CANTOR DISPROOF <<<<<<<<<<<<<<<<
in [Logic]
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From: |-|ercules on 2 Jul 2010 04:18 "|-|ercules" <radgray123(a)yahoo.com> wrote... > "Transfer Principle" <lwalke3(a)lausd.net> wrote >> On Jun 28, 9:58 pm, herbzet <herb...(a)gmail.com> wrote: >>> Tim Little wrote: >>> > In short, just another crank. >>> Yeh, well, I actually defended him from the beginning when he >>> showed up in sci.logic with his project of providing formal >>> systems in which the assertions of various cranks can be >>> demonstrated. I think that is a valid intellectual exercise, >>> at least, and could provoke some actually interesting >>> discussions about fundamental assumptions we routinely make >>> in logic/math. >> >> I actually attempted to do this a few times in this thread, but >> when Herc stated that he was trying to use some form of >> induction, all I could muster was a schema of the form: >> >> (phi(.1) & An (phi(n 1's) -> phi(n+1 1's))) -> phi(.111...) >> >> Of course, such schemata are invalid in standard theory, and I >> even attempted to warn Herc that the majority of posters in this >> thread are likely to reject such schemata. >> >> I still believe that it's possible to find a workable schema that >> describe Herc's intuitions, but it won't be easy. > > phi( <[1] 2 3 4...> ) & An (phi ( <[1 2 ... n] n+1 n+2 ...>) -> phi( <[1 2 ... n n+1] n+2 n+3 ...> )) > -> > phi( <[1 2 3 4...]> ) > > Herc I was going to check yesterday WHAT TIME I made this 'discovery'. Just realized it was 4:20! The most famous time of day, on par with 3:14! I asked my Lord if that was a good thing, and scrolled to this comment on RicksBlog.com <<is clear to those that can see that they have the tools and the products to win the war.>> I find that amazing as Vertical Horizon's song "HE SAYS ALL THE RIGHT THINGS AT EXACTLY THE RIGHT TIME" is about me. The world runs to clockwork. I hereby rename Prefix Induction Schema to Stoners Formula! There is only 1 infinity!!!!!! Herc
From: Jesse F. Hughes on 2 Jul 2010 07:47 Transfer Principle <lwalke3(a)lausd.net> writes: > On Jun 30, 5:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> To give you an example: for a while, I did a little research in ZFA, >> anti-well-founded set theory. I liked the theory. I suppose that I >> found it preferable to ZFC at the time. I did not argue that ZFC was >> wrong, or give tired, silly arguments that the axiom of regularity is >> false because it's false in ZFA. The fact that ZFC was regarded as a >> foundation of mathematics while ZFA was not did not bother me. I worked >> in ZFA because it provided a nice setting for the things I was doing. >> That's rather different than the behaviors we see here. > > As Hughes is allowed to use a theory which proves the negation of > Regularity, likewise Herc should be allowed to use a theory which > proves the negation of Cantor's Theorem. I wonder what combination > of axioms and posting behavior will lead to Herc being granted > that freedom. Pardon me? Who denies him any such freedom? You failed to understand my point entirely. When Herc says ZFC is wrong or bad or some theorem in ZFC is wrong, then an argument ensues. If Herc were to say, here is a mathematical theory and here are some of its consequences (and his claims were sensible and correct), why should anyone argue? In any case, Herc has not been denied any freedoms -- at least, not by sci.math and not for his claims regarding Cantor. But when he says something that is apparently false or groundless, then he is corrected (and often insulted). Nonetheless, he's perfectly free to persist in his mathematical delusions. -- "I'd step through arguments in such detail that it was like I was teaching basic arithmetic and some poster would come back and act like I hadn't said anything that made sense. For a while I almost started to doubt myself." -- James S. Harris, so close and yet....
From: MoeBlee on 2 Jul 2010 17:40 On Jul 2, 12:12 am, Transfer Principle <lwal...(a)lausd.net> wrote: > at what point > beyond which a theory differs from ZFC such that we should no > longer call the objects which satisfy them sets? I don't know. But we can adopt certain definitions, such as: x is a set <-> (x=0 or Eyz y in x in z)) That will work as long as the theory defines '0' appropriately. We may consider at least four predicates class (has a member or is 0) set (as defined above) urelement (has no member but is not 0) proper class (class but not itself a member of anything) Then certain theories may prove which of those exist or do not exist. > This question > has come up in other threads as well. By this line of argument, > one could even point out that there are objects satisfying _ZF_ > that are different from those satisfying ZFC, namely those > without choice functions, are infinite yet Dedekind finite, and > so on. ZF does not prove there exists an infinite yet Dedekind infinite set, right? Rather, it is undecidable in ZF whether there exists such a thing. Same with sets without a choice function, right? > So I wonder, where is the line that when crossed we can > no longer call them sets, but, to use MoeBlee's name, "zets"? Just to be clear, this has NOTHING to do with what I meant about using the term 'zet'. > I want to discuss theories other than ZFC -- and hope that I > can do so with neither the five-letter insults nor the > behavior that inspires those words appearing. What engenders such remarks is not merely discussing alternative theories, but certain OTHER behavior. This has been pointed out to you hundreds and hundreds of times (literally); I don't know why you don't get it. MoeBlee
From: Chris Menzel on 2 Jul 2010 18:58 On Fri, 2 Jul 2010 14:40:34 -0700 (PDT), MoeBlee <jazzmobe(a)hotmail.com> said: > On Jul 2, 12:12 am, Transfer Principle <lwal...(a)lausd.net> wrote: > >> at what point beyond which a theory differs from ZFC such that we >> should no longer call the objects which satisfy them sets? > > I don't know. But we can adopt certain definitions, such as: > > x is a set <-> (x=0 or Eyz y in x in z)) > > That will work as long as the theory defines '0' appropriately. And surely extensionality is essential to our conception of set. >> This question has come up in other threads as well. By this line of >> argument, one could even point out that there are objects satisfying >> _ZF_ that are different from those satisfying ZFC, namely those >> without choice functions, are infinite yet Dedekind finite, and so >> on. > > ZF does not prove there exists an infinite yet Dedekind finite set, > right? Rather, it is undecidable in ZF whether there exists such a > thing. Same with sets without a choice function, right? Of course, so all that can be said is that ZF simply cannot prove that there are no such sets. It is misleading to say that there *are* such sets that "satisfy" ZF (whatever exactly "satisfy" is supposed to mean).
From: Transfer Principle on 2 Jul 2010 23:46
On Jul 2, 3:58 pm, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > On Fri, 2 Jul 2010 14:40:34 -0700 (PDT), MoeBlee <jazzm...(a)hotmail.com> > said: > > ZF does not prove there exists an infinite yet Dedekind finite set, > > right? Rather, it is undecidable in ZF whether there exists such a > > thing. Same with sets without a choice function, right? > Of course, so all that can be said is that ZF simply cannot prove that > there are no such sets. It is misleading to say that there *are* such > sets that "satisfy" ZF (whatever exactly "satisfy" is supposed to mean). I mean that there's a model of ZF with such sets. Of course, such a model would be a model of ZF+~AC, not a model of ZFC. Note that Hughes was the first to use the word "satisfy" in this manner: Hughes: Even in a not-too-distant from standard set theory like NF, it seems fairly evident to me that objects which could conceivably satisfy NF are somewhat different from objects that would satisfy, say, ZFC. So what does Hughes mean for an object to "satisfy" a theory? I interpreted it to mean that there exists a model of NF which proves the existence of the object, but not one of ZFC, but maybe Hughes had something else in mind. Thus, a D-finite T-infinite set "satisfies" the axioms of ZF in that it satisfies Extensionality (i.e., it's determined by its elements), satisfies Powerset and Union (since it has a powerset and union), and so on. But maybe there's a better word than "satisfies" to indicate that an object adheres to each of a list of axioms. All I wanted to know is whether objects whose existence is refuted by ZFC but can exist in other theories should still be called "sets." Menzel gives some criteria, namely that it should at least adhere to Extensionality, and that sets ought to contain elements (except 0) and be elements of other sets. |