- #1
quantum123
- 306
- 1
Today while day dreaming I discovered something interesting. I can prove a=a.
Here's how:
You can prove P=>P using natural deduction rules.(=> Intro)
So you can prove that x is an element of a => x is an element of a
Hence a is subset of a, and vice versa.
By ZFC axiom of extension, a=a
So a=a need not be an axiom, because it can be proven. In this sense, equality is not the fundamental concept. Set membership is.
Here's how:
You can prove P=>P using natural deduction rules.(=> Intro)
So you can prove that x is an element of a => x is an element of a
Hence a is subset of a, and vice versa.
By ZFC axiom of extension, a=a
So a=a need not be an axiom, because it can be proven. In this sense, equality is not the fundamental concept. Set membership is.