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The notions themselves may not necessarily need to be stated; Susan Haack (1978) writes, "A set of axioms is sometimes said to give an implicit definition of its primitive terms." [7] Euclidean geometry: Under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweenness , and incidence.
For example, it is provable without the axiom of choice that every vector space of finite dimension has a basis, but the generalization to all vector spaces requires the axiom of choice. Likewise, a finite product of compact spaces can be proven to be compact without the axiom of choice, but the generalization to infinite products ( Tychonoff's ...
New terms are defined using the primitive terms and other derived definitions based on those primitive terms. In a deductive system, one can correctly use the term "proof", as applying to a theorem. To say that a theorem is proven means that it is impossible for the axioms to be true and the theorem to be false.
A good example is the relative consistency of absolute geometry with respect to the theory of the real number system. Lines and points are undefined terms (also called primitive notions) in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems. [citation needed]
A variation on the method of forcing can also be used to demonstrate the consistency and unprovability of the axiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice.
A typical example of this type of notation can be found in the work of E. V. Huntington (1874 – 1952) who, in 1913, [54] produced an axiomatic treatment of three-dimensional Euclidean geometry based upon the primitive notions of sphere and inclusion (one sphere lying within another). [42]
List or describe a set of sentences in the language L σ, called the axioms of the theory. Give a set of σ-structures, and define a theory to be the set of sentences in L σ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite fields. An L σ ...
Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. Axiom of extensionality; Axiom of empty set; Axiom of pairing; Axiom of union; Axiom of infinity; Axiom schema of replacement; Axiom of power set ...