Prove 1+1 = 2 in Fewer Pages & Win US$5 at Evo & I

[MathematicalPhysicist]

I don't understand why all this work took them something like three volumes of hundreds of pages, I read somewhere that Quine said their work could be much more brief in the number of pages, as in only 200 and something pages.

Philosophers...

 

[SW VandeCarr]

I don't understand why all this work took them something like three volumes of hundreds of pages, I read somewhere that Quine said their work could be much more brief in the number of pages, as in only 200 and something pages.

Philosophers...

What's wrong with this proof using Peano's axioms? If "one" is represented as the successor of zero, S(0), then it's sufficient to prove S(0)+S(0)=S(S(0)).

http://www.xamuel.com/proving-1-plus-1-equals-2/

 

[MathematicalPhysicist]

Well from the axioms it's just a matter of simple deduction.

But I hope you see the circularity of proving it, I mean the successor operator and its definition presuppose you know what are the natural numbers.

 

[SW VandeCarr]

Well from the axioms it's just a matter of simple deduction.

But I hope you see the circularity of proving it, I mean the successor operator and its definition presuppose you know what are the natural numbers.

I'm not sure what you mean by "circular". The Peano axioms give us a way to construct the natural numbers given the existence of zero. Does that mean it presupposes the existence of the natural numbers?

 

[MathematicalPhysicist]

You suppose that zero and one exist and that they are different, otherwise all the numbers will be equal to zero.

I mean the axiomatics just put in formal way what we know already from first grade or kindergarden, or whatever.

As Kronecker put it:"God gave us the natural numbers, all else is man made".

 

[SW VandeCarr]

You suppose that zero and one exist and that they are different, otherwise all the numbers will be equal to zero.

I mean the axiomatics just put in formal way what we know already from first grade or kindergarden, or whatever.

Be that as it may, most of mathematics is based on axioms, theorems and formal proofs. The OP asked for a proof, whether it was serious or not. 1+1=2 is a formal statement. That is, it's a well formed formula that admits a proof under the Peano axioms (using the Peano formalism). The proof may be simple, even trivial, but a proof was what was requested.

BTW, my understanding is that the Peano system only presupposes zero and a successor function exist. Under the Peano system 0 and S(0) are clearly not the same.

 

[Dmobb Jr.]

It is currently proved in x number of pages. I'm asking for the proof to be in x-1 pages. So three lines doesn't meet the criteria, sorry, FlexGunship.

(don't know the figure x, but it is certainly more than 100 - refer Principia Mathematica)

In Principia Mathematica, type theory is used, which is a very complicated set theory. In modern set theory it is quite easy to prove 1 + 1 = 2 as I showed in post 36. This is a formal proof like Russell's, except it uses the now standard ZFC axioms.Edit: I have not actually read Principia Mathematica so correct me if I am wrong.