An uncountable countable set
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From: Han de Bruijn on 29 Sep 2006 03:00 stephen(a)nomail.com wrote: > In TO-matics, it is also possible to end up with > an empty vase by simply adding balls. According to TO-matics > > ..1111111111 = 1 + 1 + 1 + 1 + ... > > and > ..1111111111 + 1 = 0 > > So if you just keep on adding balls one at a time, > at some point, the number of balls becomes zero. > You have to add just the right number of balls. It is not > clear what that number is, but it is clear that it > exists in TO-matics. Commonly known with digital computers as "overflow" ? If TO-matics is an idealization of overflow, then it _is_ consistent anyway. Sad for you :-( Han de Bruijn
From: Han de Bruijn on 29 Sep 2006 03:02 Tony Orlow wrote: > Your axiom system is a farse. I'd rather think it is a farce. Han de Bruijn
From: Han de Bruijn on 29 Sep 2006 03:05 MoeBlee wrote: > Han de Bruijn wrote: > >>It's a priorities issue. Do axioms have to dictate what constructivism >>should be like? Should constructivism be tailored to the objectives of >>axiomatics? I think not. > > Fine, but if you don't give a formal system, then your mathematical > arguments are not subject to the objectivity of evaluation that > arguments backed up by formal systems are subject to. Exactly! Constructivism is not Formalism. Han de Bruijn
From: Han de Bruijn on 29 Sep 2006 03:10 stephen(a)nomail.com wrote: > So why is it okay to end up with zero balls, when you never remove > any at all, but it is not okay to end up with zero balls when > each ball is clearly removed at a definite time? Why is it not okay to approach the infinite otherwise than via the limit concept? Applied to a _finite_ sequence of events? Han de Bruijn
From: Han de Bruijn on 29 Sep 2006 03:11
MoeBlee wrote: > Tony Orlow wrote: > >>You might want to expand your reading. > > That's rich coming from a guy who hasn't read a single book on > mathematical logic or set theory. That's rich coming from a guy who hasn't read _anything else_ than books on mathematical logic or set theory. Han de Bruijn |