Obections to Cantor's Theory (Wikipedia article)
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From: Daryl McCullough on 25 Jul 2005 11:07 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >Daryl McCullough said: >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >> >I asked for a definition of infinite, and no one could give me a >> >definition of that word. The best I could get was that an infinite >> >set can have a bijection with a proper subset, which is hardly a >> >definition of the word "infinite". >> >> On the contrary, that's a perfectly good definition of the concept >> "infinite set". >But not the word "infinite" on its own. There is no need for such a definition. Words are never used on their own, they are used in *sentences*. The role of a definition is to understand the meaning of *sentences* involving a word. Saying a set S is infinite if there is a bijection f between S and a proper subset of S is all that you need to know about the meanings of sentences about infinite sets. It doesn't tell you what "infinite natural" means, but that's actually because it is a meaningless phrase. -- Daryl McCullough Ithaca, NY
From: David Kastrup on 25 Jul 2005 11:33 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > The fact is that any set of whole numbers greater than or equal to 1 > MUST have at least one element with a value at least equal to the > size of the set. That is a fact and has been proven. Sulking won't make it so. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Daryl McCullough on 25 Jul 2005 11:23 Tony Orlow <aeo6(a)cornell.edu> said: >Daryl McCullough said: >> That's not true. If S is an infinite set of strings, then there >> is a difference between (1) There is no finite bound on >> the lengths of strings in S. (2) There is a string in S that is >> infinite. > >Yes, I understand the difference between those two statements, and >in this case the two are equivalent. Good. >If the length of strings is L and the symbol set has a >finite size of S, then you have S^L strings, which is infinite >IF AND ONLY IF L is infinite. Infinite Set <-> Infinite Element. You've just contradicted yourself. I thought you agreed that there is a difference between the two sentences (1) There is no finite bound on the lengths of strings in S. (2) There is a string in S that is infinite in length. In case (1), the set is infinite, but there is no infinite element. You seem to be thinking that there is some L characterizing the lengths of strings. There isn't. There are strings of length 1, there are strings of length 2. There are strings of length 3. Each string has a length, but there is no length associated with the set of finite strings as a *whole*. >> If you wrote these out as logical statements, you would see >> that you are mixing up the order of quantifiers: >> >> (1) forall b, exists s in S, >> (if b is a finite bound, then length(s) > b) >> >> (2) exists s in S, forall b >> (if b is a finite bound, then length(s) > b) >> >> Statement (1) says that the *set* S has no finite bound. >> Statement (2) says that S contains an *element* that has >> no finite bound. Those are two different statements. > >I don't need a lesson in logic, thanks. Yes, Tony, you certainly do. Since you are at Cornell, there are a number of courses you could take to fill the gaps in your knowledge. To learn about logic, I suggest starting with Math 281: Deductive Logic. To actually get up to speed on the theory of natural numbers, you need to take MATH 481 Mathematical Logic. Or you could just take advantage of the excellent faculty. Go to Dexter Kozen or Anil Nerode or Richard Shore and ask them about your claim that an infinite set must contain an infinite object. (But please don't tell them that I put you up to it.) -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 25 Jul 2005 11:26 Tony Orlow <aeo6(a)cornell.edu> said: >I don't need a lesson in logic, thanks. Yes, Tony. That is exactly what you need. You seem to be trying to learn some elementary facts about logic and mathematics by arguing on the internet. That isn't a very efficient way for a beginner to learn. If you're at Cornell, then go take a course in logic. It's an excellent institution. -- Daryl McCullough Ithaca, NY
From: Tony Orlow on 25 Jul 2005 12:04
Stephen Montgomery-Smith said: > Tony Orlow (aeo6) wrote: > > Stephen Montgomery-Smith said: > > > I love the way mathematikers love to spew insults instead of replies. The > > conclusion of the proof is that there isn't such a list, which is what I > > disagree with, so your statement is either totally confused, or deliberate > > obfuscation, which is more likely. > > > > Whatever you "got", you didn't catch it from me. > > I admit that I was poking fun at your argument. But I see so many other > people trying to discuss things with you at a more logical level, and > simply getting nowhere. You are definitely not "getting" what they have. > > Now let me, for a moment, put aside ideas like "correct" or "incorrect." > There is something in my very depths of my being, that when I read one > of the mathematikers proofs, that it just makes complete sense to me. I > can try to justify why I think our approach is the correct approach > using the collected thoughts of philosophers and logicians through the > ages, but in the final analysis, I just *know* that we are correct. > > You do come across as sincere in your differing opinions. I can only > suppose that in some strange manner that your brain is wired differently > than ours are. What seems completely logical and sensible to us, seems > to be nonsense to you, and conversely, what seems to be a proper > argument to you, is so weird and strange to us that we seem unable to > even know where to start it trying to disuade you from your point of view. > > Stephen > Well, it's nice to see someone who allows for different modes of thought among participants in these arenas. I do indeed think I am wired a little differently, working more with visual and constructive methods than with symbols and equations. As you say you simply "know" when what you are reading is correct, I feel the same in my position. However, that really doesn't fly in math or science, and one needs to test one's ideas to see if they pan out or fall flat. In science one can do an experiment or study. In math, one has to compare results from different methods to see if they agree, which is precisely what I am doing. Some of the conclusions of set theory directly contradict results from other areas of math, and are at the root of the problems in cardinality. When you speak about proofs "seeming" correct, you are talking about intuition, and I agree it is important and useful. This area of math is admittedly about the most counterintuitive around, and people tell me it takes a couple years to fully wrap one's head around it. That, to me, is an indication of problems. It's counterintuitive because it's incorrect. I know that rubs some the wrong way, especially if they have invested a lot into understanding it, and I'm sorry, but it's better to lance the boil thatn to let it fester any longer than necessary. As far as I can see, it has far-reaching harmful effects. But, I don't expect anyone to see what I see until after the fact. Correcting this problem, and the more general problem within the study which makes this disconnect possible, will lead eventually to the basic mathematical truths which translate directly into laws of physical and psychic reality, in my opinion. Of course, there's no point in trying to argue about that. I don't expect the world to see through my eyes, but I do expect mathematicians to be able to follow a couple of simple equations and accept minor corrections without getting nasty. Well, I did, anyway...... -- Smiles, Tony |