Obections to Cantor's Theory (Wikipedia article)
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From: Daryl McCullough on 20 Jul 2005 14:20 Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >Essentially, the proof shows that no set can have a larger number >of naturals in it than the values of all the naturals in it. What does "the number of naturals in a set mean"? >If each finite n in N is the size of the set including all m<=n, then each of >them corresponds to a finite set. Right. Each natural number n corresponds to a finite set: the set of all natural numbers less than n. >How do we get an infinite set, then, if m<=n is finite for any finite >n in N? You get an infinite set by (1) Pick some starting number a. (2) Pick an operation f(x) that, given a number n, returns a new number that is greater than n. (3) Then form the set { a, f(a), f(f(a)), ... } That's guaranteed to be infinite. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 20 Jul 2005 14:24 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > >Daryl McCullough said: >> What definition of finite are you using? >Less than any infinite number. Not infinite. No, give a definition that *doesn't* use the words "infinite", "finite", "limitless", "limit", "boundless", etc. Give a *mathematical* definition. >I am sure you know what I mean. I am sure I don't. -- Daryl McCullough Ithaca, NY
From: Virgil on 20 Jul 2005 14:40 In article <MPG.1d4833f3aa19a72a989f40(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d471fc316a53825989f29(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > > 100100...100100100100 Of course, you will argue that this infinite > > > value is not a natural number, since all naturals are finite, but > > > that is clearly incorrect, as it is impossible to have an infinite > > > set of values all differing by a constant finite amount from their > > > neighbors, and not have an overall infinite difference between some > > > pair of them, indicating that at least one of them is infinite. > > > > TO's assumption of "infinite naturals" carefully avoids any > > consideration of the concomitant necessity of at some point having to > > add 1 to a finite number to produce an infinite number. > > > That is not only not a necessity, but an impossibility, as I have said many > times, and with which you agree. The problem of the largest finite or > smallest > infinite cannot be solved, and should not be the centerpiece of a theory that > hopes to achieve anything. One of the axioms for the set of naturals says that any set which contains 1 and contains n+1 whenever it contains n MUST contain ALL naturals, i.e., must be N. That means that the set of "finite" naturals is the set of all naturals. That point seems to be the pons asinorum that TO cannot cross.
From: stephen on 20 Jul 2005 14:39 In sci.math malbrain(a)yahoo.com wrote: > step...(a)nomail.com wrote: >> In sci.math malbrain(a)yahoo.com wrote: >> >> >> > Daryl McCullough wrote: >> >> Tony Orlow writes: >> >> > >> >> >Dik T. Winter said: >> >> >> >> >> Back on your horse again. Tell me about the binary numbers (extended to the >> >> >> left with 0's) where the leftmost 1 is in a finite position. Are all those >> >> >> numbers finite? Are there only finitely many of them? >> >> > >> >> >yes and yes >> >> >> >> What definition of "finite" are you using? >> >> > Main Entry: finite >> > Pronunciation: 'fI-"nIt >> > Function: adjective >> > Etymology: Middle English finit, from Latin finitus, past participle of >> > finire >> > 1 a : having definite or definable limits <finite number of >> > possibilities> b : having a limited nature or existence <finite beings> >> >> > This definition from webster should suffice. >> >> Definitions from webster rarely suffice for mathematical >> arguments. > They are illustrative of how the set of contradictions is resolved by > the majority. >> >> > Binary numbers with ones >> > in finite positions have a limited number of possibilities. >> >> > karl m >> >> What is that limit? How is it defined? Do you seriously >> believe that there are only a finite number of finite positions? >> >> A binary number with one's in finite positions can have an arbitray >> number of one's. There is no limit on the possibilities. >> The set of f finite binary strings is infinite. > Right. We're discussing the number of permutations of each of these > strings taken individually. > karl m No, Dik is asking about the set of binary strings whose leftmost 1 is in a finite position, i.e. {1, 10, 11, 100, 101, 110, 111, .... } or in Tony-speak {000...0001, 000...0010, 000...0011, 000...00100, 000...00101, ... } Tony thinks there is only a finite number of such strings. Stephen
From: Tony Orlow on 20 Jul 2005 14:46
Daryl McCullough said: > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > >> If TO's assumprtions were actually the case, there would have to be a > >> finite natural so large that adding 1 to it would produce an infinite > >> natural. But TO cannot produce either a largest finite nor a smallest > >> infinite, so the set of all finite naturals is already big enough. > >> > >We have been through all this before. You lay these requirement on me, but when > >I say you cannot have a smallest infinite omega > > But the Peano axioms say that *all* nonempty sets of naturals have > a smallest element. So if you say that there is no smallest infinite > natural, then that implies that there are *no* infinite naturals. > > -- > Daryl McCullough > Ithaca, NY > > all nonempty sets of finite naturals have a smallest element. All non-empty sets of infinite naturals have a largest element. You have a mirro-image situation here. -- Smiles, Tony |