# 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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