An uncountable countable set
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From: Michael Stemper on 12 Oct 2006 13:24 In article <1160613324.049353.270480(a)k70g2000cwa.googlegroups.com>, Ross A. Finlayson writes: >David Marcus wrote: >> Ross A. Finlayson wrote: >> > David Marcus wrote: >> > > So, you aren't saying that ZF is inconsistent. You are saying that you >> > > prefer to use a different system of axioms. Is that correct? >> > No, David, Dave, I say ZF is inconsistent. That doesn't mean all its >> > results are false. It just lets me say whatever I want about a less >> > inconsistent set of "axioms". >> So, you are saying that ZF is inconsistent. By "inconsistent", do you >> mean that ZF proves both P and not P, for some statement P? If so, >> please be explicit: what is the statement and what is the proof in ZF of >> it? >Build a set: {x: true}, it's a set, No, "{x: true}" is gibberish. Just putting down a couple of braces and throwing random text inside them does not generate a set. At least, no according to the axioms of ZF that you're trying to refute. -- Michael F. Stemper #include <Standard_Disclaimer> There is three erors in this sentence.
From: mueckenh on 12 Oct 2006 13:44 Randy Poe schrieb: > Tony Orlow wrote: > > Likewise, adding labels to the balls in this infinite case > > does not add any information as far as the quantity of balls. > > No, but what the labels do is let us talk about a particular > ball, to answer the question "is this ball removed"? By this nice example we see that this information is completely irrelevant, as irrelevant as the possibility to find a bijection between infinite sets and the mathematics built on this facility. > > If there is a ball which is not removed, whatever label > is applied to it, then it is still in the vase. > > If there is a ball which is removed, whatever label is > applied to it, then it is not in the vase. And if three balls without labels are in the vase, then they are inside independent of someone's knowledge about their names or their different properties. Don't forget: Cantor introduced set theory for use in physics and chemistry. Don't forget: Originally mathematics was created as a meaningful tool for science and economy. > > > That is > > entirely covered by the sequence of insertions and removals, quantitatively. > > Specifically, that for each particular ball (whatever you > want to label it), there is a time when it comes out. We are lucky that atoms do not carry labels, otherwise the universe would probably be empty already. Regards, WM
From: David Marcus on 12 Oct 2006 13:53 Ross A. Finlayson wrote: > David Marcus wrote: > > Ross A. Finlayson wrote: > > > David Marcus wrote: > > > > > the null axiom theory. > > > > > > > > So, you aren't saying that ZF is inconsistent. You are saying that you > > > > prefer to use a different system of axioms. Is that correct? > > > > > > Hi David, > > > > > > As you said, it's good to take these things one at a time, so, I am > > > happy to explain my opinion about set theory to you. > > > > > > Having been explaining set theory for some time, I was able to to take > > > part in some of these discussions here for example with everybody. > > > > > > No, David, Dave, I say ZF is inconsistent. That doesn't mean all its > > > results are false. It just lets me say whatever I want about a less > > > inconsistent set of "axioms". > > > > So, you are saying that ZF is inconsistent. By "inconsistent", do you > > mean that ZF proves both P and not P, for some statement P? If so, > > please be explicit: what is the statement and what is the proof in ZF of > > it? > > Build a set: {x: true}, it's a set, with unrestricted comprehension > it's not a set, without it's not either, so there are no true, or > provable, predicates in ZF. Sorry, I don't follow. Let's go one step at a time: What is your statement P? > A lot of the axioms, of ZF(C) set theory, can be seen to enrich or > broaden the set-theoretic universe. Regularity is not one of those. > > There is no universe in ZF. -- David Marcus
From: David Marcus on 12 Oct 2006 13:56 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> Please state the problem in English ("vase", "balls", "time", "remove") > >>>>>>> and also state your translation of the problem into Mathematics (sets, > >>>>>>> functions, numbers). > >>>>>> Given an unfillable vase and an infinite set of balls, we are to insert > >>>>>> 10 balls in the vase, remove 1, and repeat indefinitely. In order to > >>>>>> have a definite conclusion to this experiment in infinity, we will > >>>>>> perform the first iteration at a minute before noon, the next at a half > >>>>>> minute before noon, etc, so that iteration n (starting at 0) occurs at > >>>>>> noon-1/2^n) minutes, and the infinite sequence is done at noon. The > >>>>>> question is, what will we find in the vase at noon? > >>>>> OK. That is the English version. Now, what is the translation into > >>>>> Mathematics? > >>>> Can you only eat a crumb at a time? I gave you the infinite series > >>>> interpretation of the problem in that paragraph, right after you > >>>> snipped. Perhaps you should comment after each entire paragraph, or > >>>> after reading the entire post. I'm not much into answering the same > >>>> question multiple times per person. > >>> I snipped it because it wasn't a statement of the problem, as far as I > >>> could see, but rather various conclusions that one might draw. > >> I drew those conclusions from the statement of the problem, with and > >> without the labels. > > > > I'm sorry, but I can't separate your statement of the problem from your > > conclusions. Please give just the statement. > > The sequence of events consists of adding 10 and removing 1, an infinite > number of times. In other words, it's an infinite series of (+10-1). Sorry, but I don't quite understand. When you stated the problem in English, it ended with a question mark. But, your statement in Mathematics does not end with a question mark. If it is a problem/question, I think it should end with a question mark. Please give the statement of the problem in Mathematics. -- David Marcus
From: imaginatorium on 12 Oct 2006 13:58
Tony Orlow wrote: > David Marcus wrote: <snip-snop: the valiant shall see> > Time is actually irrelevant. If you are trying to determine the limit of the sequence of operations, time does appear to be irrelevant, yes. > ... The sequence is measured in iterations as > n->oo, and the number of balls in the vase at iteration n is represented > by sum(x=1->n: 9). The limit of this sum as x diverges also diverges in > linear fashion. Certainly does. I mean that sum from x=1 as x increases 2, 3, 4, ... without limit of (10-1) diverges. Let me ask you another question, Tony, as I don't think you answered the last one. Here is an argument, ending with a conclusion I don't personally swallow. Can you tell me at what point it goes wrong? (Or do you think it is valid?) Consider the function step0: R -> R mapping x to 0 if x<0 and mapping x to 1 if x>=0. FWIW, we can write this function in a C-like way (taking 'TRUE' and 'FALSE' to have the numeric values 1 and 0 respectively), so it is just a simple expression: step0(x) = (x>=0) OK, for n a positive integer, now consider the sequence of values of step0(p) for p=-1, -1/2, -1/3, ... -1/n, ... without end For any n, -1/n < 0, therefore step0(-1/n) = 0. So the sequence of values is simply the constant sequence 0, 0, 0, 0, ..... without end The limit of a constant sequence of values is the single value itself. Therefore lim(n->oo) step0(-1/n) = 0 By the Orlovian limit-swapping axiom, therefore: step0(lim(n->oo) -1/n) = 0 But lim (n->oo) -1/n = 0. Thus step0(0) = 0. But by definition, step0(0) = 1 Therefore 0 = 1. Corollary: set theory is inconsistent. Brian Chandler http://imaginatorium.org |