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Zermelo-Fraenkel set theory

 
Nick George
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So, I read through the Zermelo-Fraenkel axioms, and to a certain degree I think I understand them. I'm still hung up on this concept:




Why not just have



This is certainly permissible by the axiom of pairing.

Now, I'm probably just demonstrating how little I understand, but what is the meaning of the having 3 = {1,2}?


I was thinking their method must lend themselves to proving arithmetic. So, lets see:


which is still just 3.


Let's try this:

what do we know about this set?

[ March 31, 2005: Message edited by: Nick George ]
 
Nick George
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I guess one benefit is that one could prove the existance of all the numbers inductively, i.e.,



but that doesn't seem especially exciting.
[ April 01, 2005: Message edited by: Nick George ]
 
Warren Dew
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Nick George:

I guess one benefit is that one could prove the existance of all the numbers inductively ... but that doesn't seem especially exciting.

That's probably because they made the mistake of teaching you arithmetic first, so you take it for granted. I think the idea behind "new math", back in the 1960s, was that Zermelo-Fraenkel set theory would be taught before arithmetic, since it was more fundamental. Then arithmetic would seem like something new and wonderful when the little four year olds derived it from set theory.

Unfortunately for me, new math didn't hit until I was in grade school, so I had the same reaction that you did. Thirty years later, I've rejected the axiom of choice as well.
 
Nick George
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but, what I'm wondering is, can one prove that 2 + 3 = 5 using this theory?


I read the article on the Axiom of Choice a few times, but I'm still a little fuzzy on it, I'm working on it.
 
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