An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: David Marcus on 23 Oct 2006 12:08 stephen(a)nomail.com wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > cbrown(a)cbrownsystems.com wrote: > >> What I honestly find baffling is your repeated claim that it doesn't > >> then logically follow from assumptions (2) and (5), that if a ball is > >> in the vase at /any/ time, it is a ball which is labelled with a > >> natural number; and so therefore the above statement is logically > >> equivalent to "there are no balls in the vase at t=0". > > > It is rather amazing. The logic seems to be that the limit of the number > > of balls in the vase as we approach noon is infinity, so the number of > > balls in the vase at noon must be infinity, but all numbered balls have > > been removed, therefore the infinity of balls in the vase at noon aren't > > numbered. It does have a sort of surreal appeal. > > With the added surreal twist that the limit of the number > of unnumbered balls in the vase as we approach noon is 0, > but the number of unnumbered balls in the vase at noon is > infinite. :) I guess consistency isn't required for surrealism. At least Tony doesn't draw the standard conclusion that Mathematics is inconsistent. He just pursues his contradictions wherever they may lead. -- David Marcus
From: David Marcus on 23 Oct 2006 12:14 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> (5) If a ball is placed in the vase at some time t, it must be in > >>> accordance with the description given in the problem: it must be a ball > >>> with a natural number n on it, and the time t at which it is placed in > >>> the vase must be -1/floor(n/10). > >>> > >>> <snip> > >> Those all look reasonable to me as I read them. I don't see any > >> statement regarding the fact that ten balls are added for every one > >> removed, though that can be surmised from the insertion and removal > >> schedule. That's the salient fact here. You never remove as many as you > >> add, so you can't end up empty. > > > > What about #5? It says that every ball in the vase has a natural number > > on it. Do you agree with that? > > That is in the problem statement. Therefore, nothing transpires at noon, > since -1/n<0 for all n e N. If I read this correctly, you agree that at all times every ball that is in the vase has a natural number on it, but at noon you say that there is a ball in the vase that does not have a natural number on it. Is that correct? > >> Either something happens an noon, or it doesn't. Where do you stand on > >> the matter? > > > > What does "something happens" mean, please? I really don't know what you > > mean. > > ??? Do you live in the universe, or in a static picture? When "something > happens" o an object, some property or condition of it "changes". That > occurs within some time period, which includes at least one moment. > There is no moment in this problem where the vase is emptying, > therefore, that never "occurs". If you are going to insist that time is > a crucial element of this problem, then you should at least be familiar > with the fact that it's a continuum, and that events occurs within > intervals of that continuum. Thanks. That explains what "something happens" means. Now, please explain what "emptying" means. -- David Marcus
From: MoeBlee on 23 Oct 2006 13:38 Lester Zick wrote: > >I don't insist that conversations must be about set theory. However, > >when various incorrect things are said about set theory, then I may see > >fit to comment, as well as I may see fit to mention set theoretic > >approaches to certain mathematical subjects. In the present case, a > >poster mentioned certain things about set theory, then I responded, > >then he asked me questions, then I gave answers, then you commented on > >my answers. That hardly presents me as insisting that the scope of this > >newsgroup be confined to set theory. > > Technically set "theory" is only a method not a theory since it can't > be proven true or false. By a 'theory' I mean a set of sentences closed under entailment. That's what *I* mean when I use the word in such contexts as this one. I deny that there are other senses of the word; I'm just telling you exactly what I mean in this context. > Moe, you get all bent > out of shape and claim they're spouting nonsense on the internet for > no better reason than they don't follow the party line on the subject. No, I have no problem with someone rejecting the axioms of any particular theory, thus to reject the theory. What I post against is people saying things that are incorrect about certain theories. Saying that one rejects the axioms or even that one rejects classical first order logic is fine with me (though, I'm interested in what alternatives one offers). But that is vastly different from saying untrue things about existing theories. MoeBlee
From: MoeBlee on 23 Oct 2006 13:51 Lester Zick wrote: > >I don't say that. And the definition of 'cardinality of' does use > >'equinumerosity', but the definition of 'equinumerosity' does not use > >'cardinality of', so there is not the circluarity you just arbitarily > >claim there to be. > > Arbitrarily? And maybe you'd like to apprise us of a definition for > equinumerosity which doesn't already assume cardinality for the > elements which are "equally numerous"? This conversation with you is ridiculous. Of course I can keep defining until I hit the primitives. You can keep asking me to do that, in a backwards motion, from the complex to the primitives, for a long time, and I finally will hit the primitives. But it would be much more efficient for me just to state the primitives and then demonstrate and define in a forward direction. But I'l indulge you just one more time: x is equinumerous with y <-> Ef(f is a bijection from x onto y). There is no assumption of having defined 'cardinality' prior to the above definition. Now you can say, "define 'bijection'". And then I do that. Then you say, "define 'function'". Then I do that. Finally, we reach the primitives. > I've already been over the subject of mathematical > >definitions with you in other threads. > > I've been over the issue with Randy and a few others but I can't > specifically recall visiting the issue with you. You don't recall. I do. In particular, you brought the conversation to a nadir with your mere harrumping about the criteria of non-creativity and eliminability that I mentioned. MoeBlee
From: MoeBlee on 23 Oct 2006 14:00
Lester Zick wrote: > On 19 Oct 2006 14:35:08 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >Lester Zick wrote: > >> And if set theorists ever get around to formulating a mathematical > >> theory do let us know. > > > >There are a whole bunch of recursively axiomatized mathematical > >theories that are set theories. Z set theory (and its variants) has > >been in good stead as an axiomatic theory since Skolem mentioned how to > >handle the previously too vague notion of a definite property. > > I have no idea what you think mathematics is, Moe, I can offer you many texbooks, book, articles, and journals that are just the beginning of the mathematics I would like to study. Meanwhile, I asked you for just a single reference to what you study. You replied by saying there is none (or words to that effect). > but when you use > phrases like "recursively axiomatized" they sound more like slogans > than mathematics. 'recursively axiomatized' is a precisely, rigorously defined predicate in mathematics. I can't help that it sounds to YOU like a slogan, since I can't help that you've never studied the subject. > Others that come to mind are "robust" and "well > ordered". I didn't use the word 'robust'. But 'well ordered' is a precisely, rigorously defined mathematical term. > I'm confindent sets and set methodologies have numerous > uses. I just can't tell what they might be from anything you've had to > say. All I hear are things like the "axiom of regularity" the "axiom > of infinity" and the "axiom of choice" which sound more like buzzwords > used purely to justify things no one can demonstrate true or false. They are formal axioms - precise formulas. They're definitely not just buzzwords, except as used by cranks. MoeBlee |