, 1, 2, 3, \ldots \}$$ contains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element.

Properties
Depending on the framework in which the natural numbers are introduced, this (second-order) property of the set of natural numbers is either an axiom or a provable theorem. For example:
 * In Peano arithmetic, second-order arithmetic and related systems, and indeed in most (not necessarily formal) mathematical treatments of the well-ordering principle, the principle is derived from the principle of mathematical induction, which is itself taken as basic.
 * Considering the natural numbers as a subset of the real numbers, and assuming that we know already that the real numbers are complete (again, either as an axiom or a theorem about the real number system), i.e., every bounded (from below) set has an infimum, then also every set $$A$$ of natural numbers has an infimum, say $$a^*$$. We can now find an integer $$n^*$$ such that $$a^*$$ lies in the half-open interval $$(n^*-1,n^*]$$, and can then show that we must have $$a^* = n^*$$, and $$n^*$$ in $$A$$.
 * In axiomatic set theory, the natural numbers are defined as the smallest inductive set (i.e., set containing 0 and closed under the successor operation). One can (even without invoking the regularity axiom) show that the set of all natural numbers $$n$$ such that "$$\{0,\ldots,n\}$$ is well-ordered" is inductive, and must therefore contain all natural numbers; from this property one can conclude that the set of all natural numbers is also well-ordered.

In the second sense, this phrase is used when that proposition is relied on for the purpose of justifying proofs that take the following form: to prove that every natural number belongs to a specified set $$S$$, assume the contrary, which implies that the set of counterexamples is non-empty and thus contains a smallest counterexample. Then show that for any counterexample there is a still smaller counterexample, producing a contradiction. This mode of argument is the contrapositive of proof by complete induction. It is known light-heartedly as the "minimal criminal" method and is similar in its nature to Fermat's method of "infinite descent".

Garrett Birkhoff and Saunders Mac Lane wrote in A Survey of Modern Algebra that this property, like the least upper bound axiom for real numbers, is non-algebraic; i.e., it cannot be deduced from the algebraic properties of the integers (which form an ordered integral domain).

Example applications
The well-ordering principle can be used in the following proofs.

Prime factorization
Theorem: Every integer greater than one can be factored as a product of primes. This theorem constitutes part of the Prime Factorization Theorem.

Proof (by well-ordering principle). Let $$C$$ be the set of all integers greater than one that cannot be factored as a product of primes. We show that $$C$$ is empty.

Assume for the sake of contradiction that $$C$$ is not empty. Then, by the well-ordering principle, there is a least element $$n \in C$$; $$n$$ cannot be prime since a prime number itself is considered a length-one product of primes. By the definition of non-prime numbers, $$n$$ has factors $$a, b$$, where $$a, b$$ are integers greater than one and less than $$n$$. Since $$a, b < n$$, they are not in $$C$$ as $$n$$ is the smallest element of $$C$$. So, $$a, b$$ can be factored as products of primes, where $$a = p_1p_2...p_k$$ and $$b = q_1q_2...q_l$$, meaning that $$n = p_1p_2...p_k \cdot q_1q_2...q_l$$, a product of primes. This contradicts the assumption that $$n \in C$$, so the assumption that $$C$$ is nonempty must be false.

Integer summation
Theorem: $$1 + 2 + 3 + ... + n = \frac{n(n + 1)}{2}$$ for all positive integers $$n$$.

Proof. Suppose for the sake of contradiction that the above theorem is false. Then, there exists a non-empty set of positive integers $$C = \{n \in \mathbb N \mid 1 + 2 + 3 + ... + n \neq \frac{n(n + 1)}{2}\}$$. By the well-ordering principle, $$C$$ has a minimum element $$c$$ such that when $$n = c$$, the equation is false, but true for all positive integers less than $$c$$. The equation is true for $$n = 1$$, so $$c > 1$$; $$c - 1$$ is a positive integer less than $$c$$, so the equation holds for $$c - 1$$ as it is not in $$C$$. Therefore, $$\begin{align} 1 + 2 + 3 + ... + (c - 1) &= \frac{(c - 1)c}{2} \\ 1 + 2 + 3 + ... + (c - 1) + c &= \frac{(c - 1)c}{2} + c\\ &= \frac{c^2 - c}{2} + \frac{2c}{2}\\ &= \frac{c^2 + c}{2}\\ &= \frac{c(c + 1)}{2} \end{align}$$ which shows that the equation holds for $$c$$, a contradiction. So, the equation must hold for all positive integers.

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

title: Well-ordering principle


text:
Corollary

In mathematics and logic, a corollary is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another proposition; it might also be used more casually to refer to something which naturally or incidentally accompanies something else.

Overview
In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective. More formally, proposition B is a corollary of proposition A, if B can be readily deduced from A or is self-evident from its proof.

In many cases, a corollary corresponds to a special case of a larger theorem, which makes the theorem easier to use and apply, even though its importance is generally considered to be secondary to that of the theorem. In particular, B is unlikely to be termed a corollary if its mathematical consequences are as significant as those of A. A corollary might have a proof that explains its derivation, even though such a derivation might be considered rather self-evident in some occasions (e.g., the Pythagorean theorem as a corollary of law of cosines ).

Peirce's theory of deductive reasoning
Charles Sanders Peirce held that the most important division of kinds of deductive reasoning is that between corollarial and theorematic. He argued that while all deduction ultimately depends in one way or another on mental experimentation on schemata or diagrams, in corollarial deduction:

"it is only necessary to imagine any case in which the premises are true in order to perceive immediately that the conclusion holds in that case"

while in theorematic deduction:

"It is necessary to experiment in the imagination upon the image of the premise in order from the result of such experiment to make corollarial deductions to the truth of the conclusion."

Peirce also held that corollarial deduction matches Aristotle's conception of direct demonstration, which Aristotle regarded as the only thoroughly satisfactory demonstration, while theorematic deduction is:

 * 1) The kind more prized by mathematicians
 * 2) Peculiar to mathematics
 * 3) Involves in its course the introduction of a lemma or at least a definition uncontemplated in the thesis (the proposition that is to be proved), in remarkable cases that definition is of an abstraction that "ought to be supported by a proper postulate."

title: Corollary


text:
Talk:Well-ordering theorem

A brute-force proof
It has always amazed me to see how many well educated and intelligent people objected to the Well-Ordering Theorem, calling it a "paradoxical" consequence of the Axiom of Choice. But as soon as one accepts that every set has its cardinality, the "paradox" vanishes. Let's see.

A cardinality |X| of a set X is defined as the smallest ordinal that has the same number of elements as X has. Being an ordinal, it is well-ordered by the less-than relation <. (Some prefer to name the relation < the "is-an-element-of" relation $$\in$$ in this context.)

Just take any set X and pick any 1-1 function f from X onto its cardinality |X|. It must exist by the definition of two sets having the same number of elements. Now, define the well-ordering << on X by:

x << y iff f(x) < f(y)).

<< is a well-ordering of X because < is a well-ordering of |X|.

To make the story shorter, the existence of well-ordering of any set X is entailed by the fact that its cardinality |X| is a well-ordered set (by < or by $$\in$$, as some would prefer to say).

The "brute force" descriptor can be justified as follows. One can think of the above construction of a well-ordering of X this way.

Pick up an ordinal number b of cardinality > |X| (for instance, the cardinality of the set of all subsets of X) and construct a (transfinite) sequence S of distinct elements of X using consecutive elements of b as indices of S. Informally, beginning with X = X', just keep removing the elements of X' and appending them to the end of sequence S'  constructed already until X' becomes empty. Formally, if for some ordinal d < b, the sequence S'  has already been constructed for all ordinals c < d and X' is still non-empty then remove any element z from X' and extend S' on d by putting S' (d) = z. (In addition to transfinite induction, the Axiom of Choice comes into play here, although not in a way that might be bona fide characterized as "counterintuitive".)

Here is a transfinite program that does just that:

 * X' = X;
 * for (d in b)
 * if (X' == empty) break;
 * z = choose_one_of ( X' )
 * X' = X' \ {z};
 * S' [d] = z;
 * }
 * S = S' ;
 * S = S' ;

At some point, one must run out of elements of X' (and, therefore, the break instruction must be executed, eventually) because b has the cardinality > |X| $$\geq$$ |S' |, and S' , being duplicate-free as it is, establishes a 1-1 function from the set of indices of S'  onto (the set of elements of) S' . So, there are not enough elements in X to assign them, via S' , to all indices in b without duplication, and the for-loop in the above program will break. At that point, X' is empty and S'  has all the elements of X, thus it becomes the S that we have been constructing.

So, the sequence S is well defined, has no duplicates (so that it well-orders its elements), and the set of its elements is equal to X. Thus S well-orders X. Q.E.D.

The above is by no means "simple" or "complete" proof, whatever these two adjectives mean. (Actually, it would become fairly complicated if one bothered to provide all the details of it.) It's just that it leaves little or no room for bona fide intuitive objections, unless one objects to the existence of the cardinality of an uncountable set.

For instance, in light of the above argument the existence of a well-ordering of all reals is equally "paradoxical" as the existence of ordinal numbers of cardinality equal to (or larger than) the continuum.

Marek Suchenek (July 2, 2013)

Is this technically speaking a theorem (in that it follows from the axiom of choice); or is it a principle (it is equivalent to the axiom of choice)? Chas zzz brown 03:02 Jan 14, 2003 (UTC)

It is a theorem of ZFC, in that it is logically entailed by the axioms of ZFC. One could take it to be an axiom, in which case the proposition conventionally called the "axiom of choice" would be a theorem. But that is not conventionally done. Lots of mathematicians know and use the axiom of choice every day without ever in their lives thinking about well-orderings, so I think that way of doing things is appropriate. Michael Hardy 03:06 Jan 14, 2003 (UTC)

 * Placing it as part of the default ZFC makes sense to me (I'm pro-Choice :)). I suppose the same goes for Zorn's lemma. BTW, it is helpful when adding new mathematical articles if you could place a link on the List of mathematical topics, so that these articles can be easily tracked when changes are made. Cheers Chas zzz brown 03:15 Jan 14, 2003 (UTC)

Principle/Axiom/Theorem confusion
--- I have always learned that the Well-Ordering Principle is simply that the Naturals are well-ordered, and that the Well-Ordering Principle is equivalent to the Principle of Mathematical Induction. The Well-Ordering Axiom, on the other hand, states that any set can be well-ordered, and that is what this article is about. The Well-Ordering Principle certainly doesn't deserve to be called an axiom, and although the Well-Ordering Theorem can be proved from AC, it can also receive axiom status indifferently - it is independent of ZF.

My point is, it doesn't make sense to say that the Well-Ordering Theorem is "not to be confused with" the Well-Ordering Axiom, when as far as I know, everyone who says "Well-Ordering Axiom" is referring to the Well-Ordering Theorem, not to the Well-Ordering Principle, which is not equivalent to AC. I move that the Well-Ordering Axiom redirect be pointed here, and the phrasing in the parentheses be changed. GTBacchus 07:11, 21 Jun 2004 (UTC)

When I hear "Well ordering axiom/principle/theorem", I always think of the more general one first, and the case restricted to the natural numbers is just a special case. These could actually fit on the same page, with a section about the specialisation to the naturals. (Which obviously has special importance.)

-- Cale Gibbard 18:48, 05 Oct 2007 (UTC)

well-ordering and uncountable
This theorem seems to imply that there is a well-ordering on the reals, albeit not yet conceived-of. Why doesn't this imply that the reals (or any set subject to this theorem for that matter) could be put into 1-1 correspondence with $$\mathbb{N}$$ in accordance with that particular well-ordering, contradicting the fact that many sets such as the reals are uncountable? Btyner 04:22, 16 February 2006 (UTC)

 * Because, well, it doesn't. There's no way to construct such a 1-1 correspondence. You are probably thinking of a particular way to construct one, but without knowing which, we can't tell you why it doesn't work. --Mellum 07:35, 16 February 2006 (UTC)

 * The Well-order article gives the impression that a well-order is just a list such that a given element of the set will be found on the list eventually (?). The well-ordering theorem says such a list of the reals exists. It seems like the uncountability of the reals hinges on the fact that no one has yet conceived of such a list. Btyner 15:13, 16 February 2006 (UTC)

 * I don't see where it gives this impression. It says that you can make a list and will always know what to take next, but it doesn't say nor imply that this will eventually get every element, except in the finite case. That clearly doesn't even hold for countable sets: If I take 2, 4, 6, ... I always know what to take next and never run out of elements, but I won't hit all elements. --Mellum 20:14, 16 February 2006 (UTC)

 * Thank you; I think I understand now. By the theorem, there is an ordering on $$\mathbb{R}$$ such that all subsets thereof have a least element under that ordering. Therefore, $$\mathbb{R}$$ has a least element, call it $$x_1$$. Now, $$\mathbb{R}-\{x_1\}$$ is a subset of $$\mathbb{R}$$, so it too has a least element; call it $$x_2$$, and so on. You are saying that the set $$\{x_1,x_2,\ldots\}$$, being countable, leaves out most of $$\mathbb{R}$$. Btyner 03:31, 17 February 2006 (UTC)

this article is worthless without proof
also, cocks (but seriously guys. i came here looking for a proof.  can't you at least give me a linque?) oh yeah re: unsourced: if it had a proof, that would be, well, proof, and the article wouldn't be 'unsourced' because anyone (with the proper background/sufficient time) could understand the proof and know whether the article was correct or not. —Preceding unsigned comment added by 216.15.119.166 (talk) 06:54, 31 January 2008 (UTC)

How do like my sketch of proof? —Preceding unsigned comment added by Adrionwells (talk • contribs) 08:11, 7 November 2009 (UTC)

why is there so much about game theory in this article??
There should be more about set theory. Adrionwells (talk) 15:55, 11 November 2009 (UTC)adrionwells

 * Split into a separate article Zermelo's theorem (game theory), although I doubt there's enough there to survive. It does not belong in this article.  — Arthur Rubin  (talk) 17:10, 11 November 2009 (UTC)

"Axiom of choice & its Equivalents"
That looks so ugly in this article. Can someone edit it for the better? — Preceding unsigned comment added by 108.41.98.105 (talk) 21:46, 8 January 2023 (UTC)

The proof of Zorn lemma => well ordering principle is incorrect
Let me reproduce the proof first:

 * The well-ordering theorem follows from Zorn's lemma. Take the set $$\mathbf{A}$$ of all well-orderings of subsets of X: an element of $$\mathbf{A}$$ is an ordered pair (a,b) where a is a subset of X and b is a well-ordering of a. $$\mathbf{A}$$ can be partially ordered by continuation. That means, define E ≤ F if E is an initial segment of F and the ordering of the members of E is the same as their ordering in F. If $$\mathcal{E}$$ is a chain in $$\mathbf{A}$$, then the union of the sets in $$\mathcal{E}$$ can be ordered in a way that makes it a continuation of any set in $$\mathcal{E}$$; this ordering is a well-ordering, and therefore, an upper bound of $$\mathcal{E}$$ in $$\mathbf{A}$$ . We may therefore apply Zorn's Lemma to conclude that $$\mathbf{A}$$ has a maximal element, say (M,R). The set M must be equal to X, for if X has an element x not in M, then the set M∪{x} has a well-ordering that restricts to R on M, and for which x is larger than all elements of M. This well ordered set is a continuation of (M,R), contradicting its maximality, therefore M = X. Now R is a well-ordering of X.

I take issue with this part:

 * If $$\mathcal{E}$$ is a chain in $$\mathbf{A}$$, then the union of the sets in $$\mathcal{E}$$ can be ordered in a way that makes it a continuation of any set in $$\mathcal{E}$$; this ordering is a well-ordering, and therefore, an upper bound of $$\mathcal{E}$$ in $$\mathbf{A}$$

Contrary to the above, the union of $$\mathcal{E}$$ need not be well-ordered. Let $$X = \mathbb{Z}$$. Let $$A_n = \{-n, -(n-1), \ldots, 1, 0\}$$, and let $$E_n = (A_n, \leq|_{A_n})$$, where $$\leq|_{A_n}$$ is a standard ordering on integers restricted to $$A_n$$. Finally let $$\mathcal{E} = \{E_n: n \in \mathbb{N}\}$$. Then the union of chain $$\mathcal{E}$$ is clearly not well ordered. Therefore, I'm removing the erroneous proof. Dodek (talk) 22:45, 24 January 2020 (UTC)

$$ A_n$$ is not an initial segment of $$A_{n+1}$$, since $$-(n+1) \in A_{n+1}$$ but $$-(n+1) \notin A_n$$. Therefore, your $$\mathcal{E} $$ is not a chain. 65.92.110.77 (talk) 22:21, 4 December 2021 (UTC)

"Axiom of choice & its Equivalents"
That looks so ugly in this article. Can someone edit it for the better? — Preceding unsigned comment added by 108.41.98.105 (talk) 21:48, 8 January 2023 (UTC)

title: Talk:Well-ordering theorem


text:
Lucio Gutiérrez

Lucio Edwin Gutiérrez Borbúa (born 23 March 1957) served as 43rd President of Ecuador from 15 January 2003 to 20 April 2005.

Early life
Lucio Gutierrez, in full Lucio Edwin Gutiérrez Borbua, (born 23 March 1957, Quito, Ecuador), Ecuadoran army colonel and politician who served as president of Ecuador (2003–05). Gutiérrez was raised in Tena, an Amazon basin town. He was the son of a traveling salesman and attended primary and secondary school in Tena before transferring at age 15 to a military college in Quito. Gutiérrez graduated from the Army Polytechnic School as a civil engineer after having won honor's for academic and athletic prowess. He later studied in Brazil and the United States.

Gutiérrez rose steadily through the army ranks. In 1990–92 he served with the United Nations observer mission to Nicaragua. As a young man, he demonstrated little interest in politics, but during the 1990s he sympathized with fellow Ecuadorans as they became increasingly disenchanted with corruption and poverty. In 1997, as aide-de-camp to Pres. Abdalá Bucaram Ortiz, he refused an order to use force against a crowd outside the presidential palace. Bucaram fled the palace and was later removed from office by the National Congress. In 1999 Gutiérrez repeatedly questioned the government's conduct and pointedly refused to shake Pres. Jamil Mahuad Witt's hand during a public ceremony in December.

Military career
Gutierrez was born on 23 March 1957, in the city of Quito. Although holding degrees in civil engineering, administration and physical education, he chose a career with the Ecuadorian army, where he rose to the rank of colonel before his involvement in a coup against the government.

In January 2000, thousands of Ecuadorians of Andean Indian descent massed in the capital of Quito to protest corruption in the government and economic policy. Ordered to break up demonstrations, Gutierrez instead helped feed the protesters, allowed them to occupy the country's congress building, and joined Indian leader Antonio Vargas and a judge, Carlos Solorzano in "announcing a government of national salvation."

The coup did not last long and the old government returned to power, but with a new president to replace the discredited predecessor. Gutierrez and other supporters of the coup were imprisoned for six months and expelled from the army.

As with many other military men who have entered politics in Latin American countries, Gutiérrez's first experiences in a public forum date back to a period of political and social unrest. Indigenous and poor Ecuadorans staged massive protests when the Jamil Mahuad government decided to adopt the U.S. dollar as the national currency in 2000. Gutiérrez disobeyed the government's order to repress the protestors and instead, together with fellow officers and their troops, joined the movement to depose president Mahuad. Days later, the Ecuadoran Congress deposed the president and the vice-president took the presidential chair

Political rise
Gutiérrez entered the national spotlight during the 2000 Ecuadorian coup d'état that unseated President Jamil Mahuad for three hours and forced him to abandon office after demonstrations in Quito by thousands of Indigenous Ecuadorians protested the Mahuad government's support of neoliberal economic policies, particularly the proposed dollarization plans. Instead of ordering to disperse the protesters, then Col. Gutiérrez and the army stood aside and let them take over the national parliament.

Under pressure from the United States, and lacking support from the indigenous movement, the Junta was dissolved by General Carlos Mendoza and the Congress named then Vice President Gustavo Noboa as president of the country. The armed forces jailed Gutiérrez for six months, but he was discharged, and faced no criminal prosecution despite the fact he was a direct participant.

Prior to the 2000 coup, Gutiérrez was aide-de-camp to former presidents Abdalá Bucaram and Fabián Alarcón. He claimed that, during the demonstrations that unseated Bucaram in February 1997, he also disobeyed orders to protect Carondelet Palace, leaving Bucaram no other choice but to leave.

Presidency (2003–2005)
Gutiérrez ran for President in 2002 as the candidate of the January 21 Patriotic Society Party (PSP), named for the date of the 2000 protest, and the Pachakutik Movement, on a platform of fighting corruption and reversing neoliberal economic reforms. He defeated banana magnate and wealthiest man in Ecuador, Álvaro Noboa, in the second round with 55% of the popular vote, through a partnership with the leftist and indigenous movement parties, Democratic People's Movement (Movimiento Popular Democratico/MPD) and Pachacutik, respectively.

Gutiérrez alienated many of his supporters by supporting the Free Trade Area of the Americas (FTAA) and by keeping the status-quo on economic issues. After three months of government, Gutiérrez broke his alliance with leftist parties and reached an agreement with the Social Christian Party (Partido Social Cristiano/PSC), continuing the economic policies of its predecessors and increasing bonds with the United States. The government received increased frequent accusations of corruption and nepotism. After two years, Gutiérrez broke the agreement with the PSC, further weakening the government politically.

In November 2004 his former left-wing supporters joined with the conservative PSC in launching an effort to impeach him on various charges. Gutiérrez was accused of embezzlement by the PSC for using funds, resources and public property in favor of PSP candidates in the elections of 2004, and by the Democratic Left (Ecuador) (Izquierda Democrática/ID), MPD and Pachakutik for jeopardizing the security of the state for literally inciting the people to burn the courts. Impeachment proceedings collapsed when two legislators broke party lines, and a majority (51) votes could not be reached to continue the proceedings.

In December 2004, Gutiérrez alleged that the Supreme Court of Justice was biased in favor of the PSC. His political party, PSP, together with the Institutional Renewal Party of National Action (Partido Renovador Institucional de Acción Nacional/PRIAN) of Álvaro Noboa, the Ecuadorian Roldosist Party (Partido Roldosista Ecuatoriano/PRE) of Abdalá Bucaram, independents, and MPD, voted in the Congress for the reorganization of the Supreme Court of Justice via majority resolution. His adversaries sustained that the Constitution gives autonomy to the judicial branch and does not authorize Congress to interfere in the judiciary by removing or nominating judges. Judges were replaced by allies to PRE, PRIAN and PSP political parties with the clear intention of dropping criminal charges against former president Abdalá Bucaram, accused of several acts of corruption during his presidency which led to his exile in Panama from 1997 until April 2005.

Crisis
On 15 April 2005, amid a growing political crisis and protests in the city of Quito against the Government, President Gutiérrez declared a state of emergency in Quito and revoked the newly appointed Supreme Court of Justice. This was a controversial move that provoked conflicting reactions and was seen by analysts as a dictatorial act. The state of emergency was lifted on 16 April, as the State of Emergency was disobeyed by citizens and General Aguas of the army, who refused to enforce it, and Ecuador's Congress was expected to hold a session in order to decide whether to ratify the Supreme Court's dismissal.

On 20 April 2005, following a week of massive demonstrations, the Congress of Ecuador (meeting in a special session in a private building, CIESPAL, with opposition delegates only), on the grounds that Gutiérrez had abandoned his constitutional duties, voted 60–2 (38 members, including the great majority of PRE/PRIAN/PSP deputies, did not vote) to remove Gutiérrez from office and appointed Vice President Alfedo Palacio González to serve as President. At the same time, the Ecuadorian Joint Armed Forces Command (Comando Conjunto de las Fuerzas Armadas) publicly expressed that they were withdrawing their support for Gutiérrez, who had no option but to leave the Presidential Palace on a helicopter. He sought political asylum in the house of the Brazilian Ambassador in the north of Quito, after his attempt to leave the city aboard a plane at Quito International Airport was thwarted by hundreds of angry protesters that breached airport security and blocked the airstrip.

Approval rating
Gutierrez began his presidency with an approval rating of 64%. Gutierrez left office with a 32% approval rating, according to a CEDATOS survey.

Post-Presidency
Brazil offered Gutiérrez asylum and arranged air transport out of Ecuador for the former president. He arrived via Rio Branco on 24 April 2005. He renounced his asylum, then went to Peru and the United States. In September he was reported to be seeking political asylum in Colombia. This was offered on 4 October only to be refused by Gutiérrez on 13 October. Then on 15 October he voluntarily returned to Ecuador vowing to "use all legal and constitutional means to retake power." He was arrested at the Eloy Alfaro International Airport in Manta, after arriving in a chartered SARPA plane with his brother, Gilmar Gutiérrez, his companion-in-arms Fausto Cobo, and some collaborators. He was taken to a prison in Quito and locked in a maximum security cell on charges of attempting to subvert Ecuador's internal security by repeatedly proclaiming to the international media that he continued to be the legitimate President of the Republic of Ecuador.

On 3 March 2006, a judge in Ecuador dismissed the charges against Gutiérrez. A very notorious political agreement ensued between Gutiérrez and the PSC. Gutiérrez and the deputies of PSP sided with the PSC in order to gain a majority for controlling the country's Constitutional Court. Upon his release, Gutiérrez thanked the Ecuadorian people for their support and vowed that he would participate and win the presidential election in October.

On 15 October 2006, his PSP, led by Gilmar Gutiérrez, got the third place in the national election with 17% of the total votes, with the support of the economically disadvantaged, landless farmers and indigenous population.

In the 2009 Presidential Election, Lucio Gutiérrez ran for president of Ecuador under the flag of PSP. In the presidential election held on 26 April, he came in second with 26.8% of the vote, losing to Rafael Correa. Lucio Gutiérrez declared election fraud even though all the polls conducted well before election day gave a great advantage to Rafael Correa, and even when international observers from the European Union declared total transparency of the election process.

He was again running for president in 2021, obtaining 1.8% of the vote.

Post-Election
Almost all supporters of Lucio Gutiérrez are in the Amazon region where he was born. This relationship has been used by his political party to gain the support of inhabitants of these isolated areas. Gutiérrez has said he will form a parallel cabinet to that of President Rafael Correa so that if the "people give him the opportunity, he will have an experienced cabinet."

Attempted coup
He is alleged to have instigated the 2010 police revolt against Correa. He has fervently denied the allegations claiming that he was in Washington D.C. attending an event at the Inter-American Defense College (of which he is an alumnus), then traveled to Miami to attend a conference on the Socialism of the 21st century and finally to serve as an observer in the Brazilian presidential poll. He has further stated that the whole event was fabricated.

A special commission formed by Correa in 2013 to investigate the events has claimed that he and his party plotted, along with the fugitive Isaias brothers (former owners of Filanbanco who reside in Miami after fleeing during the 1998–2000 banking crisis), former chief of intelligence Mario Pazmiño (accused by Correa of working with the CIA and of being involved in the Colombian attack against Ecuadorian territory in 2008 which killed FARC leader Raúl Reyes and sparked a regional crisis) as well as others.

title: Lucio Gutiérrez


text:
Philippe of Belgium

Philippe (born 15 April 1960) is King of the Belgians. He is the eldest child of King Albert II and Queen Paola. He succeeded his father upon the former’s abdication for health reasons on 21 July 2013. He married Mathilde d'Udekem d'Acoz in 1999, with whom he has four children. Their eldest child, Princess Elisabeth, is first in the line of succession.

Early life
Philippe was born on 15 April 1960 at the Belvédère Château in Laeken, northern Brussels. His father, Prince Albert, Prince of Liège (later King Albert II), was the second son of King Leopold III of Belgium and a younger brother of Baudouin. His mother, Paola, Princess of Liège (later Queen Paola), is a daughter of the Italian aristocrat Fulco VIII, Prince Ruffo di Calabria, 6th Duke of Guardia Lombarda. His mother descends from the French House of La Fayette, and the king is a descendant of Gilbert du Motier, Marquis de Lafayette, and Marie Adrienne Françoise de Noailles.

Philippe was baptised one month later at the Church of St. James on Coudenberg in Brussels on 17 May, and named Philippe after his great-great-grandfather Prince Philippe, Count of Flanders. His godparents were his paternal grandfather, King Leopold III, and his maternal grandmother, Donna Luisa, Princess Ruffo di Calabria.

Albert and Paola's marriage was unhappy, and they were usually absent from Philippe's life. The child neglect was so severe that child psychologist Peter Adriaenssens described it as, "something that would justify intervention by social workers."

Philippe has a half-sibling, Princess Delphine of Belgium (born 1968).

He was also the fifth cousin of the late Queen Elizabeth II since they share a common ancestor, Francis (he was the grandfather of Queen Victoria and one of his sons, Leopold I would become the first King of the Belgians). Queen Elizabeth II and King Philippe had been 3rd cousins once removed through King Christian IX of Denmark.

Education
As a child the future king was required to move from a French-language school which he liked to a secondary school in Flanders, where he found it difficult to make friends. "In my youth, I had many problems at school", Philippe told teenage dropouts in 2019. "I felt badly treated. It was not easy for me". From 1978 to 1981, Philippe was educated at the Belgian Royal Military Academy in the 118th "Promotion Toutes Armes" (Promotion All Weapons). On 26 September 1980, he was appointed second lieutenant and took the officer's oath.

Philippe continued his education at Trinity College, Oxford, and he attended graduate school at Stanford University, California, where he graduated in 1985 with an MA degree in political science. He obtained his fighter pilot's wings and his certificates as a parachutist and a commando. In 1989, he attended a series of special sessions at the Royal Higher Defence Institute. The same year, he was promoted to colonel.

In 1993, King Baudouin died in Spain, Albert became the new king, and Philippe became the new heir apparent, titled Duke of Brabant.

On 25 March 2001, Philippe was appointed to the rank of major-general in the Land Component and the Air Component and to the rank of rear-admiral in the Naval Component.

Marriage
Philippe married Mathilde d'Udekem d'Acoz, daughter of a Walloon count of a Belgian noble family and female line descendant of Polish noble families such as the princes Sapieha and counts Komorowski, on 4 December 1999 in Brussels, in a civil ceremony at the Brussels Town Hall and a religious ceremony at the Cathedral of Saint Michel and Saint Gudule in Brussels. They have four children who were all born at Erasmus Hospital in Anderlecht, Brussels:
 * Princess Elisabeth (born 25 October 2001)
 * Prince Gabriel (born 20 August 2003)
 * Prince Emmanuel (born 4 October 2005)
 * Princess Eléonore (born 16 April 2008)

Foreign trade
On 6 August 1993, the government named Philippe as honorary chairman of the Belgian Foreign Trade Board (BFTB). He succeeded his father, who had been honorary chairman of the BFTB since 1962. On 3 May 2003, he was appointed honorary chairman of the board of the Foreign Trade Agency, replacing the BFTB.

In this capacity, Philippe has headed more than 60 economic missions. Upon his accession as seventh King of the Belgians, this role was taken over by his sister Princess Astrid.

Reign
King Albert II announced on 3 July 2013 that he would abdicate in favour of Philippe on 21 July 2013. Approximately one hour after King Albert II's abdication, Prince Philippe was sworn in as King of the Belgians. His eldest child, Princess Elisabeth, became his heir apparent and is expected to become Belgium's first queen regnant.

Philippe played a role in forming a coalition government after the 2014 Belgian federal election. Political meetings with the King were moved from the Palace of Laeken to the Royal Palace of Brussels. In May 2019, Philippe met with Vlaams Belang President Tom Van Grieken, the first time the party had received a royal audience.

In 2020, Philippe announced regret for the "acts of violence and cruelty" committed in the Congo Free State under the rule of his great-great-granduncle King Leopold II.

National honours

 * BEL - Order of Leopold - Grand Cordon bar.svg Grand Master of the Order of Leopold (21 July 2013)
 * BEL Order of the African Star - Grand Cross BAR.png Grand Master of the Order of the African Star (21 July 2013)
 * BEL Royal Order of the Lion - Grand Cross BAR.png Grand Master of the Royal Order of the Lion (21 July 2013)
 * BEL Kroonorde Grootkruis BAR.svg Grand Master of the Order of the Crown (21 July 2013)
 * BEL Order of Leopold II - Grand Cross BAR.png Grand Master of the Order of Leopold II (21 July 2013)

Foreign honours

 * 🇦🇷 Argentina: Grand Cross of the Order of the Liberator General San Martín (6 May 1994)
 * 🇦🇹 Austria: Grand Star of the Decoration of Honour for Services to the Republic of Austria (21 March 2022)
 * 🇧🇴 Bolivia: Grand Cross of the Order of the Condor of the Andes (9 September 1996)
 * 🇩🇰 Denmark: Knight of the Order of the Elephant (28 May 2002)
 * 🇫🇮 Finland: Grand Cross of the Order of the White Rose of Finland (30 March 2004)
 * 🇫🇷 France: Grand Cross of the National Order of the Legion of Honour (6 February 2014)
 * 🇩🇪 Germany: Grand Cross Special Class of the Order of Merit of the Federal Republic of Germany (6 March 2016)
 * 🇬🇷 Greece:
 * Grand Cross of the Order of the Redeemer (2 May 2022)
 * Grand Cross of the Order of Honour (1 February 2005)
 * Holy See: Knight of the Collar of the Order of the Holy Sepulchre (17 November 2015)
 * 🇭🇺 Hungary: Grand Cross of the Order of Merit of the Republic of Hungary (18 April 2008)
 * 🇮🇹 Italy: Knight Grand Cross with Collar of the Order of Merit of the Italian Republic (25 October 2021)
 * 🇯🇵 Japan: Collar and Grand Cordon of the Supreme Order of the Chrysanthemum (9 October 2016)
 * 🇯🇴 Jordan: Collar of the Order of al-Hussein bin Ali (18 May 2016)
 * 🇱🇹 Lithuania: Grand Cross with Golden Chain of the Order of Vytautas the Great (24 October 2022)
 * 🇱🇺 Luxembourg: Knight of the Order of the Gold Lion of the House of Nassau (15 March 1999)
 * 🇳🇱 Netherlands:
 * Knight Grand Cross of the Order of the Netherlands Lion (28 November 2016)
 * Knight Grand Cross of the Order of Orange-Nassau (6 May 1993)
 * Recipient of the King Willem-Alexander Inauguration Medal (30 April 2013)
 * 🇳🇴 Norway: Grand Cross of the Order of St. Olav (20 May 2003)
 * 🇴🇲 Oman: Member of the Order of Oman (3 February 2022)
 * 🇵🇱 Poland:
 * Knight of the Order of the White Eagle (13 October 2015)
 * Grand Cross of the Order of Merit of the Republic of Poland (18 October 2004)
 * 🇵🇹 Portugal:
 * Grand Collar of the Order of Prince Henry (22 October 2018)
 * Grand Cross of the Military Order of Christ (18 October 2005)
 * Grand Cross of the Military Order of Aviz (18 September 1997)
 * 🇪🇸 Spain: Knight Grand Cross of the Order of Isabella the Catholic (16 May 2000)
 * 🇸🇪 Sweden:
 * Knight of the Royal Order of the Seraphim (7 May 2001)
 * Recipient of the 50th Birthday Badge Medal of King Carl XVI Gustaf (30 April 1996)
 * Recipient of the 70th Birthday Badge Medal of King Carl XVI Gustaf (30 April 2016)
 * 🇹🇷 Turkey: Member of the Order of the State of Republic of Turkey (4 October 2015)

title: Philippe of Belgium


text:
Talk:Philippe, Crown Prince of Belgium

Is the Belgian succession gender-neutral? If not, is there no chance that Prince Philippe might, at some time in the future, have a son? Is Princess Elizabeth guaranteed to become Queen unless she dies first? -- Zoe

 * Yes, since 1991 Belgium has absolute primogeniture: sex doesn't count. -- Someone else 03:15 Jan 14, 2003 (UTC)

 * Thanks, someone else. -- Zoe

 * When dealing with a Crown Prince/ss of a state, use the form "{name}, Crown Prince/ss of {state}" unless there is a clear formal title awarded to a prince which defines their status as crown prince (eg, 'Charles, Prince of Wales', 'Felipe, Prince of Asturias', etc)

From Naming conventions (names and titles). Is there a reason why this shouldn't apply to "Duke of Brabant"? --Jao 03:51, 26 Jan 2004 (UTC)

title: Talk:Philippe, Crown Prince of Belgium


text:
Blu Cantrell

Tiffany Cobb (born March 16, 1976), known professionally as Blu Cantrell, is an American R&B singer.

Cantrell rose to fame in 2001, with the release of her debut single, "Hit 'Em Up Style (Oops!)", which peaked at number two on the US Billboard Hot 100 and topped the US Mainstream Top 40 chart. The song also charted in several other countries, and appeared on her debut album, So Blu. In 2002, the song earned Cantrell a Grammy Award nomination. In 2003, Cantrell released her second album, Bittersweet, which was nominated for a Grammy Award and included the single "Breathe" (featuring Sean Paul). Written and produced by Ivan Matias, "Breathe" was a major global success in 2003, especially in the United Kingdom, where it topped the UK Singles Chart for four consecutive weeks. "Breathe" also reached the top ten of several other charts across the world, including the European Hot 100.

Early life
Cantrell was born Tiffany Cobb in Providence, Rhode Island. Her mother, former beauty queen Susi Franco, was an actress and jazz vocalist. According to Cantrell, she has African-American, German, and Italian ancestry. Cantrell's parents separated when she was a young child, and she along with her five siblings—Adam, Tino, Nick, Kelli, and Summer—were raised by her mother.

Early career
In the late 1990s, Cantrell established herself as a professional backing vocalist for artists such as Sean "Puffy" Combs. In 1999, she became a member of the girl group 8th Avenue, a protégé of singer Teddy Riley. The band recorded several songs and appeared on Blackstreet's 1999 album Finally, but their material was shelved after Riley left Blackstreet to reform his previous group Guy, and Blackstreet were dropped by Interscope Records. Shortly thereafter, 8th Avenue also disbanded. Before long, Cantrell was introduced by both a dancer friend and R&B singer Usher to music producer Tricky Stewart, the head of Red Zone Entertainment. Stewart originally wanted Cantrell to become a member of his girl group 321, but after a fruitful recording session, he offered to help develop Cantrell's solo career instead. Cantrell subsequently moved in with Stewart and his girlfriend in their house in Atlanta, and was promptly placed with Arista Records head Antonio "L.A." Reid, who offered the singer a contract with the company after hearing one song she wrote and sang in front of him and his staff. After a bidding war with several different labels, Reid's bid was the highest, prompting Cantrell to sign with them.

2001–2004: So Blu and Bittersweet
After her signing with Arista, Cantrell went straight into recording sessions with Dallas Austin and Stewart, as well as Jimmy Jam and Terry Lewis. In July 2001, her debut album, So Blu, was released. The record earned generally favorable reviews from critics and became a commercial success, particularly in North America, where it peaked at number eight on the US Billboard 200. It was eventually certified gold by both the Recording Industry Association of America (RIAA) and Music Canada. The album's lead single "Hit 'Em Up Style (Oops!)" became a top ten hit in Australia, Canada, New Zealand, and the Netherlands and peaked at number two on the US Billboard Hot 100. The song earned Cantrell Grammy Award nominations for Best Female R&B Vocal Performance and for Best R&B Song, as well as an American Music Award nomination for Favorite New Soul/R&B Artist, both in 2002. Also in 2002, Cantrell was featured in a small cameo role in Charles Stone III's musical comedy-drama film Drumline, where she could be seen singing the American national anthem. Her song "It's Killing Me (In My Mind)" was included on the soundtrack of the 2002 action comedy film Bad Company.

In 2003, Cantrell released her second album, Bittersweet which featured production from Kevin "She'kspere" Briggs, Mike City, Soulshock & Karlin, and Shep Crawford. As with her debut, the album garnered a positive reception from critics, even earning her a Best R&B Album nomination at the 46th Grammy Awards, but was less successful in the United States, peaking at number 37 on the Billboard 200. The album was a success internationally, where sales were boosted by its hit single "Breathe", a collaboration with Sean Paul. Her highest-charting single yet, it topped the charts in the United Kingdom and the Republic of Ireland, and reached the top ten of the majority of charts it appeared on, ranking among the biggest-selling releases of the year. "Breathe" was followed by the top thirty single "Make Me Wanna Scream".

2005–present
In 2005, following the formation of the joint venture of BMG and Sony Music Entertainment, Arista merged with J Records and began operating under the newly formed RCA Music Group. At about the same time, Cantrell's recording deal was up for renewal and though she was in a position to re-sign with the label, the singer chose to leave the company following the resignation of her mentor L.A. Reid. She then toured intensively as a free agent. Cantrell also starred alongside LisaRaye and Kenya Moore in the musical stage play Gossip, Lies and Secrets which ran from September until November in 2007. The following year, she appeared in NBC's Celebrity Circus, becoming the first celebrity to be eliminated from the show in the season's second week.

In 2013, Cantrell confirmed that she was working independently on her third studio album. In December 2016, she told Fuse that she was still working on a new album.

Despite the lack of US success after her one big hit, Cantrell continues to perform in Australia and Europe.

Personal life
Cantrell was taken into custody by police for a psychological evaluation on September 3, 2014. She was seen running around the streets of Santa Monica at around 2 am, screaming that someone had "poisoned her with gas". Cantrell, according to witnesses, went "berserk" and referred to herself as a "one-hit wonder", questioning authorities whether they recognized her; when her "erratic" behavior failed to cease, someone called the police. She was taken to a nearby hospital, where she was evaluated by medics.

Discography

 * So Blu (2001)
 * Bittersweet (2003)

title: Blu Cantrell


text:
Macy Gray

Natalie Renée McIntyre (born September 6, 1967), known by her stage name Macy Gray, is an American R&B and soul singer and actress. She is known for her distinctive raspy voice and a singing style heavily influenced by Billie Holiday.

Gray has released ten studio albums, and received five Grammy Award nominations, winning one. She has appeared in a number of films, including Training Day, Spider-Man, Scary Movie 3, Lackawanna Blues, Idlewild, For Colored Girls, and The Paperboy. Gray is best known for her international hit single "I Try", taken from her multi-platinum debut album On How Life Is. Gray was inducted into the National Rhythm & Blues Hall of Fame in 2014 in her hometown of Canton, Ohio.

Early life
Natalie McIntyre was born in Canton, Ohio, the daughter of Laura McIntyre, a math schoolteacher, and Otis Jones. Her stepfather was a steelworker, and her sister is a biology teacher. She has a younger brother, Nate, who owns a gym in West Philadelphia and was featured on the season five finale of Queer Eye. She began piano lessons at age seven. A childhood bicycle mishap resulted in her noticing a mailbox of a man named Macy Gray; she used the name in stories she wrote and later decided to use it as her stage name. She was late developing and did not learn to hold conversation until just before her tenth birthday.

Gray attended elementary school with Brian Warner (later known as musician Marilyn Manson) although they did not know each other. She attended more than one high school, including a boarding school which asked her to leave due to her behavior.

She attended the University of Southern California and studied scriptwriting.

Musical career
While attending the University of Southern California, she agreed to write songs for a friend. A demo session was scheduled for the songs to be recorded by another singer, but the vocalist failed to appear, so Gray recorded them herself. "I started forming bands and writing songs just for fun and then I really got into it and got attached to it. Then a friend of mine asked me to be a singer in his jazz band. He gave me all these jazz CDs and I studied all these different singers and I kind of taught myself how to sing for a gig, but I didn't take it seriously until later." She then met writer-producer Joe Solo while working as a cashier in Beverly Hills. Together, they wrote a collection of songs and recorded them in Solo's studio. The demo tape gave Gray the opportunity to sing at jazz cafés in Los Angeles. Although Gray did not consider her unusual voice desirable for singing, Atlantic Records signed her. She began recording her debut record but was dropped from the label upon the departure of A&R man Tom Carolan, who had signed her to the label. Macy returned to Ohio but in 1997 Los Angeles based Zomba Label Group Senior VP A&R man Jeff Blue, convinced her to return to music and signed her to a development deal, recording new songs based on her life experiences, with a new sound, and began shopping her to record labels. In 1998, she landed a record deal with Epic Records. She performed on "Love Won't Wait", a song on the Black Eyed Peas' debut album Behind the Front.

1999–2001: On How Life Is
Gray worked on her debut album in 1999 with producer Darryl Swann. Released in the summer of 1999, On How Life Is became a worldwide smash. The first single, "Do Something", stalled on the charts, but the second single, "I Try", made the album a success. "I Try" (which was originally featured in Love Jones and the Jennifer Aniston-starring romantic-comedy Picture Perfect in 1997) was one of the biggest singles of 1999, and subsequent singles ("Still" and "Why Didn't You Call Me") ensured the album became triple platinum in the US and quadruple platinum in the UK.

In 2001, Gray won the Grammy Award for Best Female Pop Vocal Performance for "I Try", which was also nominated for "song of the year" and "record of the year". She then collaborated with Fatboy Slim, the Black Eyed Peas, and Slick Rick (on the song "The World Is Yours", from the Rush Hour 2 soundtrack), as well as acting for the first time, in the thriller Training Day. In August 2001, Gray was booed at the Pro Football Hall of Fame exhibition game after forgetting the lyrics to the American national anthem.

2001–2005: The Id and The Trouble with Being Myself
Gray's The Id featured appearances by John Frusciante and Erykah Badu on the single "Sweet Baby" (which was co-written with longtime collaborator Joe Solo). The album peaked at number eleven on the Billboard 200. It fared even better in the UK, where it reached number one on the UK Albums Chart and was certified gold by the BPI.

In 2002, she appeared in Spider-Man and contributed a remix of her song "My Nutmeg Phantasy" to its accompanying soundtrack. Gray also worked with Santana on the track "Amoré (Sexo)", for his album Shaman.

Also in 2002, she appeared on the Red Hot Organization's compilation CD in tribute to Nigerian Afropop pioneer Fela Kuti, Red Hot and Riot. She appeared on a remake of Kuti's classic track "Water No Get Enemy" alongside prominent neo soul, hip hop and R&B artists, D'Angelo, the Soultronics, Nile Rodgers, Roy Hargrove, and Kuti's son, Femi Kuti.

She recorded a duet with Zucchero called "Like the Sun (From Out of Nowhere)", which featured Jeff Beck on guitar released in 2004 on Zu & Co., a duet collection. Her song "Time of My Life" was included in the soundtrack to 8 Mile. A cartoon based on Gray's childhood was being developed, but it never came to fruition.

In 2003, Gray released her third studio album, The Trouble with Being Myself, to rave reviews. The lead single, "When I See You", became a radio hit in the US and a top-forty hit in the UK, although the album was not as well received by fans. Nevertheless, it became Gray's third top twenty album in the UK. A greatest hits collection and a live album were subsequently released: The Very Best of Macy Gray (2004) and Live in Las Vegas (2005). Additionally, Gray was featured on Marcus Miller's 2005 album Silver Rain, on a cover of Prince's 1986 song "Girls & Boys". She also appeared on the soundtrack to the film Chicago with Queen Latifah and Lil' Kim on "Cell Block Tango/He Had it Comin'".

2007–2010: Return to music and Big
Gray began 2007 by being kicked off-stage at a concert in Barbados for profanity (which was part of the show), but she was not aware that it was against the law in that country. She gave a public apology that night to avoid arrest.

In March, Gray released her fourth studio album (sixth overall), Big. Two singles, "Finally Made Me Happy" and "Shoo Be Doo", were released from the album. "What I Gotta Do", another track from the album, is featured on the Shrek the Third soundtrack. It has been considered Gray's comeback album, after a four-year hiatus since her last studio album. The album was critically acclaimed and seen by some as her best work to date. It features collaborations with Natalie Cole, Fergie, Justin Timberlake, and will.i.am, who co-executive produced the album with Gray. It was moderately successful in the US, where it debuted and peaked at number 39 on the Billboard 200, becoming Gray's highest-charting album since The Id. Big reached number 62 on the albums chart in the UK, her lowest-charting UK album, but it achieved some success in several other countries, including Switzerland, the Czech Republic, and Finland, reaching the Top 40 on their album charts.

PBS's Soundstage live concert series premiered a Gray concert on July 5.

On July 7, 2007, Gray performed at the Brazilian leg of Live Earth at Copacabana Beach in Rio de Janeiro, Brazil. Gray and her band members wore clothes bearing political messages. Gray's dress carried the message "Darfur Red Alert".

In 2008, Gray launched a new campaign under the name "Nemesis Jaxson", with the single "Slap a Bitch".

Early in 2009, Gray recorded the song "Don't Forget Me" for the soundtrack of Confessions of a Shopaholic.

2010–2011: The Sellout
The first single from Gray's fifth studio album The Sellout, "Beauty in the World", is featured in the final sequence of the series finale, "Hello Goodbye", of the ABC television series, Ugly Betty. "Beauty in the World" was also used as the theme in multiple videos created by Microsoft to promote Internet Explorer 9. Both singles released from the album ("Beauty in the World" and "Lately") were top 10 hits on the US Billboard Hot Dance Club Songs Chart.

Upon its release, The Sellout received generally mixed reviews from most music critics. Metacritic gave it an average score of 57, based on 15 reviews; Andy Gill of The Independent gave it three out of five stars; while AllMusic writer John Bush shared a similar sentiment and panned Gray's songwriting.

However, The Boston Globe's James Reed commended its production, and Jeremy Allen of NME gave the album a 7/10 rating.

2011–2015: Covered, Talking Book, and The Way
In 2011, Gray signed a deal with 429 Records and started recording a series of covers for her next studio album Covered. The album was officially released on March 26, 2012. The album's first single was "Here Comes the Rain Again" (originally performed by Eurythmics). On February 16, 2012, Gray participated in the Sanremo Festival as a guest, performing alongside Gigi D'Alessio and Loredana Bertè.

For the 40th anniversary of the 1972 Stevie Wonder album Talking Book, Gray covered the entire record and released her Talking Book as a tribute.

In an interview with Oprah Winfrey's Where Are They Now, Gray opened up about her problems with drug abuse, stating that she was ill-prepared for the level of fame she received. The interview coincided with the release of her album The Way, released in October 2014 on Kobalt Records. A world tour was announced shortly after its release.

2016–present: Stripped and Ruby
In 2016, Gray's career continued with an album produced by Chesky Records, the record label founded by Grammy-nominated composer and musician David Chesky. Stripped was released September 9, 2016, and garnered almost unanimous praise from critics. The album debuted at No. 3 on the Billboard Jazz Chart. Earlier that year, she was featured on Ariana Grande's song "Leave Me Lonely" from her third studio album, Dangerous Woman.

Gray release her tenth studio album entitled Ruby in September 2018.

Other work
In 2001, she was the voice for Seeiah Owens in the video game SSX Tricky. Gray sang the theme song for the Nickelodeon animated series As Told by Ginger, composed by Jared Faber and Emily Kapnek. Gray also performed the song on the short-lived UPN romantic comedy Second Time Around starring Boris Kodjoe and Nicole Ari Parker in 2004. In 2006 Gray appeared in the eighth season of Bravo's Celebrity Poker Showdown, playing for Habitat for Humanity. She finished in third place.

In August 2008, Gray headlined at the 2008 Summer Sundae music festival in Leicester, England, performing cover versions of Rod Stewart's "Da Ya Think I'm Sexy?", Deee-Lite's "Groove Is in the Heart" and Radiohead's "Creep". For the gig, her band wore pink Andy Warhol wigs. On September 28, 2008, Gray sang the American national anthem as part of the Israeli flag-raising ceremony at the Israeli consulate of Los Angeles. Also in 2008, Gray collaborated with Australian DJ and singer Kaz James on the song "Can't Hold Back". The single was released in early 2009 in Australia and is credited to Kaz James featuring Macy Gray. In 2009, Gray briefly competed in season 9 of Dancing with the Stars with professional partner Jonathan Roberts. They were eliminated in a double elimination in week one. In 2012, she performed the Michael Jackson song "Rock with You" for a special performance of the West End musical Thriller – Live for BBC Children in Need Pop Goes the Musical.

In 2015, Gray was featured on the song "Into the Deep" by Galactic. In 2016, Gray was featured on the song "Leave Me Lonely" by Ariana Grande. In 2018, Gray was featured alongside Dolly Parton on a re-recording of "Two Doors Down" for the Dumplin' soundtrack. In 2020, Gray was featured on the single "Out Of Love" by Canadian band Busty and the Bass. The band also released a notable cover of Gray's "I Try" in 2016.

In 2021, Gray competed in season three of the Australian version of The Masked Singer as "Atlantis". On February 20, 2022, Gray performed The Star-Spangled Banner at the 2022 NBA All-Star Game. Gray represented her home state of Ohio at the 2022 America Song Contest with the song "Every Night".

In 2023, Gray competed in season ten of The Masked Singer as the wild card contestant "Sea Queen". She finished in 4th place.

Personal life
Gray was married to Tracy Hinds, a mortgage broker, for about two years, but they divorced prior to her rise to prominence. They have three children.

She opened the Macy Gray Music Academy in 2005.

In a July 2022 interview on Piers Morgan Uncensored, Gray was asked how she would define a woman, to which she answered "boobs and a vagina". She went on to state that she did not support the inclusion of transgender women in women's sports. In the days following the interview, she said that her words had been misunderstood, along with rejecting criticism she had received online for her views. In a subsequent appearance on the Today television show, she said, "Being a woman is a vibe and it's something I'm very proud of and something that it is very precious to me. I think that if you in your heart feel that's what you are, then that's what you are regardless of what anybody says or thinks".

In an opinion piece published in MarketWatch in June 2021, Gray called for a redesign of the United States flag stating that it "no longer represents democracy and freedom". Her suggestions included two additional stars for Washington, D.C., and Puerto Rico, the white stripes to be recolored off-white and the stars to be given a variety of colors.

Discography
Studio albums
 * On How Life Is (1999)
 * The Id (2001)
 * The Trouble with Being Myself (2003)
 * Big (2007)
 * The Sellout (2010)
 * Covered (2012)
 * Talking Book (2012)
 * The Way (2014)
 * Stripped (2016)
 * Ruby (2018)

Awards and nominations
Gray won five of seventeen nominations, including Grammy Awards, MTV Video Music Awards and Brit Awards.

title: Macy Gray