How would you explain [one of these theorems in the foundations of mathematics](http://en.wikipedia.org/wiki/Category%3ATheorems_in_the_foundations_of_mathematics) to a fellow colleague outside the field of logic (or rather to an undergraduate mathematics student) handwaving over the details?


The aforementioned link is by no means complete and following are some of the missing ones:


* [Beth definability theorem](http://en.wikipedia.org/wiki/Beth_definability)
* [Brouwer's fan theorem](http://en.wikipedia.org/wiki/K%25C3%25B6nig%2527s_lemma#Relationship_to_constructive_mathematics_and_compactness)
* [Craig interpolation theorem](http://en.wikipedia.org/wiki/Craig_interpolation)
* [Mostowski collapse lemma](http://en.wikipedia.org/wiki/Mostowski_collapse_lemma)
* [Omitting types theorem](http://en.wikipedia.org/wiki/Omitting_types_theorem#The_omitting_types_theorem)
* [Ramsey's theorem](http://en.wikipedia.org/wiki/Ramsey_theorem)
* [Shoenfield absoluteness theorem](http://en.wikipedia.org/wiki/Absoluteness#Shoenfield.27s_absoluteness_theorem)
* [Tarski's undefinabilty theorem](http://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem)
* [Vaught conjecture](http://en.wikipedia.org/wiki/Vaught_conjecture)
* [Well-ordering theorem](http://en.wikipedia.org/wiki/Well-ordering_theorem)


Due to the dearth of lucid papers on these topics -admittedly which is by no means easy to understand without pursuing years in college- my search has only yielded the following accounts:


* [(The Cartoon Guide to) Löb's Theorem by Eliezer Yudkowsky](http://yudkowsky.net/rational/lobs-theorem)
* [Konig's infinity lemma](http://www.maa.org/devlin/devlin_01_06.html)
* Zorns Lemma ([Axiom of Choice](http://www.math.vanderbilt.edu/~schectex/ccc/choice.html))


I am hoping to get a better understanding in expository terms of these logical theorems. Any reference that sheds light targeting students who had at most an introductory symbolic logic course at college level (such as myself) would be appreciated.



An undergraduate's guide to the foundational theorems of logic
**Edit:** This answer was given to the original formulation of the question, which asked for five-minute explanations for laypersons met on the street, rather than handwavy introductions for undergraduates. Maybe it still works though.




---


Since I have only 5 minutes to tell a layperson, I'd channel the late George Boolos and explain the second incompleteness theorem using only one-syllable words (*Mind* 103, pp. 1-3).



> 
> First of all, when I say "proved", what I will mean is "proved with the aid of
> the whole of math". Now then: two plus two is four, as you well know. And,
> of course, it can be proved that two plus two is four (proved, that is, with the
> aid of the whole of math, as I said, though in the case of two plus two, of
> course we do not need the whole of math to prove that it is four). And, as
> may not be quite so clear, it can be proved that it can be proved that two plus
> two is four, as well. And it can be proved that it can be proved that it can be
> proved that two plus two is four. And so on. In fact, if a claim can be proved,
> then it can be proved that the claim can be proved. And that too can be
> proved.
> 
> 
> Now, two plus two is not five. And it can be proved that two plus two is not
> five. And it can be proved that it can be proved that two plus two is not five,
> and so on.
> 
> 
> Thus: it can be proved that two plus two is not five. Can it be proved as well
> that two plus two is five? It would be a real blow to math, to say the least, if
> it could. If it could be proved that two plus two is five, then it could be
> proved that five is not five, and then there would be no claim that could not
> be proved, and math would be a lot of bunk.
> 
> 
> So, we now want to ask, can it be proved that it can't be proved that two plus
> two is five? Here's the shock: no, it can't. Or, to hedge a bit: if it can be
> proved that it can't be proved that two plus two is five, then it can be proved
> as well that two plus two is five, and math is a lot of bunk. In fact, if math is
> not a lot of bunk, then no claim of the form "claim X can't be proved" can be
> proved.
> 
> 
> So, if math is not a lot of bunk, then, though it can't be proved that two plus
> two is five, it can't be proved that it can't be proved that two plus two is five.
> By the way, in case you'd like to know: yes, it can be proved that if it can be
> proved that it can't be proved that two plus two is five, then it can be proved
> that two plus two is five.
> 
> 
> 


But if you were to start saying this to someone unsolicited, you might raise some eyebrows and be asked to leave the store or exit the bus. Proceed with care.


