nate2823

It depends on what you mean by "simple". The Diagonal Lemma is extremely easy to state and prove (by which I mean that the proof itself has very few steps), but the proof looks like magic. That is to say, the standard proof doesn't really reveal how the Lemma was discovered in the first place.

Gödel Numbering, on the other hand, isn't too difficult to understand, but actually proving the Incompleteness Theorems (or whatever) usually requires pages and pages of boring, combinatorial proofs that one's Numbering works the way one wants it to. Conceptually, however, Gödel Numbering was a massive leap forward. As I understand it, before Gödel's paper in 1931, no... (read more)

There is a difference between the brain encoding concepts a certain way and concepts themselves being a certain way (or best studied at a certain level of abstraction, or best characterized in terms of necessary and sufficient conditions, etc.). Analogously, when I think of the number 2, I might associate it with certain typical memories, perceptions, other mathematical ideas, etc. etc. None of this has anything (well, almost anything) to do with the number 2 itself, but rather merely with my way of grasping it.

Concepts, like the number 2, are so-called “abstract objects”. They do not have spatio-temporal location. If your philosophical view implies that the question "Where is the number... (read more)