Obections to Cantor's Theory (Wikipedia article)
in [Logic]
|
Prev: Derivations
Next: Simple yet Profound Metatheorem
From: malbrain on 27 Jul 2005 19:12 malbr...(a)yahoo.com wrote: > Tony Orlow (aeo6) wrote: > > So, the sum of a finite number of finite terms is infinite. Sure. > > Sometimes. The sum of infinite 1/2^n is one, for example. I gave you > this example yesterday in Barb's java program that generates all the > natural numbers. Perhaps you weren't paying attention? Ooops, I misread what you wrote as, "the sum of an infinite number of finite terms in finite." Negation dyslexia, again. karl m
From: malbrain on 27 Jul 2005 19:20 Tony Orlow (aeo6) wrote: > malbrain(a)yahoo.com said: > > Tony Orlow (aeo6) wrote: > > > malbrain(a)yahoo.com said: > > > > Virgil wrote: > > > > > In article <MPG.1d4863d52071fde5989f51(a)newsstand.cit.cornell.edu>, > > > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > > > > > and that I was trying to prove > > > > > > things about sets, not numbers, which is also bullshit, since I was > > > > > > proving a property regarding a set DEFINED by a natural number, which > > > > > > is ultimately a property of that number. > > > > > > > > > > But the set N is not defined by any one natural number > > > > > > > > Under Tony's theory, the number representing N is defined by a string > > > > of an infinite number of ones. Yes, more than one Turing machine can > > > > produce this. karl m > > > > > > > > > > > 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. > > > > There's already a STANDARD method for coding real numbers using the > > integers. It's published by the Institute of Electrical and Electronic > > Engineering. > Those aren't real numbers. Their infinite whole numbers. Actually, they're "floating point" numbers which model the real numbers for my purposes. karl m
From: malbrain on 27 Jul 2005 19:24 Tony Orlow (aeo6) wrote: > malbrain(a)yahoo.com said: > > Tony Orlow (aeo6) wrote: > > > I am not an expert in rings, nor am I going to sketch the axioms that you know > > > better than I, nor does it help the conversation when you snip the original > > > statement was repsonding to, which had soemthing to do with numbers being their > > > own multiples of more than 1, or something. It didn't make sense in the context > > > of normal quantitative addition. It was a vague guess as to what the idea was. > > > > > > The sum of an infinite number of 1's is infinite. That's all. > > > > Here's what Pascal had to say about his triangle in the 1600s: > > > > "Even though this proposition may have an infinite number of cases, I > > shall give a very short proof of it assuming two lemmas. The first, > > which is self evident, is that the proposition is valid for the second > > row. The second is that if the proposition is valid for any row then it > > must necessarily be valid for the following row. From this it can be > > seen that it is necessarily valid for all rows; for it is valid for the > > second row by the first lemma; then by the second lemma it must be true > > for the third row, and hence for the fourth, and so on to infinity" > > > > karl m > > > > > That sounds pretty inductive to me. Pascal was alright. You know, Pascal's > triangle is a table of the number of boundary features of each dimension on a > triaguloid of a given dimension? Like, how many tetrahedrons are on the > boundary of a 9D triaguloid......like that. Just an aside. So far I've only needed to work the generation of Pascal's triangle into a recursive program to enter Cal's CS61A course. karl m
From: Barb Knox on 27 Jul 2005 19:28 In article <MPG.1d51a192a95c0c4f989fba(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >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. It sounds like you're saying that it runs for an infinite amount of time (producing all the finite naturals) AND THEN it starts producing infinite ones. Right? -- --------------------------- | 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: malbrain on 27 Jul 2005 19:39
Tony Orlow (aeo6) wrote: > Chris Menzel said: > > On Tue, 26 Jul 2005 16:39:58 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > > > ... > > > The big problem in transfinite cardinality is the assumption that a > > > bijection necessarily indicates equal sizes for infinite sets, as it > > > does for finite sets. > > > > It's not an assumption, it's a definition, one that you reject. That's > > fine, but until you provide equally rigorous alternatives, it's the only > > game in town. And frankly, you've got no business rejecting it just > > because some of the consequences conflict with your intuitions. The > > mathematics of the infinite is occasionally surprising, especially when > > one doesn't have a complete understanding of the subject matter. > > > > > When the only way to form a bijection is with a mapping function, > > > > How else? > > > > > then that function needs to be taken into account. This nonsense > > > about an infinite set of finite whole numbers is pretty bad too, but > > > probably without any real consequences. > > > > You seem to agree that the set of whole numbers is infinite. But there > > was an inductive argument a few posts back that all the whole numbers > > are finite, and hence that the set of finite whole numbers is infinite. > > There was some real mathematics there. Why have you not responded to a > > mathematical proof that all the members of the infinite set of whole > > numbers are finite? It would be your chance to show everyone where the > > error in the argument lies. > > > > Chris Menzel > > > > > I have repsonded to that proof over and over. Where were you when we were > discussing the nature of inductive proof, and the implicit infinite loop in the > construction that no one seems to have considered, and the fact that adding 1 > an infinite number of times, as is done in the generation of the infinite set > of naturals, produces an infinite sum? If you look at Barb's java program that generates all the natural numbers, you don't find any infinite sums computed -- just an endless stream of natural numbers, each one of which is finite. Now, her "while" loop needs a little (external) work. Since it is written for the "accelerating virtual-java-machine" and the OPERATING SYSTEM can remember the START TIME and the timing of the program's FIRST ITERATION, then the OS knows when to cut loose with the BOLT-OF-LIGHTNING. karl m |