wood burning stoves 2.0*
The moose likes Meaningless Drivel and the fly likes Zermelo-Fraenkel set theory Big Moose Saloon
  Search | Java FAQ | Recent Topics | Flagged Topics | Hot Topics | Zero Replies
Register / Login


JavaRanch » Java Forums » Other » Meaningless Drivel
Bookmark "Zermelo-Fraenkel set theory" Watch "Zermelo-Fraenkel set theory" New topic
Author

Zermelo-Fraenkel set theory

Nick George
Ranch Hand

Joined: Apr 04, 2004
Posts: 815
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 ]

I've heard it takes forever to grow a woman from the ground
Nick George
Ranch Hand

Joined: Apr 04, 2004
Posts: 815
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
blacksmith
Ranch Hand

Joined: Mar 04, 2004
Posts: 1332
    
    2
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
Ranch Hand

Joined: Apr 04, 2004
Posts: 815
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.
 
wood burning stoves
 
subject: Zermelo-Fraenkel set theory