Peano arithmetic in a sentence
1) Peano arithmetic is equiconsistent with several weak systems of set theory.
2) This is the language of Peano arithmetic .
3) Peano arithmetic is also incomplete by Gödel's incompleteness theorem.
Peano arithmetic example sentences4) 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 arithmetic16) 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 sentence29) 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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