Peano Arithmetic
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A Peano Arithmetic is a Mathematical Logic that ...
- See: Axiom Schema, Mathematical Logic, Peano Axiom, Giuseppe Peano, Metamathematics, Number Theory, Consistency Proof, Completeness (Logic).
References
2023
- (Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/Peano_axioms#Peano_arithmetic_as_first-order_theory Retrieved:2023-6-19.
- All of the Peano axioms except the ninth axiom (the induction axiom) are statements in first-order logic.Template:Sfn The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction above is second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-order axiom schema of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom.Template:Sfnp The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property).
First-order axiomatizations of Peano arithmetic have another technical limitation. In second-order logic, it is possible to define the addition and multiplication operations from the successor operation, but this cannot be done in the more restrictive setting of first-order logic. Therefore, the addition and multiplication operations are directly included in the signature of Peano arithmetic, and axioms are included that relate the three operations to each other.
- All of the Peano axioms except the ninth axiom (the induction axiom) are statements in first-order logic.Template:Sfn The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction above is second-order, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-order axiom schema of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom.Template:Sfnp The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property).