An uncountable countable set
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From: mueckenh on 25 Jul 2006 17:32 Michael Stemper schrieb: > >Is July 11, 2002 contemporary enough? Thomas Jech: Set Theory > >Stanford.htm, Stanford Encyclopedia of Philosophy: Until then, no one > >envisioned the possibility that infinities come in different sizes, and > >moreover, mathematicians had no use for "actual infinity." > > Okay, so your own reference shows that philosophers don't think that > mathematicians use the term "actual infinity". That sounds correct. T. Jech is one of the leading set theorist. He says up to the advent of set theory there was no use for actual infinity. Learn set theory and then come back to us. Regards, WM
From: Franziska Neugebauer on 25 Jul 2006 17:55 mueckenh(a)rz.fh-augsburg.de wrote: [mathematical use of "potential" vs. "actual" infinity (sic) in Fraenkel/Levy (1976)] >> > But he frequencay of mentionings may be high though unimporant. >> >> How will you know if you don't have it at hand? > > Because I read it. Why didn't you write "But the frequency _is_ high but unimportant"? Let me ask you a simple question: Is an author _using_ the term "wife" mathematically if he thanks his wife in the preface of his recent book on set theory? >> > The meaning of "infinity" in set theory is clearly the actual one. >> >> The other way round. Students first get in touch with plain infinite >> sets > > As you might obtain from the quoted statements, these sets are > actually infinite. > >> and then read from your posts about the ancient terms. To them >> the new terms which appear are the ancient ones. Hence from modern >> set theoretic standpoint one would clearly define: "Actual infinite" >> simply means infinite. > > Of course. A last you got it. You won't get a "t" from me. F. N. -- xyz
From: Franziska Neugebauer on 25 Jul 2006 18:20 mueckenh(a)rz.fh-augsburg.de wrote: > Michael Stemper schrieb: ,----[ <200607251652.k6PGqLb119264(a)walkabout.empros.com> ] | That's one possible interpretation. Another one is that modern books | don't use the terms "actual infinity" and "potential infinity" because | they aren't meaningful terms. `---- >> >Is July 11, 2002 contemporary enough? Thomas Jech: Set Theory >> >Stanford.htm, Stanford Encyclopedia of Philosophy: Until then, no >> >one envisioned the possibility that infinities come in different >> >sizes, and moreover, mathematicians had no use for "actual >> >infinity." >> >> Okay, so your own reference shows that philosophers don't think that >> mathematicians use the term "actual infinity". That sounds correct. > > T. Jech is one of the leading set theorist. > He says up to the advent of set theory there was no use for actual > infinity. 1. This utterance is not set theroretic use. 2. There is also no use of "actual" vs. "potential" infinity after the advent of set theory either. > Learn set theory and then come back to us. If you can't take criticism you'd better not post. F. N. -- xyz
From: Dik T. Winter on 25 Jul 2006 18:52 In article <1153829965.348565.324380(a)s13g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > Indeed, it is not a unary natural number. Perhaps it is the unary > > representation of something else, but not of a natural number. > > As 0.111... according to your assertions can be indexed by the set of > naturals and as this set has omega members, 0.111... is the unary > representation of omega. And so not of a natural number. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 25 Jul 2006 19:26
In article <1153861897.581734.123090(a)s13g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > If *every* digit can be indexed, then I need not put "finite" between > > > "every" and "sequence", because every indexing sequence is finite. A > > > digit which is not a finite sequence remote from the first digit cannot > > > be indexed and cannot be covered. > > > > Your deleting of "finite" (or "indexing") here is precisely what I said, > > the negation of the axiom of infinity. Any indexing sequence is finite. > > Right. But the sequence of *all* digits of 0.111... is neither an > > indexing sequence, nor a finite sequence. And it is precisely *that* > > sequence that can not be covered by a finite natural. Remember: there > > is a set that contains all naturals. There is (clearly) no natural that > > is the largest in that set. So there is also no last digit in the > > sequence of digits of 0.111..., where the digits are indexed by naturals. > > But every natural is finite. Every digit which can be indexed by > naturals is *by definition* in the true list (not the extended list) > and every digit which cannot be indexed is not in the true list. So > your attempt to explain that a number which is not in the true list > nevertheless could be indexed is completely unfounded. Let's state that more proper. The index of every digit that can be indexed is in the true list (of natural numbers). But later you use the word "index" to mean something completely different. You are stating there that if every digit of a number can be indexed the number *itself* is in the list. But that does not follow. You need a proof for that. And there is no such proof. As I stated earlier, every *terminating* number of which all digits can be indexed through numbers in the list is itself in the list. That is trivial. The last digit index position is the number itself. But that is *not* true for a non-terminating list, as it does not have a last index position. > > > A union of finite sequences is a finite sequence. > > > > An arbitrary union of finite sequences is not necessarily a finite > > sequence. This is similar to: the union of finite sets is a finite set. > > But this argues that the union of {1}, {1, 2}, {1, 2, 3}, ... is a > > finite set. Consider the elements of those sets as indices to the > > elements of the sequences. This is only true when N is finite, but that > > is the negation of the axiom of infinity. > > True is that the elements of n are finite. Hence there union cannot be > infinite, because every union of natural numbers is a natural number. Now you are talking about unions of natural numbers. Makes no sense to me in this context. I am not talking about unions of natural numbers, but about unions of sets of natural numbers. And telling that every union of natural numbers is a natural number presupposes some model of the natural numbers. What I am saying is independent of the model of natural numbers you use. > And every natural number is finite. You always intermingle size and > number (Anzahl). I was not speaking about ordinal number here, but about cardinal number. And, no, you intermingle set with element. According to the axiom of infinity the set of natural numbers does exist. It is easy to show that it can not be finite, and so it must be infinite. > > > If you believe that not the whole infinte sequence 0.111... can be > > > completely covered by a natural number, then please name a 1 in it, > > > which cannot be covered, be it more at the front edge or more in the > > > back area. Isn't this is a fair offer? > > > > No. Because I never argued such. > > You argued that every digit of K can be indexed. Right. > This means every digit > can be covered. Right. > K is nothing else than every digit. Right. > Hence your claim > unavoidably implies that K can be covered. Wrong. Suppose K can be covered by An, but we know that the n+1-st digit of K is *not* covered by An, so K is clearly not covered by An. Your reasoning is like this: for all p in N, K[p] = 1 (every digit of K can be indexed) for all p in N, for all n <= p, K[n] = Ap[n] (every digit of K can be covered) your conclusion is that there is an n such that for all p in N, K[p] = An[p] (K can be covered) but that is a false conclusion, not supported by any logic, because (obviously) K[n+1] != An[n+1]. > This implies that K is in > the true list. So the final conclusion is false. > Again: Your claim that ***every*** digit of K can be indexed but K > cannot be covered cannot hold unless you show a finite natural number > which indexes a digit n but does not cover all digits less than n. Wrong, see above. And: every digit of K can be indexed: for all p in N, K[p] = 1 K can not be covered: there is no n in N such that for all p in N, K[p] = An[p] there a natural number that does not cover all digits: there is a n such that for some p < n, K[p] != An[p] The first is obviously just a statement. The second is true from the definitions of K and An. The third is false, it is a clear contradiction of the second. Now show by what logic reasoning the third statement necessarily follows from the second. > Suppose I want to cover the n-th digit > > of 0.111..., I simply take An. That does *not* show that 0.111... can > > be covered by some An. Name 0.111... K. When K can be covered by any An > > that would mean: > > there is an n such that for all p in N, An[p] = K[p] (1) > > the negation of that is (and that is what I argue): > > there is no n such that for all p in N, An[p] = K[p] (2) > > but that does *not* mean: > > there is a p such that for all n in N, An[p] != K[p] (3) > > which you seem to imply, by some garbled logic. > > My logic is very clear: But unproven. > (every digit of n can be indexed) ==> (n is in the list) Where is the *proof* of this? The left hand states: for all n in N, K[p] = 1 your conclusion reads: there is an n such that An = K As I said before, this |