Learn English – What does “you’re the one” mean in this context

For one ordinarily modifies the preceding NP. In practice it has much the same meaning as at least: that the NP is one entity occupying the particular syntactic/semantic role which the sentences assigns, and there may be other entities occupying that role.

I, for one, dislike the RNR construction. ... I dislike that construction, and there may be others who also dislike it.
I dislike the RNR construction, for one. ... I dislike that construction, and I may dislike others as well.

The phrase is always ‘parenthetical’: it’s really a shorthand way of qualifying your implication without making a big deal of explaining the qualification. In your sentence, for instance, the author is just making sure that we do not understand his particular notice of Gödel's response to this possibility as an indication that Gödel was the only mathematician who did not expect this result.

Mycroft Holmes: What do you think of MI5 security?
Mary Morstan: I think it would be a good idea.

What Mycroft means is

What do you think of the current level of security at MI5?

Mary understands the question, but she believes that materials at MI5 are not really protected very well. She may even think that the protection from espionage at MI5 is so poor that there's "no security" at all.

The meaning of her reply is:

I think that it would be a good idea for MI5 to set up a security system to protect their documents.

She is being sarcastic with this answer. She subverts the meaning of Mycroft's question in order to express her disdain regarding the abysmal level of security at MI5.

She reframes Mycroft's question as if it meant:

What do you think of the idea of setting up a security system at MI5?

It in her answer means "setting up a security system".