Obections to Cantor's Theory (Wikipedia article)
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From: imaginatorium on 21 Jul 2005 02:34 David Kastrup wrote: > 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. Seems harsh to niggle with what is a pretty clear exposition of the basic problem, but you have to be ultra-careful with words hereabouts. The "arbitrarily large" is a property not of the *collection of elements*, but rather a property of the individual elements, under some quantification - well, that's it: Tony has quantifier dyslexia (I really like that term!), so normal expressions don't work. Tony at some point will extract your statement, claim that "arbitrarily large" is the same as "unboundedly large", which is the same as "infinite", because Webster says the last term means "unbounded". No, sorry, I have no idea what the answer is - I don't think there is one. How could anything be clearer than what you wrote? Brian Chandler http://imaginatorium.org
From: Han de Bruijn on 21 Jul 2005 03:00 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
From: David Kastrup on 21 Jul 2005 03:07 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. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Robert Low on 21 Jul 2005 03:34 Jeffrey Ketland wrote: > One can define x<y in the first-order language of PA by the formula > (Ez)(z =/=0 & y = x+z). > The least number principle does express that (N,<) is a well-ordering. But in the weakened sense that it's only sets of naturals which make some P(n) true that have to have a least element. For example, the set of infinite elements of a non-standard model has no least element, but that's OK because there's no way of expressing 'n is infinite' in the language.
From: Han de Bruijn on 21 Jul 2005 03:38
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 |