Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 26 Jul 2005 10:36 David Kastrup said: > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > N is finite if every member of N is finite. Show me how you get > > infinite S^L with finite S and L. (silence) > > By not having a limit on S and L. They can get arbitrarily large > values, but all of them are finite. There is no maximum. THERE IS NO > MAXIMUM!!!!!!! And that's what causes a set consisting just of finite > values to have an infinity of values. There is always one more value, > even though all of them are finite. > > Here is a game for you: you name a finite number, and then I name a > finite number. If I can't name a higher number, you win. > > You'll always lose, because the set we draw my values from is infinite > and I can always add 1 to the number you are naming. Yet the numbers > we name are all finite. > > We don't need infinitely large values for the natural numbers to be an > infinite set. All we need is arbitrarily large values. > > "infinitely large" would be a property associated with a single > member. "arbitrarily large" is a property associated with a > collection of members that form an infinite subset. > > The Peano axioms don't give us "infinitely large", but they _do_ give > us "arbitrarily large". > > Right. The Peano axioms don't fully address this issue, hence the confusion. However, you cannot argue that any particular S^L can be infinite with finite S and L, can you? If S is finite (2 for binary, etc) then we must allow infinite strings, in order to have infinite numbers of strings. If you want to say "unbounded", fine. The you have an "unbounded" set. But this term, "unbounded", is simply part of the Cantorian mind-bend, as I have come to realize. This distinction between "unbounded" and "infinite", when both mean "without end or bound", is purely artificial, and only serves to cover up the inconsistencies in the theory as a whole. So, whatever term you want to use for the natural numbers, whether they are finite, unbounded, potentially infinite, or whatever, your set size cannot be LARGER than the range of your values by more than 1, so it cannot be infinite and still have a finite range of values. -- Smiles, Tony
From: Tony Orlow on 26 Jul 2005 10:41 Han de Bruijn said: > Tony Orlow (aeo6) wrote: > > > In my book, there are 2^N > > log2's of natural numbers. That doesn't make it uncountable. It just makes it a > > bigger set. > > Please tell us what the title of your book is. > > Han de Bruijn > > Sorry, Han. That's an English colloquialism. It's as if everyone has a book they are writing, or taking notes in, in their head. I simply meant, the way I see it, it is perfectly easy to enumerate the members of a powerset of a set that is countable. But, I think I will title my book: "0 1 oo: The Square Circle". ;) -- Smiles, Tony
From: Tony Orlow on 26 Jul 2005 10:42 David Kastrup said: > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > Tony Orlow (aeo6) wrote: > > > >> In my book, there are 2^N log2's of natural numbers. That doesn't > >> make it uncountable. It just makes it a bigger set. > > > > Please tell us what the title of your book is. > > "Increasing your member size beyond natural". > > Just guessing. > > I don't need help with my member size, thanks. ;) -- Smiles, Tony
From: Tony Orlow on 26 Jul 2005 10:48 Han de Bruijn said: > Daryl McCullough wrote: > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > >>You ask this as if any set that is larger than any other set is > >>"uncountable". Do you consider the two terms, "larger" and > >>"uncountable" to be synonymous? > > > > Not quite. "Uncountable set" means "set with a larger cardinality > > than the set of naturals". > > > > Look, Tony. Your objections to standard mathematics all seem > > to revolve around disagreements about the meanings of words. > > Words such as "infinite", "finite", "larger", etc. *Real* > > mathematics doesn't depend on word choice. > > > > Can you express what you are trying to say without using > > any of those controversial words? Normal mathematics can. > > The use of the word "larger" to mean "having a greater > > cardinality" is just terminology. All of mathematics would > > go through just as well without *ever* using the word "larger". > > You could just as well use the word "more bloppity": > > > > By definition, a set S is said to be more bloppity than a set R > > if there is a 1-1 function from R to S, but there is no 1-1 function > > from S to R. > > > > Instead of using the term "size" to refer to sets, we could > > refer to the "bloppitude". > > > > Instead of using the words "infinite", we could use the term > > "mega-bloppity". > > > > Nothing of any importance about mathematics would change > > if we substituted different words for the basic concepts. > > > > In contrast, your arguments are about nothing *but* terminology. > > To me, that shows that there is no actual content to your arguments. > > An actual mathematical argument does not depend on word choice. > > > > As a challenge, see if you can express your claims about > > infinite sets, or infinite naturals, or set size, or whatever, > > *without* using the words "infinite", "larger", "size", etc. > > This clearly represents the formalist (Hilbertian) view on mathematics > as a "senseless game with symbols". > > The problem is that pro-Cantorians think that it is the only possible > view. Mathematicians like Brouwer, on the other hand, have repeatadly > emphasized that mathematics should have a _meaning_. But such a meaning > can only be attached if it is _outside_ the formalism. > > Han de Bruijn > > Thank you Han. That comes very close to what I have been feeling is a real problem in math these days. It seems like axioms are given a status that makes them unquestionable and almost incomprehensible, beyond the application of them. For instance, everyone's dismissal of the infinity inherent in the recursive nature of inductive proof is a sign that the axiom is not really understood, but accepted without question. For me, mathematics without meaning is unsatisfying, and symbolic manipulation without understanding is boring. With a closer examination of axioms and a willingness to revise them, I would hope that mathematicians could develop a fully integrated set of universal axioms which would cover all of math. But, that won't happen as long as axioms are taken without question, just because they "work". -- Smiles, Tony
From: David Kastrup on 26 Jul 2005 10:51
Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > David Kastrup wrote: > >> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: >> >>>And this is precisely what anti-Cantorians find unacceptable. IMHO >>>an "infinite" set cannot consistently have a "size". > >> This is a _perfectly_ valid point of view (as opposed to the views >> of those you sympathize with which are wildly inconsistent). > > Uh, uh. How do you know with which I symphathize? Do you keep a record > of my responses to everybody? Well, you use the word "anti-Cantorians" above. If this was not intended to mean mostly a particular set of some outspoken people in this Usenet group, it would appear that I misinterpreted this. > [ .. rest deleted .. ] > > OK. It seems that we finally have arrived somewhere. I have one > final question, though. Is it "legal" (according to mainstream > mathematics) to call a set "countable" if it can be brought in 1:1 > correspondence with the naturals? That's the usual usage of the word. > And uncountable otherwise? Well, in one direction: {1} can't be brought into 1:1 correspondence with the naturals, either. But if you can't map the naturals to cover the set exhaustively, than the usual term would be "uncountable". It may be that there is a bit of excluded middle: I think finite sets are usually classed neither as countable nor uncountable. Not sure about that, though. > Perhaps it's trivial for anyone, but I'm a bit at lost (due to those > heated "Cantor's diagonal argument" threads I think :-) Tony Orlow appears to assign some other meaning to those words, but he has not yet come up with a useful definition. Apart from him, I think most of the participants in the group seem to agree on the meaning. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum |