details re Z-R |- PA consistent
in [Math]
|
From: herbzet on 11 Jul 2010 17:20 Transfer Principle wrote: > MoeBlee wrote: > > Transfer Principle wrote: > > > > > Here we go again with a five-letter insult! But, as I've said many > > > times before, all the five-letter insults in the world aren't going to > > > convince me to abandon my raison d'etre for posting to sci.math. > > But you don't say now, > > > > "Okay, I see that I really went past what is reasonable and instead > > claimed something about MoeBlee that I had no basis to claim," as > > instead you slather on yet more of your self-justifications. > > My claim about MoeBlee's reason for posting has faced so much > hostility that I feel compelled to take it back. And so let me do so > right now: > > I unequivocally take back what I wrote about MoeBlee's reason > for starting a new thread. > > So how can I correct what I wrote? <Sigh> There's no fixing stupid. -- hz
From: Transfer Principle on 12 Jul 2010 19:57 On Jul 11, 9:55 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 10, 10:36 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > On the other hand, I do need to give credit where credit's due, and > > praise those who do overcome their biases to become more > > open-minded about alternate set theories. And so, in the most > > recent Herc thread (where he discusses his newly found finitism), > > I must commend the posters Fred Jeffries > I am curious. Exactly what biases have I overcome? (This is an honest > question: Since they are my own biases it is difficult for me to see > them on my own.) I assume that Jeffries believes in infinite sets (i.e., he regularly works in theories which prove their existence). So this represents a bias towards infinite sets. Yet he is able to have a civil discussion about finitism in the Herc thread without resorting to the usual five-letter insults. So Jeffries is a counterexample to my claim that those who are biased towards infinite sets consider those who are biased towards finite sets to deserve five-letter insults. For this, Jeffries should be commended. Thank you. > > and David Libert. In > > response to Dan Christensen's question about how finitists deal > > with sqrt(2), Libert provided some links to relevant discussions, > > while Jeffries a nearly 30-year old article. Although there's no > > guarantee that Herc has access to the article, I find this far > > preferable to doling out five-letter insults. > Seems to me that if there is a certain kind of poet [sic] that you prefer to > another kind that you might encourage more of the former by > intelligently responding to THEM rather than wasting your life posting > more of the kind you claim to not prefer Which kind is that? Biased posts? I do agree that I could find a way to write posts that are less biased. Indeed, I start out on the way to this goal by commending the posters who write unbiased posts, as I'm doing now. Note that I don't respond to Jeffries directly, since I don't have access to the journal that he mentions in that post. But Libert provides clickable links, and Srinivasan just added a post that mentions some of the same issues touched upon by Libert. So I can read those posts and be able to make that intelligent response to their posts.
From: Transfer Principle on 12 Jul 2010 21:04 On Jul 11, 12:28 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jul 10, 10:36 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > MoeBlee calls the other posters "dogmatic," and they call > > MoeBlee "dogmatic" in return. Also, both sides regularly > > insult each other all the time. Once again, that would make > > those on both sides "cranks." > Not by me. If Joe says Bob is dogmatic, that doesn't in itself make > Joe a crank. But if Joe says Bob is dogmatic, and Bob says Joe is dogmatic, why should I believe Joe that Bob is dogmatic? For I could equally believe Bob's claim that Joe is dogmatic. If two posters make symmetrical yet opposing claims, then absent any supporting information I see no reason to favor one claim over the other. > > On the other hand, I do need to give credit where credit's due, and > > praise those who do overcome their biases to become more > > open-minded about alternate set theories. And so, in the most > > recent Herc thread (where he discusses his newly found finitism), > Herc has an alternative set theory? Pray tell its language, axioms, > and definitions. In the current Herc thread, Herc states that he is replacing the Axiom of Infinity (i.e., he is starting with a standard theory such as ZF and replacing Infinity) with his "Axiom of No Infinity": "Herc's Axiom Of No Infinity If the Qth element of a sequence is the natural number Q, then the size of the sequence equals some element of the sequence." He also has what he calls his "Axiom of Pseudo Infinity": "Herc's Axiom Of Pseudo Infinity (based on above equation AOF) There is a set, I, that includes all the natural numbers that could physically be computed (before the end of the computer sustainable Universe)" Right now, I'm guessing (not _lying_, but _guessing_) that MoeBlee is going to criticize these axioms because they aren't symbolic enough. Fine then -- this makes me the bridge between the formalist MoeBlee and the informalist Herc. Earlier, back before Herc had converted to finitism, I was playing around with the following schema (in the language of ZFC plus an additional primitive constant symbol, which we might as well call "I" since this is what Herc calls it): (phi(0) & (Ax (phi(x) -> phi(xu{x})))) -> phi(I) Jesse Hughes pointed out that the schema proves that I is finite. It now appears that I equals some (von Neumann) natural number, though we can't be sure which one. But of course, what's to stop MoeBlee from forming the set Iu{I} to form a natural number not in I? Since all the axioms of ZF-Infinity are available, anyone can just take the successor to I to form a new natural number. In other words, it doesn't force I (the set of all Herc naturals) to equal N (the set of _all_ naturals). But we still might be able to salvage this theory, borrowing ideas from NBG or NFU. In NFU (where V is a _set_ and Frege's naturals are used instead of von Neumann's), the Axiom of Infinity can be stated as: {} is not a natural number. For using Frege naturals, the natural number n is the set of all sets of cardinality n. Since {} is not a natural number, it means that for every n, the set of all sets of cardinality n is not {} -- i.e., there is a set (a subset of V) of every finite cardinality n -- therefore the universe V must be infinite. In this case, we can write Herc's Axiom of No Infinity as: {} _is_ a (Frege) natural number. This forces the universe to be finite, which is Herc's stated purpose. But I'm not sure whether I want to resort to NFU to make Herc's theory work. There might also be a trick using NBG and Srinivasan to declare some object similar to D -- such as the class of all sets of cardinality greater than n -- to equal the empty set 0.
From: Jesse F. Hughes on 12 Jul 2010 22:29 Transfer Principle <lwalke3(a)lausd.net> writes: > But I'm not sure whether I want to resort to NFU to make Herc's > theory work. There might also be a trick using NBG and Srinivasan > to declare some object similar to D -- such as the class of all sets > of cardinality greater than n -- to equal the empty set 0. Herc doesn't have a theory. Herc is, as we all know, a radically deluded individual. Of all the persons that you "defend", Herc is clearly the worst choice. I'm not sure why you think that he is a mathematician at all. He is honestly incapable of rational argument. I don't say this to be mean or to defend my own biases regarding the existence of infinite sets (about which I really have no coherent philosophical views) but rather because we can all see that Herc is a disturbed individual who believes that he is Adam, that an unfortunate lady is his Eve and that satellites are tormenting him with sonar. These assertions are not wholly irrelevant to understanding his "mathematical" claims. -- "You are simply one person who persists in delusive thinking about your own relative importance, as you also rationalize data you do not wish to accept. I, unlike you, am a worldwide figure." -- James S. Harris, on self-delusions
From: Daryl McCullough on 12 Jul 2010 23:04
Jesse F. Hughes says... >Herc doesn't have a theory. Herc is, as we all know, a radically >deluded individual. > >Of all the persons that you "defend", Herc is clearly the worst choice. >I'm not sure why you think that he is a mathematician at all. He is >honestly incapable of rational argument. I don't say this to be mean or >to defend my own biases regarding the existence of infinite sets (about >which I really have no coherent philosophical views) but rather because >we can all see that Herc is a disturbed individual who believes that he >is Adam, that an unfortunate lady is his Eve and that satellites are >tormenting him with sonar. These assertions are not wholly irrelevant >to understanding his "mathematical" claims. But there must be an alternative foundation for mathematics and logic according to which Herc's beliefs are perfectly sensible. -- Daryl McCullough Ithaca, NY |