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It is known that there are non-standard models of Peano Arithmetics when it is described using first order logic. My question is if there is standard model (one which does not contains non-standard elements) of PA described in FOL. What is example of such canonical model?

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up vote 7 down vote accepted

It isn't clear what "described in FOL means".

The first-order theory of Peano Arithmetic certainly has a standard model, which consists of the standard natural numbers with their usual arithmetical operations. Peano Arithmetic also has many nonstandard models.

The thing that cannot be done in first-order logic is to make a theory $T$ such that the only model of $T$ is the standard model of arithmetic. This has nothing to do about arithmetic per se; a first order theory that has an infinite model has infinitely many other infinite models, regardless of the subject matter of the first order theory.

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So do I suspect right that $\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, ...$ is standard model of Peano Arithmetic formalized in First Order Logic? – Trismegistos Jan 8 '13 at 15:42
That's not usually the way it is defined in ZFC, but that way of defining the numbers (together with the usual interpretations of the addition and multiplication operations) would work. There are many equivalent (isomorphic) ways of defining the standard model. The usual way of doing it in ZFC is to let the standard model consist of the finite von Neumann ordinals, which are $0 = \emptyset$, $1 = \{0\}$, $2 = \{0, 1\}$, $3 = \{0,1,2\}$, ... – Carl Mummert Jan 8 '13 at 18:22

By Gödel's Incompleteness Theorems any construction of a model for PA must transcend PA. Using a bit of set theory (some small fragment of ZF) one can show that $\omega$ (the first transfinite von Neumann ordinal) together with appropriate definable arithmetic operations forms a model of PA and has no non-standard elements.

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