Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 27 Jul 2005 16:48 In article <MPG.1d518e578b8d660d989faa(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > 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. Does TO claim that there are natural numbers which are not real numbers?
From: Tony Orlow on 27 Jul 2005 16:53 Daryl McCullough said: > Tony Orlow (aeo6) wrote: > > >Robert Low said: > > >> Are there rationals in [0,1) where p and/or q have to be infinite? > >> > >Interesting question. In order to have an infinite number of them, yes, you > >would need either an infinite number of numerators or an infinite number of > >denominators, both of which are whole numbers. > > Tony, you are just winging it, making up your answers as you go, > without any regard to whether they are consistent with what you've > already said. That's the point of rigorous definitions and axioms--they > keep the mathematician honest. He can't just make stuff up, he has to > follow the rules---even if they are rules that the mathematician made > up himself. If I am just winging it, I'm doing pretty good. Do you have a specific incosistency you'd like to mention? I don't see it. > > Your basic claim is that if S is a set of naturals, then either > S is finite, or S contains an infinite natural. Why in the world > do you claim that? It's provably false. I have explained it. Hopefully you have seen the explanations today, again. > > Let's go through the steps, and you can say which ones you > disagree with: > > 1. First, do you agree that if S is a finite set, then > the number of elements in S must be some (finite) natural > number? That's the definition of a finite set, more or less. (yeay, yeah bijections, I know) > > If that's not the case, then I don't know what you mean by > finite (and I suspect that neither do you). > > 2. Second, do you agree that if n is any natural number, > and S has exactly n+1 elements, then S has a largest element? Sure, if you can identify the elements in it. If you don't know any upper bound then you can't know it. There is a difference between finite and entirely known, as far as I can see. > > I don't know how you could believe otherwise. It's provable > by induction on n. Maybe you want to say that induction > is only sometimes true and sometimes false, depending on how > you feel at the moment? No, induction is good if it doesn't try to prove finiteness over an infinite number of implicit iterations that involve constant increase. What is this proof, now? I suppose I should take a look at it and tell you where the flaw is, not that you'll agree with anything I have to say. > > So, from 1 and 2, it follows that > > Let FN = the set of finite natural numbers. If FN is finite, > then FN has a largest element. I disagree with that blanket statement, as I have said many times. > > So your claim that FN is finite implies that there is some number > n that is the largest finite natural number. That means that > > n is finite > n+1 is infinite > > By any sane definition of "infinite" I would think that would be > impossible. Virgil has the same problem. In the case of the finite naturals, it simply does not hold true. Give the proof and I'll take a look at it. > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony
From: Virgil on 27 Jul 2005 16:52 In article <MPG.1d518ef9bea7a235989fac(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > malbrain(a)yahoo.com said: > > Dik T. Winter wrote: > > > In article <1122347583.518181.245300(a)g14g2000cwa.googlegroups.com> > > > malbrain(a)yahoo.com writes: > > > > > > > > The C language is defined by the C standard, as defined by ISO. There > > > > are no "unbounded" standard types in the C language. karl m > > > > > > Who is talking about C? > > > > Of the billions of computer systems deployed since the micro-computer > > revolution, the overwhelming majority are programmed with C. karl m > > > > > C IS the best language. I love C. But C is not mathematics. Knowledge of C no more makes one competent in mathematics than knowledge of French makes one competent in Chinese.
From: Virgil on 27 Jul 2005 16:54 In article <MPG.1d519043816d5e6989fad(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Obviously, i am rejecting some of your axioms and assumptions, so I > am not working with the same set as you are. Mathematics as a whole > should be viewed as a system of axioms that should be consistent > overall. That is the real litmus test of a given subsystem: external > consistency within the broader field of all math. By that test, Z, ZF, ZFC, NBG, etc, pass, at least conditionally, and TO's system fails miserably.
From: Daryl McCullough on 27 Jul 2005 16:36
Tony Orlow (aeo6) says... >Martin Shobe said: >> There is no finite upper bound on how many times I can apply it. >> However, each application occurs after only a finite number of >> previous applications. >Do you, or do you not, have an infinite set? Yes, and what that means is that it goes on without end. So if you generate the elements one at a time, there is never a point in which the entire infinite set will be generated. >Do you, or do you not, egenrate one member per iteration. >How many iterations are requires to produce an infinite set, one at a time? The process of generating an infinite set, one member at a time, never finishes. That's what it means to have an infinite set. But each *element* of the set is generated at some finite time. Once again, you are confused about quantifiers. Do you understand the difference between the following two statements: (Suppose that I'm generating the elements of some set, one at a time, one second per element): 1. For every element m, there exists a time t such that m is generated by time t. 2. There exists a time t such that for every element m, m is generated by time t. Do you think that 1 means the same thing as 2, in spite of the fact that the order of quantifiers is different? If you agree that they mean different things, can you think of a way that 1 can be true, but 2 is false? -- Daryl McCullough Ithaca, NY |