An uncountable countable set
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From: Lester Zick on 8 Oct 2006 17:10 On Sun, 08 Oct 2006 13:27:00 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <1160309951.808867.33730(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > >> > The "number" of naturals is not a natural. And a "sum" such as the one >> > suggested, need not exist at all. >> >> Every natural is the sum of 1 + as many 1 at it has predecessors. An >> infinite sequence of predecessors gives an infinite result. If no >> infinite sequence of predecessors exists, then only a finite sequence >> does exist. > >To claim that the number of naturals must be a natural is to require >indiscriminately that sets be allowed to be members of themselves, which >causes worse logical problems than any "actual" infinities have ever >caused. How so pray tell? ~v~~
From: Tony Orlow on 8 Oct 2006 17:13 Virgil wrote: > In article <452948bf(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <45287184(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> > >>>> Strings with only finite bit positions. >>> Wrong!!! Strings with only finite bit positions can still have >>> infinitely many bit positions as there are infinitely many finite >>> naturals. Finite naturals always have a finite most significant bit >>> position and only finitely many non-zero digits. >> Incorrect. If every bit is in finite position, then there is no location >> in the string where it can be said to have an infinite value. > > But there can still be infinitely many bit positions. No, that would be a contradiction, since an infinite number of bits times a positive power of 2 yields an infinite product. > > In order to prohibit infinitely many bit positions one must have a most > significant bit position as well as a least significant bit postion. In the Dedekind sense, yes, but quantitatively, the string cannot assume an infinite value without a significant bit in a positively infinite position. > > A string is only finite if it has two ends, and a digit string without a > most significant bit position has, at most, one end, and is therefore, > endless and infinitely long. Given quantitative methods, Dedekind's definition of infinite in insufficient. >>>>> So obviously this rule, given a starting point of 0, a finite natural >>>>> and a finite-length bitstring, can never produce anything but another >>>>> finite-length bitstring as a successor. So you've proven that N >>>>> can contain only finite naturals. >>>>> >>>>> Unless you think that your rule allows an infinite bitstring successor >>>>> to be formed from some finite bitstring? >>>> You will not produce 1 bits in infinite positions without an infinite >>>> number of successions. >>> Which the naturals forbid. >> Correct. Given the restriction of finiteness on the values of the >> elements and the constant finite difference between successors, it is >> impossible for N to have "an infinite number of successions", even if >> the bound on the successions is undefinable. > > To say that each is finite does not require that there be only finitely > many of them. To have a linearly ordered set, with all elements within a finite number of steps from every other, does. > > That particular conflation of senses has been TO's stumbling block from > the beginning. he cannot conceive of an endless sequence of naturals > without having some member itself being endless, despite the many > contrary examples such as {1/n: n in N} I have specifically addressed such cases. You have consistently been a numbskull in forgetting the fact that each natural is separated from its closest neighbors by 1 unit. > >>>> 1) ....00000 is a number. >>> It is a digit string, and might be a numeral, but without a suitable >>> context it is not a number. >>> >> That's the equivalent of Peano Axiom 1. I declared it a number, and a >> member of the set. So, it's a number. > > Call it a TO-number, if you like, but you do not get to call things > numbers for others. I can declare it as a Peano set. Then it's a model of the naturals, eh? It's stupid to argue that binary numbers aren't numbers. Do you argue that ....0000 is anything other than 0? >>>> 2) If x is a number, then the successive number, formed by inverting the >>>> rightmost 0 and all 1's to the right of it, is also a number. >>> And if it is NaN then neither are any of those other things. >> I declared it a number. What makes Peano's '0' a number? > > Peano has considerably more clout than TO. At least among mathematicians. Peano has my respect.
From: Tony Orlow on 8 Oct 2006 17:36 David Marcus wrote: > Tony Orlow wrote: >> David R Tribble wrote: >>> Virgil wrote: >>>>> Except for the first 10 balls, each insertion follow a removal and with >>>>> no exceptions each removal follows an insertion. >>> Tony Orlow wrote: >>>>> Which is why you have to have -9 balls at some point, so you can add 10, >>>>> remove 1, and have an empty vase. >>> David R Tribble wrote: >>>>> "At some point". Is that at the last moment before noon, when the >>>>> last 10 balls are added to the vase? >>>>> >>> Tony Orlow wrote: >>>> Yes, at the end of the previous iteration. If the vase is to become >>>> empty, it must be according to the rules of the gedanken. >>> The rules don't mention a last moment. >>> >> The conclusion you come to is that the vase empties. As balls are >> removed one at a time, that implies there is a last ball removed, does >> it not? > > 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? With just this information, it is rather clear that this constitutes an infinite series, +(10-1)+(10-1)+(10-1)+..., or +9+9+9+..., which clearly diverges to some infinite number, of balls, that is. However, the gedanken as described contains other information. Each ball is marked with a unique natural number, so that there is a ball for every natural. Given this identification scheme, we are given two possible sequences of inserting 10 and removing 1 which differ only in the labels on the balls. In both, we insert 1-10 on the first iteration, 2-20 on the second, etc, or balls 10n+1 through 10n+10 for each iteration (starting at 0). The difference is in the removal part of the iteration, where in the first case we remove 1, then 2, then 3, etc, or ball n+1 in iteration n. In the second case we remove ball 1, then 11, then 21, etc, or ball 10n+1 in iteration n. Because transfinite set theory makes no distinction between aleph_0 and aleph_0/10, no distinction is made between the number of iterations it takes to add all the balls, and how many it takes to remove them. Erroneously, the unending sequences of required iterations are considered the same. So, in the first case, it is reasoned that all balls are inserted before "noon" and all balls are removed before "noon", and therefore nothing remains at noon. In the second case, balls 2 through 10 clearly remain, and all n+2 through n+10, for all n in N, which is an infinite set of balls remaining. So, apparently, one might surmise that you could perform experiment 2, have some infinite number of balls in the vase, and then switch the labels around afterwards and make them all disappear. It's just silliness. If the answer is clear with limited information, then additional information is irrelevant. Without the labels, we clearly have a divergent infinite series. With them, well, we have a bunch of hocus pocus, and I can't even really give any mathematical expression to that. It was easy without the labels. "Pay no attention to the man behind the vase. I am the Great and Powerful Aleph" ;) Tony
From: Lester Zick on 8 Oct 2006 17:40 On Sun, 08 Oct 2006 15:12:28 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Sat, 07 Oct 2006 23:45:23 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> mueckenh(a)rz.fh-augsburg.de wrote: >>>> Tony Orlow schrieb: >>>> >>>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>>> Tony Orlow schrieb: >>>>>> >>>>>> >>>>>>>>> Why not? Each and every number of the list terminates. That one is a number >>>>>>>>> that does *not* terminate. >>>>>>>>> >>>>>>>>> > If you think that 0.111... is a number, but not in the list, >>>>>>>> It is me who insists that it is not a representation of a number. >>>>>>> Well, Wolfgang, that sets us apart, though I agree it's not a "specific" >>>>>>> number. It's still some kind of quantitative expression, even if it's >>>>>>> unbounded. Would you agree that ...333>...111, given a digital number >>>>>>> system where 3>1? >>>>>> That is the similar to 0.333... > 0.111.... But all these >>>>>> representations exist only potentially, in my opinion. The difference >>>>>> is, that 0.333... can be shown to lie between two existing numbers, so >>>>>> we can calculate with it, while for ...333 this cannot be shown. >>>>> I think it can be shown to lie between ...111 and ...555, given that >>>>> each digit is greater than the corresponding digit in the first, and >>>>> less than the corresponding digit in the second. >>>> Yes, but only if we define, for instance, >>>> >>>> A n eps |N : 111...1 < 333...3 where n digits are symbolized in both >>>> cases. >>>> >>>> This approach would be comparable with the "measure" which gives >>>> >>>> A n eps |N : |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. >>>> >>>> I don't know whether these definitions are of any use, but I am sure >>>> that they are not less useful than Cantor's cardinality. >>>> >>>> Regards, WM >>>> >>>> . >>>> >>> My opinion about that is, if one wants to talk about what happens "at >>> infinity", that's the way that makes sense, not the measureless way of >>> abstract set theory. I trust limit concepts, but not limit ordinals. >> >> Tony, would it be fair to characterize what you're trying to say as >> that there is some kind of positive/negative crossover at infinity >> such that {-00, . . .,-1, 0, +1, . . . +00}? I haven't really been >> following this thread too closely so I'm trying to understand what >> you're after here in basic terms instead of the exact arguments >> involved. >> >> ~v~~ > >Hi Lester, how's thangs? Hey, Tony, pretty much as usual. >I wasn't saying that right here, but agreeing with Wolfgang that limit >concepts make sense, while the transfinitological approach doesn't. What >I did say was that there are two ways to view the number line, one where >oo and -oo are polar opposites, and a number circle where they are the >same. But there is no single real number line. There's a single rational/ irrational line but not even a single transcendental line, Tony. So on the surface I don't see what this speculation has to recommend it. And more to the point I don't see any way to effect a crossover in mechanical terms. > When it comes to reality, pretty much everything is in circles, >including finite number systems so this model makes some sense, even if >it doesn't in terms of limits of, say, powers. lim(x->-oo)=0 and >lim(x->oo)=oo for n^x where n>1, so there the two are different. Well in a way I'm not sure I disagree. I'm interested in this aspect of the problem mainly because I think that perhaps the issue both of you may be trying to address is the closing of otherwise open sets. Remember x->0 or plus or minus 00 doesn't mean x ever gets there. >When it comes down to this argument, Wolfgang's argument, I agree with >his logic concerning the naturals and the identity function between >element count and value. To me "element count" and the number of commas are the same. > He chooses then to reject infinities, while I >choose to retool them. I think the problem he and I both see clearly is >that using finiteness as a "bound" on the set simply doesn't tell you >anything, but rather clouds the entire subject. > >Did that answer your question? Kinda, Tony, although I really don't get all the arguing over vases and balls. And I don't really see finiteness as an issue if you view infinity as the number of infinitesimals between limits. Then infinities are contained whatever their magnitude. ~v~~
From: Virgil on 8 Oct 2006 18:07
In article <45296779(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452946ad(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >>>> Less than any finite distance. Silly! > >>> And what is the smallest finite distance? > >>> > > Note question not answered!! But the correct answer of zero would have > > blown TO's argument to blazes, so one can see why he would not care to > > answer it. > > I was away a couple days, but I answered this, not paying attention to > that apparent contradiction, since I don't consider 0 really a finite > number at all. It's a point with no measure, as every number is measured > relative to that point. If zero is not a number, how does TO keep the positive numbers separated from negatives? > > >> When you claim that there are ordinals greater than any finite ordinal, > >> are you obligated to name the largest finite ordinal? > > > > When you claim there is a LUB to the reals strictly between 0 and 1, > > are you required to name the largest real strictly between 0 and 1? > > No. That's my point. Why should I name the smallest object which is not > infinitesimal? That is not at all what I asked. So TO is doing his STRAW MAN fallacy thing again. > > > > > A "LUB" of the naturals does not have to be a natural any more than the > > LUB of the the reals strictly between 0 and 1 has to be a real strictly > > between 0 and 1. > > If it's a discrete set, then I disagree. The set {(n-1)/n: n in N} is a discrete set with a LUB which is not a member of the set. In fact every strictly increasing sequence having a LUB has a LUB which is not a member of the sequence. |