2
$\begingroup$

Good evening,

Can someone explain to me the notion of internality in model theory (what it is, where it comes from...) ?

Thank you

$\endgroup$
5
  • $\begingroup$ 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 $\endgroup$ Nov 30, 2011 at 16:44
  • 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
    Nov 30, 2011 at 21:29
  • $\begingroup$ 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. $\endgroup$
    – user19660
    Dec 1, 2011 at 10:50
  • $\begingroup$ 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 Dean
    Dec 1, 2011 at 16:54
  • 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? $\endgroup$ Dec 1, 2011 at 17:26

3 Answers 3

9
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
1
  • 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. $\endgroup$ Dec 6, 2011 at 3:50
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.