Good evening,
Can someone explain to me the notion of internality in model theory (what it is, where it comes from...) ?
Thank you
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$\begingroup$ I'm no in logic too muchm, but I guess that you have to know the internal logic in some categories/topos, see the P. Johnstone book o on topos theory $\endgroup$– Buschi SergioCommented Nov 30, 2011 at 16:44
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4$\begingroup$ @Buschi - I disagree. There is 'model theory' which has no connections to category theory whatsoever: en.wikipedia.org/wiki/Model_theory @Alphonse - is this what you meant? And please provide a reference as to where you heard about internality. People need to know at what level to pitch their answers. $\endgroup$– David Roberts ♦Commented Nov 30, 2011 at 21:29
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$\begingroup$ 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. $\endgroup$– user19660Commented Dec 1, 2011 at 10:50
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$\begingroup$ I'm entirely guessing what you're after of course, but are you perhaps thinking of inner model theory? en.wikipedia.org/wiki/Inner_model_theory $\endgroup$– Ed DeanCommented Dec 1, 2011 at 16:54
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2$\begingroup$ Internality can also show up if you're dealing with a model of a theory which can define notions like "finite," "well-ordered," etc. So, for example, we can have a model $M$ of ZFC set theory and some $x\in M$ such that $M\models$ "$x$ is finite," and yet still have infinitely many $y\in M$ with $M\models y\in x$; this would be an example of an infinite internally finite object. In nonstandard analysis, there is a similar distinction between internal sets, which are sets that a given model "sees," and sets in general. Is this what you're interested in? $\endgroup$– Noah SchweberCommented Dec 1, 2011 at 17:26
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3 Answers
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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 $q$.
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.
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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.
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1$\begingroup$ John, welcome to MO! Backticks play a special role in the markdown syntax that MO uses. Unfortunately, this conflicts with ordinary LaTeX usage. The rule of thumb is to only use LaTeX syntax when in math mode. $\endgroup$ Commented Dec 6, 2011 at 3:50
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Could you be thinking of Skolem's Paradox?
It is sometimes explained in terms of "internal sets", the sets that a model can "see". Example:
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".
If that's what you're looking for, then http://math.stackexchange.com is probably a better place than here for follow-up discussion.
