Uncountability of the Rationals?
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From: Brian Chandler on 28 Jun 2010 06:37 Transfer Principle wrote: > On Jun 27, 12:23 am, Brian Chandler <imaginator...(a)despammed.com> > wrote: > > MoeBlee wrote: > > > I thought Orlow doesn't accept limit ordinals. Are there new updates > > > released on that matter this week? > > Under Transfer's new regime, it's possible we are only supposed to > > remember the last thing our Dear Leader said on any particular topic > > The point I'm trying to make is not that we are only supposed to > remember the last thing that TO said, but instead, if he made a > claim and was convinced that he was wrong, give him credit for > having learned that he was wrong. Grief, do you have to be so tedious? **Of course**, if Tony accepted that anything he said had been wrong, and presented a new idea, we could look at it. But in fact he has simply been regurgitating the same old stuff, with a few names changed, but absolutely nothing defined clearly. > > Definition schmefinition. Faced with my example sets: > > > > A = { "0", "10", "11", "100", ... } of (two-ended!) strings over > > > > alphabet {0,1} starting with 1 > > > > B = N ... the set of naturals (including 0), which we might represent > > > > in binary > > > > C = { 0, 10, 11, 100, ... } of integers whose decimal representation > > > > only includes digits 0 and 1 (no sign) > > The reason that Chandler brings up sets A, B, and C is in hopes > that TO would see that his Bigulosity can't assign a sensible > set size to these three sets, and so cardinality is a much more > sensible set size. This is not how I see it. Can't you write without putting words in other people's mouths? I do _not_ think "cardinality is a [better] set size". Sometimes people (non-cranks, I mean) say things like this, but I do not think it is true. The fact is that our two most obvious intuitions about the "size" of a finite set are: (a) if two sets match up they have the same "size"; (b) if we can pair off two sets so that one of them has elements "left over", then they have a different "size". The equivalence of these two intuitions for finite sets is captured by the pigeonhole principle. For infinite sets the pigeonhole principle does not apply; therefore the two intuitions are not equivalent; therefore if the name for that which is captured by the two equivalent intuitions is "size", we had better not use the notion of "size" for infinite sets. OF COURSE, a competent mathematician can define "size" in any particular context as something to do with either but not both of the intuitions, but in discussions with cranks many many problems will be avoided by not talking about "size" of infinite sets. I mention sets A, B, C because any sensible response from Tony ought to give us clues on whether his ideas make any real sense. So far, they don't seem to. (Repetition snipped) Brian Chandler
From: Jesse F. Hughes on 28 Jun 2010 08:42 Transfer Principle <lwalke3(a)lausd.net> writes: > In 2010, TO admits that these lead to inconsistency, > since they can lead to tav being both finite and > infinite or both in N+ and not in N+. And so, we avoid > the inconsistency by noting that tav is _not_ a number > (natural, nonstandard, or otherwise). You grant Tony too much. Recall the induction principle? That if phi is any formula not involving Tav, ( (E n in N+)phi(n) & (An)(phi(n) -> phi(n+1)) ) -> phi(tav)? Let phi(x) stand for "x is a number". Since the "fact" that tav is not a number plays so much work in Tony's reasoning, it must be the case that "x is a number" is expressible in Tony's (non-existent) theory. Work with me here, Walker. What follows from this? -- "1. Identical sets are identical. 2. Different sets are different. 3. Statements contradicting axioms 1 or 2 are false or malformed." -- James S. Harris offers some axioms
From: David R Tribble on 28 Jun 2010 13:07 Brian Chandler wrote: >> But anyway, we've also been told that the number of naturals varies >> anyway, depending on what base they're written in. > Transfer Principle wrote: > As I wrote earlier, Chandler's purpose of his sets A,B,C is to convince > TO that this is silly. The set of naturals ought to have the same size > no matter what base they're written in, but that the only way to > accomplish this is to allow sets to have the same size as some of > their proper subsets. > > But this does raise an interesting question that I've asked before, but > not received a sufficient answer for. Suppose we wanted to define a set > size in ZF with certain reasonable properties (that I'll delineate > later in this post). Can one prove that therefore, there must exist a > set with the same size as one of its proper subsets? Look, it is unacceptable to Tony that a (truly) infinite set a) does not contain an infinite member, or b) does not have an infinite "distance" separating at least two of its members, or c) is the same "size" as any of its proper subsets, or d) has an "absolute" size (member count). The last one is pretty well ingrained in Tony's thinking, and he is unlikely ever to change his mind about it. To do so, he would have to accept the existence of "absolute" infinite set sizes, i.e., omega and Aleph_0, and he just can't seem to jump that conceptual hurdle. All of his ideas about set sizes and unit infinities are based on mapping the elements of a set to the number line and then comparing that to the mapping of some other set (e.g., N+). In other words, for Tony there must be a geometrical basis for sets. On top of all of that is Tony's personal acceptance or rejection of a concept as the litmus test as to whether it is ultimately true or not. He's just as bad as JSH, AP, MH, Finlayson, and a few other cranks seen around these parts.
From: Transfer Principle on 29 Jun 2010 22:00 On Jun 28, 10:07 am, David R Tribble <da...(a)tribble.com> wrote: > Look, it is unacceptable to Tony that a (truly) infinite set > a) does not contain an infinite member, or > b) does not have an infinite "distance" separating at least two > of its members, or > c) is the same "size" as any of its proper subsets, or > d) has an "absolute" size (member count). > The last one is pretty well ingrained in Tony's thinking, and he > is unlikely ever to change his mind about it. To do so, he would > have to accept the existence of "absolute" infinite set sizes, > i.e., omega and Aleph_0, and he just can't seem to jump that > conceptual hurdle. This post, as well as some of the other recent posts in this thread, implies that the reason for bringing up posts from 2005 is that he still believes him in 2010. In other, words, no one gives TO credit for having learned anything in the past five years is that he _hasn't_ learned anything (at least not about a)-d)) in that time. OK, I'll grant that. If there were evidence that TO doesn't still hold his 2005 beliefs, then no one would be bringing up his 2005 posts. The fact that TO complains about the 2005 posts doesn't matter. He might claim that he doesn't believe what he wrote in 2005 anymore, but actions speak louder than words, and his actions show that he still believes now what he believed in 2005. But if we are going to judge TO based on his actions and not his words, then let me judge _Tribble_ and the other posters in this thread based on their actions. Based on their actions, I consider majority of posters to be closed-minded about theories other than ZFC -- especially theories which refute Tribble's a)-d) above. They tend to use five-letter insults against those who contradict a)-d) above, just as Tribble does in this very post: > [TO]'s just as bad as JSH, AP, MH, Finlayson, and a few > other cranks seen around these parts. They can _claim_ all they want to about how open-minded they are about theories other than ZFC refuting a)-d), but based on their behavior, they aren't. For example, WM often uses the negation of a) in his argument that infinite sets don't exist at all, and of course, we know how WM is usually treated in this newsgroup. I think that TO should have the same freedom to reject a)-d) that Tribble has to accept a)-d). And if we want to look for theories in which the negations of a)-d) are provable (or at least as many of the negations as possible without leading to inconsistency), then we should be able to do so without five-letter insults. But of course, fat chance of that ever happening. That those who reject a)-d) deserve insults is deeply ingrained in the thinking of those who so insult them.
From: Transfer Principle on 29 Jun 2010 22:50
On Jun 28, 5:42 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > In 2010, TO admits that these lead to inconsistency, > > since they can lead to tav being both finite and > > infinite or both in N+ and not in N+. And so, we avoid > > the inconsistency by noting that tav is _not_ a number > > (natural, nonstandard, or otherwise). > You grant Tony too much. > Recall the induction principle? That if phi is any formula not > involving Tav, > ( (E n in N+)phi(n) & (An)(phi(n) -> phi(n+1)) ) -> phi(tav)? > Let phi(x) stand for "x is a number". Since the "fact" that tav is not > a number plays so much work in Tony's reasoning, it must be the case > that "x is a number" is expressible in Tony's (non-existent) theory. > Work with me here, Walker. What follows from this? A contradiction? I suppose that's the answer that Hughes is looking for. So now we must ask ourselves the following questions: 1) Does this schema truly represent TO's current beliefs? 2) Does this schema truly lead to a contradiction? 3) Is TO intentionally trying to create an inconsistent theory? So let's consider these questions in turn: 1) When I first came up with this schema, it was based on what I was told was TO's beliefs. But this was based on what TO had posted back in 2005, not 2010. (See my response to Tribble for more info on the 2005 vs. 2010 issue.) So does the schema represent TO's 2010 beliefs? Assuming the answer is "yes," we can move on to: 2) Earlier, we mentioned some of the instances of this schema and their results: -- The schema proves that tav is a number. -- The schema proves that tav is finite. This may sound like a contradiction, but so far it isn't, since we haven't proved that tav is infinite yet. -- For each natural number n, the schema proves that n is in tav. This may sound like a contradiction, but there's a loophole. We think back to Ross Finlayson and his attempted proof that ZFC is inconsistent, as follows: ZFC proves that the set R of real numbers is uncountable, but by Lowenheim-Skolem, R has a countable model. RF claims that therefore ZFC is inconsistent. Of course, the usual response to RF is that _outside_ the model, R is countable, but _inside_ the model, R is uncountable. Thus, RF has not proved ~Con(ZFC). So perhaps it's possible that tav can contain all the natural numbers, yet still be finite. _Outside_ a model of the theory, tav is infinite, but _inside_ the model, tav is finite. And this would be the loophole that we seek. (In an old thread, there was discussion of a theory that is consistent as long as ZFC itself is, which proves the existence of a set which, from outside a model of the theory, contains all the naturals, all the reals, and even all the _ordinals_, yet inside the model, the set is finite!) But based on a current discussion between MoeBlee and Charlie-Boo in another thread, this loophole might not work. Once again, we have proved from this schema that: -- For each natural number n, it is provable that n is in tav. According to MoeBlee, we have: "(1): |- (allX)P(X) (2*): For all variables 'x', we have |- Px (1) iff (2*)" If we let P be the formula "x natural number -> xe(tav), then since we have already proved (2*), we can therefore conclude that (1) holds, and thus: -- It is provable that for each natural number n, n is in tav. But then this proves that _omega_ is a subset of tav. And any superset of omega must be infinite. Yet we already proved that tav must be finite. Hence a contradiction. Is my reasoning correct, and that I have really derived a contradiction from this schema? Assuming that the answer is "yes," we move on to question 3: 3) So far, we have established that this schema represents TO's beliefs and that it's inconsistent. But what we want to know is, is this intentional by TO? Is it true that TO knows that the existence of his tav leads to an inconsistent theory, yet he doesn't care? If this is so, then this is one of the few times that even _I_ will call TO "wrong." No one is going to be convinced to use an _inconsistent_ theory, not even _me_. But nonetheless, I still _refuse_ to believe that _any_ theory which refutes properties a)-d) from Tribble's post, or proves (b) from Chandler's post, must be inconsistent, just because the theory based on TO's posting so far is inconsistent. At this point, what I'd like to do is find a theory which proves as many of Chandler's (b) and the negations of a)-d) as possible without introducing an inconsistency. If we can do so, then perhaps the resulting theory would be one that TO is willing to accept, yet avoids the inconsistency that was found in TO's current theory. This is what I seek to do with the post in which I mention axioms for the relations "<=" and "~=". If the source of the inconsistency is this object called "tav," then let's first consider comparing sets via Chandler's (b) without worrying about any object called "tav." |