Proving 2+2=4 Using Field Axioms

[rafasaur]

I've looked around but haven't found anyway to prove 2+2=4. I'm pretty sure you need to use field axioms, but I just haven't found it yet. Is there a way to do it? Like showing a+a=2a? Or a+b=c? Like 1+1=2. Something like that.

Thanks!

 

[Hurkyl]

Directly.

What is the definition of 2? Of 4?

 

[ejnorman]

I've looked around but haven't found anyway to prove 2+2=4.

See Foundations of Analysis by Landau.

 

[mathwonk]

You just need a definition of 2, of 4 and of +.
Defn: 0 = empty set. 1 = {0}. 2 = {0,1} = {0,{0}}, 3 = {0,1,2} = {0,{0},{0,{0}}},
4 = {0,1,2,3} = {0,{0},{0,{0}}, {0,{0},{0,{0}}}}.
Addition is defined recursively. I.e. first adding one is defined. n + 1 = n union {n}.

i.e. 1+ 1 = {0} union {{0}} = {0,{0}} = 2.
2+1 = 2 union {2} = {0,1} union {2} = {0,1,2} = 3.
3+1 = {0,1,2}+1 = {0,1,2} union {3} = {0,1,2,3} = 4.
Assuming we have defined n+m then n + (m+1) = (n+m)+1.

now you have enough to do it. or keep reading.
So 2 + 2 = (2+1)+1 = 3+1 = 4.

Tata!

Aren’t you glad you asked? Basically it seems 4 = ((1+1)+1)+1. and 2 = (1+1).
so 2+2 = (1+1)+(1+1), so it boils down to associativity of +.

 

[battousai]

(1+1+1+1)=1+1+1+1
(1+1)+(1+1)=4(1)
2+2=4

Does that work? Sorry I'm a beginner when it comes to proofs.

 

[HallsofIvy]

You haven't said why "1+ 1+ 1+ 1" would be equal to "4(1)" or why 4(1) would be equal to 4. That was Mathwonk's point- this whole thing depends upon exactly how you define "+", "1", "2", and "4". You have just assumed basic arithmetic without giving any definitions. That is no different from just assuming that 2+ 2= 4.