An uncountable countable set
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From: Han de Bruijn on 28 Sep 2006 04:11 Randy Poe wrote: > What I would say about emptying is that the vase is empty > at noon, but there is no identifiable time before noon at which > we can say "the last ball was taken out then". > > At any time before noon, there are balls in the vase. There > is no time we can say "there goes the last ball out" since there > is no last ball in. What I would say about emptying is that the balls must have been filled with liquor. And that you must have swallowed them all before you wrote this post. Han de Bruijn
From: Han de Bruijn on 28 Sep 2006 04:19 Virgil wrote: > In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Randy Poe wrote, about the Balls in a Vase problem: >> >>>It definitely empties, since every ball you put in is >>>later taken out. >> >>And _that_ individual calls himself a physicist? > > Does Han claim that there is any ball put in that is not taken out? Nonsense question. Noon doesn't exist in this problem. Han de Bruijn
From: mueckenh on 28 Sep 2006 05:51 Virgil schrieb: > Several sets may all have the common property of being pairwise > bijectable, but if any of their members are distinguishable from those > of another set then the sets are equally distinguishable. Each one of the sets expresses, represents, and *is* the same (cardinal) number. This is but *one* possible method, or better "schema of methods", to realize 3. > > > > > > You stated that you needed counting to determine the successor. That is > > > false. The successor is defined without any reference to counting. > > > > The successor function *is* counting (+1). > > Not to those who can't count. Successorship does not require numbers, it > only requires "next". How far would those who cannot count be able to find "the next"? Regards, WM
From: mueckenh on 28 Sep 2006 05:52 Virgil schrieb: > In article <1159186907.615747.304410(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > 1/3 is a number, properly defined, for instance, by the pair of numbers > > 1,3 or 2,6 or 3,9 etc. But 0.333... is not properly defined because you > > cannot index all positions, you cannot distinguish the positions of > > this number from those with finite sequences (and you cannot > > distinguish them from other infinte sequences which could exist, if one > > could exist). > > Def: 0.333... = lim_{n -> oo} Sum_{k = 1..n} 1/3^n Definitions (even correct definitions unlike this one) don't guarantee existence (I used above "to be properly defined" but I meant "to exist"). Example: The set of all sets is defined but is not existing. Regards, WM
From: mueckenh on 28 Sep 2006 06:00
Dik T. Winter schrieb: > > > Because there is only one set that contains *all* natural numbers. > > > > Why? Your assertion is without proof. > > I should have stated: "there is only one set that contains *all* natural > numbers and no other numbers". But that is easily proven once you > understand set theory. And you will easily see that this proof is rubbish once you understand a little more than set theory. "All natural numbers and some more natural numbers" is the same as "all natural numbers". For instance: Does the set o all natural numbers include 0? In old Greek it did not even include 1. In future it may include even -1. And in all these versions you insist that there be only one number 0.111...? > > > Why should there be only one number > > 0.111... ? By what property is this 0.111... different from all the > > numbers in the list? > > Because it has infinitely many digits? That is not a property which can distinguish it from the numbers of my list. If you think that 0.111... is a number, but not in the list, then you must be able to find a position which is different from those of the numbers of my list. But you can't. Why? Because all positions are in my list already - by definition. Hence your claim is null and void. > > > And why can't there be more than one number with > > infinitely many digits? You cannot answer these questions because > > already one infinite set is a contradiction. > > No, I can not answer this question because I have no idea what you mean > with a number with more than omega digits. Consider K = 0.111... . What > is K+1? Can you provide a definition (as I did for K)? k + omega is omega. And -k + omega is omega too. There is no well defined set. > > > Which index distinguishes 0.111... from all the numbers > > of the list? You cannot answer? > > I can. None. The axiom extensionality tells us that two sets are different, if they differ in at least one element. If 0.111... differs from number n, then it differs from all numbers m < n. As 0.111... is different from each number of the list, it also differs from each one which is smaller than another one. As every number of the list is smaller than another one, 0.111... cannot be covered by all numbers. Hence, it cannot be indexed by all list numbers. > So the number can be indexed. It is curious. You could also assert something like "I can shout louder than anybody else but there is nobody who cannot shout louder than me". But it is impossible to try to exorcise your burnt-in anti-logicism. > > > So we cannot answer which index > > distinguishes the many different infinite digit sequences 0.111... from > > each other. > > What different infinite digit sequences? I note that digit sequences are > countable, and so there is only one infinite digit sequence. I note that some sequences are finite, and so there is only one finite digit sequence? Is that really an argument? > > > > > Either: There is an index which distinguishes 0.111... from any number > > of the list. > > Or: There is a number which cannot be distinguished by indexes. > > But if there was one such number admitted, how could the existence of > > many of them be excluded? > > Because there is a single minimal countable but infinite set? The set with twice (or half) as many elements is not called a minimal countable but infinite set? Regards, WM |