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<p>arithmetic with ~G is also consistent. However, since ~G states that arithmetic is inconsistent, arithmetic with ~G will not be <a href="page.php?w=%3F-consistent">?-consistent</a> (because ~G is false and this violates ?-consistency).</p>

<p><big> From an ultraproduct </big></p>
<p>Another method for constructing a non-standard model of arithmetic is via an <a href="page.php?w=ultraproduct">ultraproduct</a>. A typical construction uses the set of all sequences of natural numbers, . Choose a <a href="page.php?w=Ultrafilter_on_a_set">free ultrafilter</a> on</p><p>
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