## An undergraduate’s guide to the foundational theorems of logic [closed]

How would you explain one of these theorems 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:

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:

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.

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This should at best be a community wiki question, except I haven't figured out what is the question. – Benjamin Steinberg Jan 18 2012 at 4:22
@Benjamin: I think the question is "does anyone want to make Stone Soup"? – Yemon Choi Jan 18 2012 at 5:09
My own question is: isn't there a line in the FAQ about MO not being an encyclopaedia? – Yemon Choi Jan 18 2012 at 5:10
Even if one would accept the basic question, shouldn't logic be taken in a somewhat more narrow way. The Heine-Borel theorem, the Stone-Weierstrass theorem, and Urysohn's lemma are not results in logic. – Michael Greinecker Jan 18 2012 at 6:18
I have removed those three theorems. – Mahmud Jan 18 2012 at 7:36

## closed as not a real question by Henry Cohn, Benjamin Steinberg, Yemon Choi, Dmitri Pavlov, Emil JeřábekJan 18 2012 at 11:35

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.

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 A gem! Many thanks. – Mahmud Jan 18 2012 at 3:40

If you will accept a description by analogy, here is one for Craig's interpolation theorem: If there is a bridge needed to prove one statement from another, that bridge can be as narrow as the language which is shared by the two statements. I don't know if the proof can be conveyed in a manner as accessible as the description.