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The 21 assumptions which underlie the geometry published in Hilbert's classic text Grundlagen der Geometrie. The eight incidence axioms concern collinearity and intersection and include the first of Euclid's postulates. The four ordering axioms concern the arrangement of points, the five congruence axioms concern geometric equivalence, and the three continuity axioms concern continuity. There is also a single parallel axiom equivalent to Euclid's parallel postulate.

The axiom of Zermelo-Fraenkel set theory which asserts the existence for any sets and of a set having and as its only elements. is called the unordered pair of and , denoted . The axiom may be stated symbolically as

One of the Zermelo-Fraenkel axioms which asserts the existence for any set of the power set consisting of all the subsets of . The axiom may be stated symbolically as(Enderton 1977). Note that the version given by Itô (1986, p. 147),is confusing, and possibly incorrect.

One of the Eilenberg-Steenrod axioms which states that, if is a space with subspaces and such that the set closure of is contained in the interior of , then the inclusion map induces an isomorphism .

1. A straight line segment can be drawn joining anytwo points. 2. Any straight line segment can be extended indefinitelyin a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent.5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely..

The axiom of Zermelo-Fraenkel set theory which asserts the existence for any set and a formula of a set consisting of all elements of satisfying ,where denotes exists, means for all, denotes "is an element of," means equivalent, and denotes logical AND.This axiom is called the subset axiom by Enderton (1977), while Kunen (1980) calls it the comprehension axiom. Itô (1986) terms it the axiom of separation, but this name appears to not be used widely in the literature and to have the additional drawback that it is potentially confusing with the separation axioms of Hausdorff arising in topology.This axiom was introduced by Zermelo.

One of the Zermelo-Fraenkel axioms which asserts the existence for any set of a set such that, for any of , if there exists a satisfying , then such exists in ,This axiom was introduced by Fraenkel.

1. Zero is a number. 2. If is a number, the successor of is a number. 3. zero is not the successor of a number. 4. Two numbers of which the successors are equal are themselves equal. 5. (induction axiom.) If a set of numbers contains zero and also the successor of every number in , then every number is in . Peano's axioms are the basis for the version of numbertheory known as Peano arithmetic.

The axiom of Zermelo-Fraenkel set theorywhich asserts the existence of a set containing all the natural numbers,where denotes exists, is the empty set, is logical AND, means for all, and denotes "is an element of" (Enderton 1977). Following von Neumann, , , , , ....

Let represent "or", represent "and", and represent "not." Then, for two logical units and ,These laws also apply in the more general context of Boolean algebra and, in particular, in the Boolean algebra of set theory, in which case would denote union, intersection, and complementation with respect to any superset of and .

One of the Zermelo-Fraenkel axioms, also known as the axiom of regularity (Rubin 1967, Suppes 1972). In the formal language of set theory, it states thatwhere means implies, means exists, means AND, denotes intersection, and is the empty set (Mendelson 1997, p. 288). More descriptively, "every nonempty set is disjoint from one of its elements."The axiom of foundation can also be stated as "A set contains no infinitely descending (membership) sequence," or "A set contains a (membership) minimal element," i.e., there is an element of the set that shares no member with the set (Ciesielski 1997, p. 37; Moore 1982, p. 269; Rubin 1967, p. 81; Suppes 1972, p. 53).Mendelson (1958) proved that the equivalence of these two statements necessarily relies on the axiom of choice. The dual expression is called -induction, and is equivalent to the axiom itself (Itô 1986, p. 147)...

Given any straight line and a point not on it, there "exists one and only one straight line which passes" through that point and never intersects the first line, no matter how far they are extended. This statement is equivalent to the fifth of Euclid's postulates, which Euclid himself avoided using until proposition 29 in the Elements. For centuries, many mathematicians believed that this statement was not a true postulate, but rather a theorem which could be derived from the first four of Euclid's postulates. (That part of geometry which could be derived using only postulates 1-4 came to be known as absolute geometry.)Over the years, many purported proofs of the parallel postulate were published. However, none were correct, including the 28 "proofs" G. S. Klügel analyzed in his dissertation of 1763 (Hofstadter 1989). The main motivation for all of this effort was that Euclid's parallel postulate did..

"The" continuity axiom is an additional Axiom which must be added to those of Euclid's Elements in order to guarantee that two equal circles of radius intersect each other if the separation of their centers is less than (Dunham 1990). The continuity axioms are the three of Hilbert's axioms which concern geometric equivalence.Archimedes' Axiom is sometimes also known as"the continuity axiom."

The axiom of Zermelo-Fraenkel set theorywhich asserts that sets formed by the same elements are equal,Note that some texts (e.g., Devlin 1993), use a bidirectional equivalent preceding "," while others (e.g., Enderton 1977, Itô 1986), use the one-way implies . However, one-way implication suffices.Using the notation ( is a subset of ) for , the axiom can be written concisely aswhere denotes logical AND.

An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice. It was formulated by Zermelo in 1904 and states that, given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. The axiom of choice is related to the first of Hilbert's problems.In Zermelo-Fraenkel set theory (in the form omitting the axiom of choice), Zorn's lemma, the trichotomy law, and the well ordering principle are equivalent to the axiom of choice (Mendelson 1997, p. 275). In contexts sensitive to the axiom of choice, the notation "ZF" is often used to denote Zermelo-Fraenkel without the axiom of choice, while "ZFC" is used if the axiom of choice is included.In 1940, Gödel proved that the axiom of choice is consistent with the axioms of von Neumann-Bernays-Gödel set theory (a conservative extension of Zermelo-Fraenkel..

One of the Eilenberg-Steenrod axioms. It states that, for every pair , there is a natural long exact sequencewhere the map is induced by the inclusion map and is induced by the inclusion map . The map is called the boundary map.

An axiomatic system is said to be categorical if there is only one essentially distinct representation for it. In particular, the names and types of objects within the system may vary while still being considered "the same," e.g., geometries and their plane duals.An example of an axiomatic system which isn't categorical is a geometrydescribed by the following four axioms (Smart): 1. There exist five points. 2. Each line is a subset of thosefive points. 3. There exist two lines. 4. Each line contains at least two points.One way to see that this is a non-categorical axiomatic system is to note that one can form a compatible system from two fundamentally different models, e.g., 1. Two disjoint lines each containing two points plus a separate point not on either line. 2. Two lines containing three pointseach which intersect in one of the points. The presence of an intersection in one model andnot the other implies that the models are fundamentally..

For any set theoretic formula ,In other words, for any formula and set there is a subset of consisting exactly of those elements which satisfy the formula.

The law appearing in the definition of Boolean algebrasand lattice which states thatfor binary operators and (which most commonly are logical OR and logical AND). The two parts of the absorption law are sometimes called the "absorption identities" (Grätzer 1971, p. 5).

Propositional calculus, first-order logic, and other theories in mathematical logic are defined by their axioms (or axiom schemata, plural: axiom schemata) and inference rules. An axiom schema is a sentential formula representing infinitely many axioms. These axioms are obtained by replacing variables in the schema by any formula. For example, the axiom schema(1)in propositional calculus represents theaxioms (2)(3)and so on.It is typical to define a theory by axiom schemata rather than axioms. If axioms but not their schemata are utilized, then substitution for variables should be incorporated into inference rules.

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