details re Z-R |- PA consistent
in [Math]
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From: FredJeffries on 16 Jul 2010 10:01 On Jul 15, 7:00 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > > Attempt #2: > > > {} _is_ a (Frege) natural number. > > > (This axiom has been challenged by Jeffries, though.) > > In term of rigorousness, this is even worse than your attempt #1 > > above! > > With Jeffries, Nguyen, and MoeBlee all questioning this axiom, > it's best that I drop it. I did not question the axiom. I asked for clarification of how your conclusion followed from the axiom. I have nothing against the axiom. It's unfortunate that you are dropping it because it's the only interesting thing that you've posted this month. http://en.wikipedia.org/wiki/New_Foundations#Models_of_NFU describes "a fairly simple method for producing models of NFU" (which I don't claim to understand on first reading) which concludes "If alpha is a natural number n, we get a model of NFU which claims that the universe is finite (it is externally infinite, of course)", which seems similar to another possibility I should have asked you about: even if the existence of a natural number H which is equal to {} implies that all "greater" natural numbers are also equal to {} (and hence equal to each other?) couldn't it be the case that H is some kind of non- standard number, so that there are infinitely many natural numbers none of which are equal to {}? Another intriguing sentence (which I also don't understand yet) I've come across in http://stanford.library.usyd.edu.au/entries/settheory-alternative/#CriNFU is that "the ordinals are not well-ordered in any set model of NFU".
From: FredJeffries on 16 Jul 2010 10:13 On Jul 15, 8:19 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Still, if anyone is still interested, here is how the set theorist > Randall > Holmes defines natural numbers: <SNIP> > To repeat, Holmes uses all these definitions in NFU. In ZFC, > none of these objects (except {{}}) are sets. Holmes's > definitions can be found at his website: > > http://math.boisestate.edu/~holmes/holmes/nf.html I had to dig a bit, but I eventually found Holmes's textbook "Elementary Set Theory with a Universal Set" at http://math.boisestate.edu/~holmes/holmes/head.pdf where the natural numbers are defined in chapter 12.
From: FredJeffries on 16 Jul 2010 10:26 On Jul 14, 2:42 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > the number googolplex exists, but not all the > classical naturals less than googolplex exist. I do not find it inconceivable that some non-peano-an system could be of some value, perhaps drawing on (or contributing to) the notion of Kolmogorov complexity -- as you have pointed out, numbers used in RSA cryptography are not arrived at by starting at 0 and adding 1 repeatedly.
From: FredJeffries on 16 Jul 2010 11:01 On Jul 14, 3:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 14, 9:01 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > On Jul 12, 4:57 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > So this represents a bias towards infinite sets. > > Elsewhere you state there is nothing infinite in the real world. So > > doesn't everyone have a natural bias towards finitism? Therefore by > > your principle shouldn't anyone who is able to overcome this naive > > finitism and intelligently discuss infinite sets be supported and > > commended rather than those who have hitherto been unable to overcome > > their finitistic bias? > > Touche. > > So Jeffries cleverly finds a way to make my argument sound as if > the _finitists_ are closed-minded, while the _infinitists_ are the > ones > who really are open-minded. Oh, man! Why do you always have to come with the negativity?! I said nothing about being close-minded (I apologize if it came across that way). Just because person A should be considered open-minded does not imply that person B should be considered close-minded. Just because B should be commended does not mean that A should be condemned. Throw off the shackles of the false dichotomy! And it wasn't all that clever because at least a half dozen other posters have noted the same thing. > > Thus, how can I tell who really are the open-minded posters? You can't. Get over it. > > On the surface, one might argue that infinitists accept finite sets > (since, after all, their infinite sets have finite subsets), but > finitists > don't accept infinite sets. But then by this logic, we should call > anyone who doesn't accept the largest of the large cardinals > (including those whose existence contradicts AC) closed-minded. <sarcasm> Instead of wasting your time whining, you ought to formalize this alternative logic of yours. You could get it published, make a name for yourself. </sarcasm> > > Instead, I want to consider those who shut out their opponents ideas > to be closed-minded. You want to put people in boxes? Fine. But may I point out that in the year or so that I've been responding to your posts you've made a number of factual claims about me, my qualifications, my job,... ALL of which are verifiably wrong. So why should you or I or anyone put any faith in your psychological assessments? > Presumably, there exist finitists who at least > are open-minded to infinite sets <caricature> Hey! Guess what! The original point of finitism (cf. Hilbert) was to prove the consistency of the use of the transfinite in mathematics. </caricature> >, even if the finitists themselves > can't formalize infinite ideas. Who says that they can't? According to you, at one time everyone was a finitist. Therefore it must have been a finitist who first formalized an infinite idea. <SNIP> > idea that proper subsets can have different "sizes" (Bigulosities)? I addressed this issue a couple weeks ago to which you have made no reply. Perhaps you didn't see it: http://groups.google.com/group/sci.math/msg/c39c4f58b2b37d91
From: FredJeffries on 16 Jul 2010 11:07
On Jul 14, 3:39 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > So now what? My stated reason for posting is to discuss theories > other than ZFC. If, as it turns out, the only posters discussing > alternatives to ZFC are dogmatists, and that the non-dogmatists are > not posting alternatives to ZFC, then I'm still going to stick with > those > who are dogmatic. I'd much rather have a discussion that I enjoy with > a dogmatist than one that I don't with a non-dogmatist. > > And so I'm going to continue to have discussions about alternatives > to ZFC, no matter how dogmatic the posters discussing it become. Hey! George Greene made six posts commenting on the Ed Nelson pager that you refer to so much. Why haven't you responded and discussed it? (After all, you don't find many people more dogmatic than George). |