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  "body": "\n## Example\n\nLet $\\mathsf{PA}$ be first-order [[Peano arithmetic|0QFT7Z76N5]] in the language with $0$, $S$, $+$, $\\cdot$ and $\u003c$.\nA *non-standard model of arithmetic* is a [[model|MVCH6N9SNW]] of $\\mathsf{PA}$ that is not isomorphic to the standard model $\\N$ of the [[natural numbers|0MWKJ44V5A]].\n\nSuch a model exists.\nAdd a new constant symbol $c$, and let $T = \\mathsf{PA} \\cup \\set{c \u003e \\underline{n} : n \\in \\N}$, where $\\underline{n}$ is the term $S S \\cdots S 0$ with $n$ copies of $S$.\nEvery finite subset of $T$ mentions only finitely many $n$, so it has a model: $\\N$ with $c$ read as a large enough number.\nBy the [[compactness theorem|1NR3HAWG9D]], $T$ has a model $M$.\nIn $M$ the element $c^M$ is greater than the value of every numeral, so it is not of the form $\\underline{n}^M$.\nIn a model isomorphic to $\\N$ every element is the value of a numeral, so $M$ is a non-standard model of $\\mathsf{PA}$.\nBy the downward Löwenheim–Skolem theorem, $M$ can be taken countable.\n\n## Remarks\n\nEvery non-standard model $M$ starts with a copy of $\\N$, the values of the numerals, followed by the non-standard elements.\nThe order type of every countable non-standard model of $\\mathsf{PA}$ is $\\omega + (\\omega^* + \\omega) \\cdot \\eta$, where $\\omega^*$ is the reversed order of $\\N$ and $\\eta$ is the order type of $\\Q$.\nBy [[Tennenbaum's theorem|W6D5ZJAY9D]], no countable non-standard model of $\\mathsf{PA}$ has computable addition or multiplication.\n\n## Explanations\n\n### L1 human\n\nThe axioms of arithmetic, written in first-order logic, do not pin down the natural numbers.\nThere are other structures that satisfy every one of the axioms but contain \"infinite\" numbers larger than $0, 1, 2, \\ldots$.\nThe compactness theorem builds one by asking for a number bigger than each ordinary number, one at a time.\nInside such a model, the infinite numbers look like ordinary numbers: each has a successor, a predecessor, and so on.\n",
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