# Peano axiom in a sentence

1) The Peano axioms contain three types of statements.

2) The Peano axioms can also be understood using category theory.

axiom collocations
3) A logical system is sufficiently strong if it can express the Peano axioms.

4) But we can do this for systems far beyond Peano's axioms.

## Peano axiom example sentences

5) The Peano axioms are the most widely used "axiomatization" of first-order arithmetic.

6) This illustrates one way the first-order system PA is weaker than the second-order Peano axioms.

7) Derives the Peano axioms (called S) from several axiomatic set theories and from category theory.

8) Getzen's 1936 proof of Peano's axioms using combinatorial methods marked the beginning of ordinal proof theory.

9) All of the Peano axioms except the ninth axiom (the induction axiom) are statements in first-order logic.

10) Dedekind (1888) proposed a different characterization, which lacked the formal logical character of Peano's axioms.

11) There are also non-standard models of the Peano axioms, which contain elements not correlated with any natural number.

12) The set N together with 0 and the successor function "s" : N → N satisfies the Peano axioms.

13) The respective functions and relations are constructed in second-order logic, and are shown to be unique using the Peano axioms.

14) However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0.

### example sentences with axiom

15) The small number of mathematicians who advocate ultrafinitism reject Peano's axioms because the axioms require an infinite set of natural numbers.

16) All models that are isomorphic to the one just given are also called standard; these models all satisfy the Peano axioms.

17) The principle of mathematical induction is usually stated as an axiom of the natural numbers; see Peano axioms . However, it can be proved from the well-ordering principle.

18) Indeed, "zero is an even number" may be interpreted as one of the Peano axioms, of which the even natural numbers are a model.

19) This article provides detail of that debate and discovery from Peano's axioms in 1889 through recent discussion of the meaning of "axiom".

20) David Lewis employed plural quantification in his "Parts of Classes" to derive a system in which Zermelo-Fraenkel set theory and the Peano axioms were all theorems.

21) He argued that Peano's axioms cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is "a priori" synthetic and not analytic.

22) Nor does "WMCF" say enough about the derivation of number systems (the Peano axioms go unmentioned), abstract algebra, equivalence and order relations, mereology, topology, and geometry.

23) For example, the formal definition of the natural numbers by the Peano axioms can be described as: "0 is a natural number, and each natural number has a successor, which is also a natural number.

24) The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen's proof.

25) In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε0.

26) When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number".

27) The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as the ZF.

28) The second-order Peano axioms are thus categorical; this is not the case with any first-order reformulation of the Peano axioms, however.

29) The second-order Peano axioms are thus categorical; this is not the case with any first-order reformulation of the Peano axioms, however.

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