# Zermelo-Fraenkel set theory

Nick George

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posted 11 years ago

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 ]

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 ]

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Nick George

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Warren Dew

blacksmith

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2

posted 11 years ago

Nick George:

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.

*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.