> On Wednesday, 27 August 2014 17:57:25 UTC+2, Ben Bacarisse wrote: > >> >> I claim there is no bijection from N to the set of all paths in the >> >> complete binary tree. <snip> >> Is the non-existence of a bijection from >> N to the set of all paths a theorem of WMaths or not? > > In mathematics every infinite set is in bijection with every infinite > set except sets which contain elements that cannot be defined. But > they do not belong to mathematics.
Interesting how you just won't answer direct questions. Is the non-existence of a bijection from N to the set of all paths a theorem of WMaths or not? Am I being unclear? Would you like me to define my terms? Shall I define the tree? The set of all paths? The set N? What a bijection is? You are being unclear -- I need to know if the set of all paths is covered by your get-out-of-jail card ("sets which contain elements that cannot be defined"). If you just said yes or no, I'd know.
But if anyone is still in any doubt that set theory and WMaths are the same (except for the words) here is the last piece of the puzzle. There *are* infinite set that don't biject with N, just as everyone has been saying all along. And, guess what? I bet they will turn out to be the same ones we all expect!
<snip> >> If it is not, then you've hit the jackpot: a formula true in set theory >> and false in WMaths. I will then gladly go cut and paste the proof I >> gave however many months ago it was, and you can show your proof of >> existence. Deal? > > I have show it often enough: Every item that can be defined or > expressed in mathematics is in the list of all finite expressions.
This is vacuous. All finite expressions are finite. So what?