most recent 30 from http://mathoverflow.net 2013-05-22T20:30:37Z http://mathoverflow.net/feeds/user/19753 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82289/notion-of-internality-in-model-theory/82724#82724 none 2011-12-05T19:12:53Z 2011-12-05T19:19:30Z

<p>Could you be thinking of Skolem's Paradox? </p> <ul> <li><a href="http://en.wikipedia.org/wiki/Skolem%27s_paradox" rel="nofollow">http://en.wikipedia.org/wiki/Skolem%27s_paradox</a></li> </ul> <p>It is sometimes explained in terms of "internal sets", the sets that a model can "see". Example:</p> <ul> <li><a href="http://en.wikipedia.org/wiki/Second-order_logic#Non-reducibility_to_first-order_logic" rel="nofollow">http://en.wikipedia.org/wiki/Second-order_logic#Non-reducibility_to_first-order_logic</a></li> </ul> <p>That's a description of why a theory containing the power set of the integers still has a countable model. The model doesn't actually contain the full powerset, but it also doesn't contain a bijection between its integers and its sets of integers, so "internally" the power set is uncountable even though it's countable "externally".</p> <p>If that's what you're looking for, then <a href="http://math.stackexchange.com" rel="nofollow">http://math.stackexchange.com</a> is probably a better place than here for follow-up discussion.</p>