# Peano arithmetic in a sentence

1)

**Peano**

**arithmetic**is equiconsistent with several weak systems of set theory.

arithmetic collocations

2) This is the language of

**Peano**

**arithmetic**.

3)

**Peano**

**arithmetic**is also incomplete by Gödel's incompleteness theorem.

## Peano **arithmetic** example sentences

4) There are many different, but equivalent, axiomatizations of **Peano**

**arithmetic**.

5) First-order axiomatizations of

**Peano**

**arithmetic**have an important limitation, however.

6) For example, the system ACA0 described below is equiconsistent with

**Peano**

**arithmetic**.

7) PRA is much weaker than

**Peano**

**arithmetic**, which is not a finitistic system.

8) Second-order

**arithmetic**includes, but is significantly stronger than, its first-order counterpart

**Peano**

**arithmetic**.

9) It is provably consistent, as is RCA0, in full first-order

**Peano**

**arithmetic**.

10) One definition is "a proof that can be carried out in first order

**Peano**

**arithmetic**.

11) Unlike

**Peano**

**arithmetic**, second-order

**arithmetic**allows quantification over sets of numbers as well as numbers themselves.

12) In the resulting system, the axioms of ZFC (and of

**Peano**

**arithmetic**) are theorems.

13) In the context of

**Peano**

**arithmetic**, it consists of the natural numbers with their ordinary arithmetical operations.

14) For example, the intended interpretation of

**Peano**

**arithmetic**consists of the usual natural numbers with their usual operations.

15) Examples of effectively generated theories with infinite sets of axioms include

**Peano**

**arithmetic**and Zermelo-Fraenkel set theory.

### example sentences with **Peano** **arithmetic**

16) The various properties like associativity can be proved from these and the other axioms of **Peano**

**arithmetic**including induction.

17) The corresponding theory ACA, consisting of ACA0 plus the full second-order induction scheme, is stronger than

**Peano**

**arithmetic**.

18) Some theories like

**Peano**

**arithmetic**are limited by much smaller ordinals (__FORMULA__ in the case of

**Peano**arithmetic).

19) Some theories like

**Peano**

**arithmetic**are limited by much smaller ordinals (__FORMULA__ in the case of

**Peano**

**arithmetic**).

20) For example, first-order

**Peano**

**arithmetic**(PA) can prove that the largest consistent subset of PA is consistent.

21) A set is definable in first order

**arithmetic**if it is defined by some formula in the language of

**Peano**

**arithmetic**.

22) The first-order part of ACA0 is exactly first-order

**Peano**

**arithmetic**; ACA0 is a "conservative" extension of first-order

**Peano**arithmetic.

23) The first-order part of ACA0 is exactly first-order

**Peano**arithmetic; ACA0 is a "conservative" extension of first-order

**Peano**

**arithmetic**.

24) There is a relation between computable ordinals and certain formal systems (containing arithmetic, that is, at least a reasonable fragment of

**Peano**

**arithmetic**).

25) Because RA can express

**Peano**

**arithmetic**and set theory, Gödel's incompleteness theorems apply to it; RA is incomplete, incompletable, and undecidable.

26) The set of first-order consequences of __FORMULA__ is the same as those of the subsystem IΣ1 of

**Peano**

**arithmetic**in which induction is restricted to Σ01 formulas.

27) This follows from the fact that the axioms of

**Peano**

**arithmetic**with the second-order induction axiom have only one model under second-order semantics.

28) A complete proof would show that the property displayed in quotes in the previous sentence is definable in the language of

**Peano**

**arithmetic**by a __FORMULA__ formula.

#### How to use **Peano** **arithmetic** in a sentence

29) To do so, fix a set of integers "Y" and add a predicate for membership in "Y" to the language of **Peano**

**arithmetic**.

30) The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as

**Peano**

**arithmetic**.

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