An uncountable countable set
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From: mueckenh on 20 Aug 2006 09:52 Tony Orlow schrieb: > Hi WM. I agree entirely with your analysis regarding the interdependence > of the existence of the infinite set of naturals and the existence of > naturals of infinite value. This leads to the conclusion that the set of > finite naturals is finite, though unbounded, since as you say, there > cannot be an infinite number of differences of 1 where no two elements > are infinitely different. In order for an infinite difference to occur > between two natural, one must have an infinite value. Hi TO. This opinion is easily proved by Franziska's recognition that there are omega differences. It is simple to see that omega*1 = omega. > However, I wonder whether you would ever consider the existence of > infinite natural numbers. Then these numbers would not deserve the name "natural number". > After all, your argument says that either you > have a finite set OR you have infinite values in the set. Is the second > option objectionable for you? What would infinite values be good for? Distinguishable infinite values are provably inconsistent (see my binary tree) and to have one infinity the approved potential infinity is sufficient. Regards, WM
From: mueckenh on 20 Aug 2006 09:54 Tony Orlow schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Franziska Neugebauer schrieb: > > > >> You have agreed to that this limit does not exist ("There is no L in > >> N"). So the sum is at least _not_ _finite_ (not in omega). > >> > > There is no L, nowhere. It is wrong to say that L is in |N and it is > > wrong to say that L is larger than any n e |N. Actual infinity does not > > exist! I told you that already several times. > > But refuse to understand what "to exist" means. In no case infinitely > > many differeces of 1 can exist unless infinitely many diferences of 1 > > do exist. But that means an infinite size. > > > > Regards, WM. > > > > Well, here you answer my question just posed to you in my last post. > There is no infinite number, in your opinion. But, how do you reconcile > this with the infinity of reals in any unit interval? If, between any > two distinguishable reals there is a third distinguishable from the > first two, then does this not imply that we can subdivide the unit > interval infinitely, yielding an infinite set of interval endpoints, aka > reals? Isn't the set of reals in (0,1] infinite, and how do you > characterize this number? Their number is potentially infinite, i. e., you can "create" (Dedekind) and use as many as you want (but not more than 10^100 numbers because of practical reasons). Regards, WM
From: mueckenh on 20 Aug 2006 09:57 Virgil schrieb: > There are as many differences, n - 0, as there are naturals n \in N. > > Therefore one of the assumptions is wrong: Either there are not > > infinitely many numbers or there is at least one number which is > > infinite by size. > > False dichtomy, it ignores the actual case: that there are infinitely > many finite numbers. There are omega pairs which produce omega differences of 1. And according to set theory omega * 1 = omega and not less than omega. > > > > > P has the same cardinality as |N. But either there are not infinitely > > many numbers or there is at least one number which is infinite by size. > > That is Tony Orlow's delusion, too. > > > > > Do you agree that card (omega) /e omega? > > > > card (omega) = aleph_0 = the number of natural numbers and that cannot > > be infinite, as we have seen. > > WE have not seen any such thing. > > It is fairly easy to see that if there is no injection from the set of > naturals to any proper subset, then there must be a largest natural. > > And if there is such an injection, then the set is Dedekind infinite. There is an injection from the number of natural numbers into the sizes of natural numbers. You see it here: 1 11 111 .... The number of natural numbers (without *any* non-natural number) is omega. Regards, WM
From: Tony Orlow on 20 Aug 2006 09:58 Dik T. Winter wrote: > In article <ec8gln$oc9$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > ... > > > What is the wrongness of the definition? Let me refrase: > > > Let L be the list of all natural numbers > > > Let K be the sequence of digits such that for each p in L, K[p] = 1. > > > What is *wrong* with that definition? > > > > Sorry to interject, but could you tell me how long a sequence K is? > > Can you not determine it? I would think it follows immediately from the > definition. > > > If > > it's finite, then you only have a finite list L, and if it's countably > > infinite, then you have an uncountable list of naturals? > > What is your problem? From the definition I would say that the size > of the sequence of digits in K is equal to the size of the sequence > of natural numbers in L. Here is my problem. Given normal combinatorics, there are 2^n unique bit strings of length n. If you want any infinite set of bit strings, n must be infinite. So, you say you have a countably infinite sequence of bit positions, or strings of length aleph_0, and 2^aleph_0 strings in the set. Like the binary reals in [0,1), you end up with an uncountably infinite set of bit strings. But, you only want a countably infinite set of naturals, so, some vast majority of the strings produced with the countably infinite sequence must be eliminated. For each bit eliminated, the number of strings is halved, but no finite number of halvings will reduce c to aleph_0. So, you would have to remove some infinite number of bits. But, since you only started with a countably infinite sequence of bits, that leaves you with no bit positions. There is no K for which 2^K=aleph_0, but this is the requirement for the number of bits needed to list the naturals. It's not that there isn't an EXACT K such that 2^K=aleph_0, but there is nothing within a factor of 2, or even any TYPE of a number which K could be, since there is nothing between finite and countably infinite, and neither works. This is one major gaping hole in transfinite mathology that no amount of duct tape can fix. Smiles, TO
From: mueckenh on 20 Aug 2006 10:00
Virgil schrieb: > > There is no question that only countable sets of numbers can be > > constructed in the mathematical sense. > > That depends on what construction methods are allowed. No. > Inherent in the > axioms of ZF and beyond are the existences of uncountable sets. Therefore ZF and beyond is inherently false. > > > And as only a countable set of such sets can be constructed in > > reality > > As no part of mathematics can be "constructed in reality", All correct mathematics can be constructed in reality. But not that, of course, what you regard as mathematics. Regards, WM |