
Re: An alternative Axiom of Infinity?
Posted:
Feb 3, 2014 3:09 PM


On Monday, February 3, 2014 2:39:50 PM UTC5, Peter Percival wrote: > Dan Christensen wrote: > > > On Monday, February 3, 2014 1:34:56 PM UTC5, Peter Percival wrote: > > >> Dan Christensen wrote: > > >> > > >>> On Monday, February 3, 2014 1:00:14 PM UTC5, Peter Percival > > >>> wrote: > > >> > > >>>> Dan Christensen wrote: > > > > >>>>> As I have told you elsewhere today, ZF theorists use '2' and > > >>>>> '3' as convenient abbreviations for '{{}, {{}}}' and '{{}, > > >>>>> {{}}}, {{}, {{}}}}' respectively. > > > > >>>> Only those who choose the von Neumann definition. Zermelo > > >>>> didn't. > > > > >>> Again, as I also said there, get with the times, Peter. That was > > >>> 106 years ago! Time to move on. > > > > >> And yet you quote with approval Cauchy (1789  1857) on 0^0. > > > > > > Only because his recommendation is still widely used to today. Ask > > > just about any educated person. On the other hand, as far as I know, > > > your outdated definition of the ZF successor function is now no more > > > than a historical curiosity. > > > > I haven't defined anything. I've just been trying to point out to you > > that the von Neumann ordinals (or any others) are an addons to ZF not > > an unchallengeable part of it. >
Be that as it may, these days, the standard ZF Axiom of Infinity uses a the successor function S(x)={x,{x}}.
But even if you use the original definition of 106 years ago, you get wonky results like 2 is an element of 3.
Dan Download my DC Proof 2.0 software at http://www.dcproof.com Visit my new math blog at http://www.dcproof.wordpress.com

