Is 1+1 always 2? I tried to prove that 1+1 is not always 2 with applying Goedel's incompleteness theorem as;

 

http://hecoaustralia.fortunecity.com/incompleteness/incomplete.htm

why not post it here instead of continually linking to your own site.

 

also, if you used the search function you would find that we have been over this several times.

Is the definition of 2 not that it is always 1+1?

Is the definition of 2 not that it is always 1+1?

 

Not always; in an axiomatic definition of the integers, you can prove that; it is defined in that case as the least integer greater than 1.

=Uncool-

And in what system is that not DEFINED as 2?

And in what system is that not DEFINED as 2?

2 is sometimes defined as 1+1, and sometimes as the successor of 1. The two must be proven equivalent. Otherwise, I don't think I understand your question...

=Uncool-

Is the definition of 2 not that it is always 1+1?

Yes, it's a definition. But it might be necessary to verify that definition is well-defined. Otherwise we could define 1+1=3 (it's obviously not well-defined, though.)

I don't understand why one would assume 1 and 1 wont equal 2. If you have 1 apple and 1 plum, you have 2 fruits. I don't know.. maybe I'm overlooking something obvious.

Btw this is my first post so Hello SFN!!