Non-standard model of arithmetic

<p>In <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a>, a non-standard model of arithmetic is a model of <a href="/facts/Peano_axioms/DhhDnNug">first-order Peano arithmetic</a> 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 <a href="/facts/Linearly_ordered/xnTtmuYJ">linearly ordered</a> and possess an <a href="/facts/Initial_segment/hSyGJ2WB">initial segment</a> <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> 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 <a href="/facts/Thoralf_Skolem/M7t2mfGx">Thoralf Skolem</a> (1934).
</p><p>Non-standard models of arithmetic exist only for the first-order formulation of the <a href="/facts/Peano_axioms/DhhDnNug">Peano axioms</a>; for the original second-order formulation, there is, up to isomorphism, only one model: the <a href="/facts/Natural_numbers/u65og8JE">natural numbers</a> themselves.
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Non-standard model of arithmetic open-in-new

Non-standard model of arithmetic