Could Set Theory Actually Prove 1+1=3?

[ilmareofthemai]

Hello all!
I recently read A Universe in Zero Words (it actually has words), a book about the history and influence of important equations. It discussed (if I understood correctly) that our current arithmetic operations are based on set theory, and that since set theory isn't entirely consistent, that a proof of the sum of one and one being equal to three might be produced.
Thoughts?
R

 

[MLP]

2+2

I would question what exactly is meant by saying that "set theory isn't entirely consistent".

 

[micromass]

Hello all!
I recently read A Universe in Zero Words (it actually has words), a book about the history and influence of important equations. It discussed (if I understood correctly) that our current arithmetic operations are based on set theory, and that since set theory isn't entirely consistent, that a proof of the sum of one and one being equal to three might be produced.
Thoughts?
R

That book is not entirely correct then. Set theory might be completely consistent, but the problem is that we don't know. We can never actually prove that set theory is consistent or not. So while most mathematicians guess that set theory is consistent, we can never know for certain. This is one of Godel's incompleteness theorems.

So if the book says that set theory isn't entirely consistent, then that is false. The right thing to say is that we don't know whether it is consistent or not. And if it is consistent: then we will never be able to prove that it is consistent. But yes, it can be that set theory is inconsistent. So it might happen that we produce a proof of 1+1=3.

 

[0xDEADBEEF]

Well and then maybe they meant naive set theory, which is inconsistent due to the "set of all sets... " stuff. But that has kind of been resolved.