I am to hold a short (10-15 min) talk that is supposed to inspire first-year undergraduate students to (in the future) take on courses in mathematical logic. They have only been introduced to the very basics of propositional logic. An obvious candidate would be Gödel's incompleteness theorems, but there is already so much accessible material produced on this topic, so I would rather choose something else. I have considered something on the arithmetical hierarchy, but I find it difficult to point to a result in this topic that is mindblowing in the same sense as Gödel's theorems are.
Furthermore, I think it would be especially nice to introduce them to some topic in constructive mathematics. I have considered explaining why the seemingly controversial statement every function on the reals is continuous is not as crazy as it sounds. I'm thinking that perhaps this should catch their attention.
To summarise i have three questions (independent of each other)
- Are there any other counterintuitive results in general mathematical logic that you think would be met with as much enthusiasm as I believe that Gödel's incompleteness theorems would be (sorry, I do realise that this question is somewhat vague).
- What would be a good way to introduce the fundamental concepts of the arithmetical hierarchy in a captivating way that does not require any background in logic? Do you have any suggestions of results I can present that will both be accessible and (hopefully) interesting to a non-specialised undergraduate student.
- Do you have any suggestions for how I can introduce the concepts and ideas behind constructive mathematics? It would be nice to include some counterintuitive result, like the one I mentioned.
Any suggestions would be very much appreciated!
EDIT: I do realise that there are thousands of threads like this, but there is little on mathematical logic in these. In any case, if this has been asked before, then I apologise in advance.