Obections to Cantor's Theory (Wikipedia article)
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From: Robert Low on 26 Jul 2005 17:19 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?
From: Robert Low on 26 Jul 2005 17:22 Tony Orlow (aeo6) wrote: > Bijection between infinite sets as proving equal size. I disagree with that > assumption. That isn't an assumption, it's a definition. Until and unless you come up with an alternative definition, *and tell us what it is*, we have nothing meaningful to say to each other.
From: Barb Knox on 26 Jul 2005 17:30 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). -- --------------------------- | BBB b \ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | -----------------------------
From: Virgil on 26 Jul 2005 17:31 In article <MPG.1d504f1f6453d9f1989f98(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > I don't know what that means, but it sure sounds useless. Where ignorance is bliss, TO is estatic. > I am talking about actual quantities Like infinite naturals? > Gee, I wonder....
From: Virgil on 26 Jul 2005 17:35
In article <MPG.1d504f8d399cdb59989f99(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > That's what Tony is illustrating. Each of his infinite strings has a > > finite value. karl m > > > > > Not if it has a non-zero digit infintiely far to the left of the digital > point. > Then the value is infinite. To what number with only finitely many non-zero digits did TO add 1 in order to get this infinite value? Since adding one to a previous natural is the only way of getting a new natural, it must have occurred. |