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Let $L$ be a language for arithmetic on the natural numbers $\mathbb{N}(=\{0,1,2,...\})$ including $0,1,+,\cdot$ and $>$. Let $Th(\mathbb{N})$ be the set of all sentences of $L$ true in $\mathbb{N}$. Show that there is a nonstandard model of $Th(\mathbb{N})$, i.e., a structure $M$ for $L$ in which every sentence of $Th(\mathbb{N})$ is true but in which there is an element $c$ greater than every $n\in \mathbb{N}$.

I think I'm a bit confused on what this question is asking me. I understand the base assumption that $L$ is a language and $Th(\mathbb{N})$ is the set of sentences that are true under natural numbers. But I don't grasp the example of what a nonstandard model is.

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  • $\begingroup$ A model which is not isomorphic to $\mathbb N $. $\endgroup$ – Andrés E. Caicedo Jan 8 '18 at 14:19
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Here is a standard compactness construction.

Let $c$ be a new constant symbol and form a new theory $W$ which consists of $Th(N)$ together with the infinitely many formulas ${0<c, 1<c, 1+1<c, 1+1+1<c, ...}$.

This new theory $W$ is consistent since any finite subset $F$ of it's axioms contains only finitely many of the new axioms and so a model for $F$ is the natural numbers with $c$ interpreted sufficiently large to accommodate the finitely many new axioms in $F$.

A model $M$ of $W$ is a model of $Th(N)$ and can't be isomorphic to $N$ because the interpretation of $c$ is unequal to any of the standard numbers 0, 1, 2, ... (to be picky, the reduct of $M$ to the language of $Th(N)$ that omits the symbol $c$ is the precise example you need).

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