25
votes
5answers
2k views

Examples of “Non-Logical Theorems” Proven by Logic

I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. Question: What are some examples of mathematical logic solving open problem outside of ...
1
vote
2answers
131 views

Mathematical statements that cannot be proved or disproved [closed]

I've recently been reading about the continuum hypothesis and am fascinated by the fact that it cannot be proved or disproved, despite the fact that the statement itself is either true or false. What ...
1
vote
1answer
45 views

Interesting Identities in First-Order Logic

Are there more identities of this sort (http://en.wikipedia.org/wiki/First-order_logic#Provable_identities) that are interesting/non-trivial? It seems that most further work in first-order logic is ...
3
votes
1answer
253 views

Good book for logic self-study

I know a similar question has already been asked, but can anyone suggest a good book on mathematical logic that includes answers to exercises? I am looking for something that is conducive to ...
1
vote
3answers
122 views

Paradoxes without self-reference?

Most paradoxes involves self-reference, the only exception known to me is Yablo's paradox, however it is still debated if it is really without self-reference. So, I was wondering, are there other ...
0
votes
5answers
428 views

Unprovability results in ZFC

I am looking for real-world examples of unprovable statements in ZFC. So, not contrived logical formulae but statements that are of importance for ordinary mathematicians. Could you please point to ...
7
votes
5answers
464 views

Most astonishing applications of compactness theorem outside logic

The compactness theorem has a lot of applications to logic and model theory. I'm looking for applications. I'm looking for theorems in other areas of mathematics which seem at first sight to have ...
5
votes
7answers
564 views

What are some examples of subtle logical pitfalls?

Here's an example: Demonstrating that the assumption $A=B$ leads to a true statement is a vacuous truth. In order the show that $A=B$, prove that the difference $\Delta =A-B$ is zero. The subtle ...
14
votes
6answers
686 views

What are the theorems of mathematics proved by a computer so far?

By theorems, I mean the ones you can find in an undergraduate course of mathematics, not the ones you can find in a textbook of automated proofs. I mean by "proved by a computer" that an existing ...
3
votes
1answer
246 views

List of important and influential publications in logic

Wikipedia surprisingly does not contain a good list. Under philosophy of mathematics it is rather vacant too. A graduate level list is here, but I was curious more about the most influential papers. ...
3
votes
1answer
151 views

A Formal and Precise treatment of Simplification?

I am looking to gain a deeper understanding of, and increase my own skill in "Mathematical Simplification". But I've been finding the concept overly vague and haven't been able to find any good ...
3
votes
1answer
378 views

Short open problems or undecidable statements in ZFC

I guess Goldbach's conjecture is a good example of a short open problem in number theory, and "Goodstein sequences reach 1" is a good example of a statement undecidable from first-order Peano ...
26
votes
10answers
9k views

Good books on mathematical logic?

I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation?