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https://en.wikipedia.org/wiki/Peano_axioms
Peano axioms
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4 Models
4.1 Set-theoretic models
4.2 Interpretation in category theory
5 Nonstandard models

https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic
Non-standard model of arithmetic
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In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers.
The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, ….
The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers.
A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

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