Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 27 Jul 2005 14:16 Virgil said: > In article <MPG.1d5028071ff97742989f8f(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > Actually there are two ways to look at it. In unsigned binary, yes, an > > infinite > > number of 1's is the largest number possible. Since we start with all 0's > > representing 0, the size of the set, N, will be one more than 111...111. It > > will be 000...001:000...000, or one unit infinity. > > Until TO can produce unambiguous rules comparing sizes for all of these > alleged pseudostrings of binary digits, it is all garbage. Unless TO > wants to say that his "naturals" are not naturally ordered and not > orderable. > > For example, given 1000...010...001 and 100...0110...001, where each > ellipsis represents a psuedostring of infinitely many zeros, by what > rule does one compare the composite psuedostrings to determine which is > larger? > > Note that were the ellipses to each represent only finitely many zeros, > the answer would be relatively trivial. > > And what is the effect of concatenation of two or more such allegedly > infinite psuedostrings? > None of that affects the requirement that they be included in order to have an infinite set of whole numbers. None of it. -- Smiles, Tony
From: Tony Orlow on 27 Jul 2005 14:20 Virgil said: > In article <MPG.1d503908f774ebb9989f91(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > Not to mention the Banach-Tarski spheres. Doesn't that derivation > > constitute a disproof by contradiction? > > Not if one understands the distinction between matheamatical 3-space and > physical 3-space. The difference being that mathematical 3-space doesn't model anything real? Bull. > > > > Isn't the result absolutely nonsensical? > > It is certainly counter-intuitive, but so is the statement that any two > (purely geometric) line segments, regardless of length, contain exactly > the same "number" or points, which as know to and accepted by the > classic Greeks. That idea is also rejected by me, especially when one line segment is a subsegment of the other. It's nonsensical and unrealistic. > > Which is why intuition about infinities should be mistrusted. Uh, no, they shouldn't. > > > And yet, it is accepted, somehow, as truth that one can > > chop a ball into a finite number of pieces and reassemble them into > > two solid balls, each the same size as the original, despite all > > evidence and logic to the contrary. > > There is no "evidence" to the contrary, since no one claims that there > is any physical way of doing this. Superficially, logic might seem to > require that everything in mathematical geometry coincide with physical > reality, but there are hundreds of ways in which it does not. The > Banach-Tarski result is just another in a long line of such differences. Derived from this ill infinite set theory nonsense. > > > It's a clear sign of something > > wrong in the system, when it produces results like that. > > The thing that is "wrong" is that our "intuition" often misleads us. Maybe yours does. > > If intuition were perfect, we would not need logic to correct it when it > goes wrong. Much intuition is based on logic, in conjunction with non-verbal forms of thought. > > So that across the gates of mathematics there might well be written: > > "Abandon intuition, all ye who enter here." > That's just stupid. No comment. -- Smiles, Tony
From: Tony Orlow on 27 Jul 2005 14:26 Robert Low said: > Tony Orlow (aeo6) wrote: > > > No, I misread those statements. The first, if you are referring to my > > arguments, should be that there is an infinite set of whole numbers, and the > > second is that there is an infinite whole number in the set. Those two > > statements imply each other because of the constant finite difference between > > whole numbers. > > OK, so how many elements are there in the set of all finite > natural numbers? > Some finite, indeterminate number. You tell me the largest finite number, and that's the set size. It doesn't exist? Well, then, I can't help you. -- Smiles, Tony
From: malbrain on 27 Jul 2005 14:42 Tony Orlow (aeo6) wrote: > Robert Low said: > > Tony Orlow (aeo6) wrote: > > > > > No, I misread those statements. The first, if you are referring to my > > > arguments, should be that there is an infinite set of whole numbers, and the > > > second is that there is an infinite whole number in the set. Those two > > > statements imply each other because of the constant finite difference between > > > whole numbers. > > > > OK, so how many elements are there in the set of all finite > > natural numbers? > > > Some finite, indeterminate number. You tell me the largest finite number, and > that's the set size. It doesn't exist? Well, then, I can't help you. >From websters 1913 dictionary: De*ter"mi*nate (?), a. [L. determinatus, p. p. of determinare. See Determine.] 1. Having defined limits; not uncertain or arbitrary; fixed; established; definite. Thus "indeterminate" is the exact opposite of fine. You can't have it both ways. karl m
From: Tony Orlow on 27 Jul 2005 14:50
Barb Knox said: > In article <MPG.1d503dd82269292f989f93(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > [snip] > >> > >> >> >Barb Knox wrote: > >> > >> >> >> In article > >> > >> >> >> <MPG.1d4ecd45545679a8989f6b(a)newsstand.cit.cornell.edu>, > >> > >> >> >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > >> > >> >> >> [snip] > >> > >> >> >> > >> > >> >> >> >keep in mind thatinductive proof IS an infinite loop, > >> > >> >> >> >so that incrementing in the loop createsinfinite values, > >> > >> >> >> >and the quality of finiteness is not maintained > >> > >> >> >> >over thoseinfinite iterations of the loop. > >> > >> >> >> > >> > >> >> >> Using your computational view, consider the following > >> > >> >> >> infinite loop (using some unbounded-precision arithmetic > >> > >> >> >> system similar to java.math.BigInteger): > >> > >> >> >> > >> > >> >> >> for (i = 0; ; i++) { > >> > >> >> >> println(i); > >> > >> >> >> } > >> > >> >> >> > >> > >> >> >> Now, although this is an INFINITE loop, every value printed will > >> > >> >> >> be FINITE. Right? > [snip] > > >In any case, sure, the program will spit out finite numbers, > > Right. > > >snce it is a finite machine running in finite time. > > Wrong. Just like your notion of "infinite loop" over the naturals is > not encumbered by limited memory, neither is this program. Consider it > to be running on a machine with unbounded memory, so it could literally > run forever. > > >If the machine had infinite capacity > > Yes, that's the appropriate model. > > >and infinite funtime, it could conceivably produce infinite results. > > HOW?? No matter how long it runs, EVERY printed value is finite. > WHEN EXACTLY do you think it would start producing "infinite" values > (whatever those might look like). > > At the point that the runtime became infinite, which is obviously not an identifiable point. At what point DOES the runtime become infinite? A million years? a billion? If you have infinite runtime, starting with a finite amount of time, then you have infinite numbers, starting with finite ones. You can't have it both ways. -- Smiles, Tony |