Cantor's Diagonalization Proof of the uncountability of the real numbersby Leucippus Tags: cantor, diagonalization, numbers, proof, real, uncountability 

#91
Feb712, 03:17 PM

Mentor
P: 16,570

This thread will deal with the theorem that states: under the given axioms of set theory, it is not true that the reals are countable. So in this thread we will accept the currect axiom system and deduce Cantor's theorem. This thread will not be used to question the axioms. If you want to challenge the axioms, you are free to do so in another thread. 



#92
Feb712, 03:18 PM

P: 1,025





#93
Feb712, 03:32 PM

P: 39

I agree that I was wrong. How's that? I AGREEnow that Cantor's proof is restricted by the the assumption of these axioms (although in truth, those axioms weren't in place in Cantor's day). They actually evolved out of the original intuitive work of Cantor. And this contributed to them becoming the formalized axioms that they have become today. Let's not forgot that there didn't even exist any such things as a formal set theory until the turn of the 20th century and Cantor's ideas played a very large role in that development. I have serious concerns with the whole development of set theory from that time period forward. And of course my ideas are necessarily going to need to be based on intuitive ideas in order to address these concerns. How could they be anything other than this? That can't very well be based on formally accepted axioms that haven't yet been written. 


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