Notion of internality in model theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:53:05Z http://mathoverflow.net/feeds/question/82289 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82289/notion-of-internality-in-model-theory Notion of internality in model theory Alphonse Dalbin 2011-11-30T16:40:12Z 2011-12-06T12:33:14Z <p>Good evening,</p> <p>Can someone explain to me the notion of internality in model theory (what it is, where it comes from...) ?</p> <p>Thank you</p> http://mathoverflow.net/questions/82289/notion-of-internality-in-model-theory/82364#82364 Answer by Alphonse Dalbin for Notion of internality in model theory Alphonse Dalbin 2011-12-01T10:50:03Z 2011-12-01T10:50:03Z <p>Thank you Buschi for your help but David Roberts is right en.wikipedia.org/wiki/Model_theory is what I meant. @David Roberts Thank you for your answer. I heard this term in a seminar. I can't give specific reference but I'm a beginner.</p> http://mathoverflow.net/questions/82289/notion-of-internality-in-model-theory/82724#82724 Answer by none for Notion of internality in model theory 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> http://mathoverflow.net/questions/82289/notion-of-internality-in-model-theory/82760#82760 Answer by John Baldwin for Notion of internality in model theory John Baldwin 2011-12-06T02:17:24Z 2011-12-06T03:48:07Z <p>The use of 'internality' in model theory that is most familiar to me is its use in nonstandard analysis. Look at 'internal' in the wikipedia article on non-standard analysis and see if that is what you remember.</p> http://mathoverflow.net/questions/82289/notion-of-internality-in-model-theory/82766#82766 Answer by Dave Marker for Notion of internality in model theory Dave Marker 2011-12-06T03:52:46Z 2011-12-06T12:33:14Z <p>The standard use in model theory is something like this. A partial type $p$ is internal to a type $q$ if there is a definable function $f$ such that any realization of $p$ is equal to $f(c_1,\dots,c_m)$ where $c_1,\dots,c_m$ are realizations of $q$.</p> <p>A typical example from differential fields: Let $X$ be the set of solutions of a linear differential equation of order $n$. Then $X$ is internal to the constants. Let $a_1,\dots,a_n$ be a fundamental system of solutions. Let $f(c_1,\dots,c_n)=\sum c_ia_i$. Then every element of $X$ is the image of an $n$-tuple of constants.</p>