Obections to Cantor's Theory (Wikipedia article)
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From: malbrain on 20 Jul 2005 16:32 aeo6 Tony Orlow wrote: > malbrain(a)yahoo.com said: > > > > > > Tony Orlow (aeo6) wrote: (...) > > > Now, I have heard the argument that inductive proof does not prove things for > > > an infinite number of steps, but only a finite number, but if this is the case, > > > then it does not prove anything for an entire infinite set of natural numbers. > > > > That's not what our axiom says. We get to prove things about the > > infinite set of natural numbers in a finite number of steps. Applying > > the axiom of induction is a single step in a leap of faith that we > > agree to before hand. > > > > > Either you agree that there are an infinite number of steps involved, or that > > > the set of naturals is finite, or that inductive proof does not prove a > > > property true for all natural numbers as Peano stated. > > > > That's not what our axiom says. It says that induction covers all the > > natural numbers in a single step, a single leap-of-faith. > > That's really not the case. It is a recursive proof where the property is > proven true for each element depending on its truth for the preceding element. > f(n)->f(n+1), for n=1 to oo. Otherwise, how do you think it proves things for > each and every n in N? We agree to the axiom of induction as a thing-for-itself. There is no "for n = 1 to oo" involved. karl m
From: Daryl McCullough on 20 Jul 2005 16:17 Robert Low says... >Funnily enough, there is a similar sounding statement that >is true in non-standard analysis: any set containing arbitrarily >large finite integers must also contain an infinite integer. >But in that game, the class of all finite integers isn't >a set :-) > >I only mentioned this because I thought it might muddy the >waters in an entertaining way... Actually, maybe Tony would be happier with nonstandard analysis... -- Daryl McCullough Ithaca, NY
From: Tony Orlow on 20 Jul 2005 16:38 Virgil said: > In article <MPG.1d4816be96c6f78d989f36(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > Cantorians seem to think infinity is simply infinity, > > > > > > Hard to fault them on that score. The fact that you have doubts > > > about whether infinity is infinity (what does "simply" add here?) > > > says quite a lot -- though *perhaps* if you were to try to > > > distinguish different senses of "infinite" with any sort of > > > mathematical precision there might be a point there. > > > That was a dishonest snip, really. Notice the comma? That means the > > sentence was not finished. The rest of it said that you do this, even > > in the midst of proving that it is not the case. > > There is a difference between even naturals and odd naturals, but there > are also differences between one even number and another. > > So to say that there are differences between infinite sets is equivalent > to saying that there are different even naturals. > > > > The diagonal argument assumes that the list is essentially square, > > Geometry does not enter into it. Such a "list" assumes that for every > natural there is a real. Decimal reprsentation assumes that for every > real and every natural, n, there is an nth decimal digit to the right of > the the decimal point for that real. > > > and can be > > traversed diagonally, but then proves that this is not the case, and > > yet, Cantorians still seem to insist that the diagonalization proves > > the reals cannot be listed, when what it proves is that there are > > more reals than naturals. > > The whole point is to prove that there are more reals than naturals. If > it does that, nothing else is needed. > If that's all you claimed, then I wouldn't be arguing, but that larger infinity is placed in some different class of "uncountables", and all "countables" are declared erroneously equal. That's where I diverge from the standard. besides, while there are certainly more reals than wholes, as can be clearly seen on the number line without all this bijectional blooplah, it is not necessarily the case that there are more reals between any two whole numbers than there are whole numbers in general. You could just as easily have the reals using half as many digits, or any other smaller infinity. The interpretation of the result is what is really at issue here. -- Smiles, Tony
From: Tony Orlow on 20 Jul 2005 16:40 malbrain(a)yahoo.com said: > > > Daryl McCullough wrote: > > Tony Orlow writes: > > > > > >Dik T. Winter said: > > > > >> Back on your horse again. Tell me about the binary numbers (extended = > to the > > >> left with 0's) where the leftmost 1 is in a finite position. Are all = > those > > >> numbers finite? Are there only finitely many of them? > > > > > >yes and yes > > > > What definition of "finite" are you using? > > Main Entry: fi=B7nite > Pronunciation: 'fI-"nIt > Function: adjective > Etymology: Middle English finit, from Latin finitus, past participle of > finire > 1 a : having definite or definable limits <finite number of > possibilities> b : having a limited nature or existence <finite beings> > > This definition from webster should suffice. Binary numbers with ones > in finite positions have a limited number of possibilities. > > karl m > > Thank you Karl. So it means "having a limit", which is pretty much an end. I have no problem with that definition. -- Smiles, Tony
From: David Kastrup on 20 Jul 2005 16:38
Helene.Boucher(a)wanadoo.fr writes: > Daryl McCullough wrote: > >> 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. >> > > Shouldn't there be one "p" in "blopptitude?" > > Anyway, nothing in mathematics would change, but surely the interest > of the resulting propositions would diminish, should the word "size" > disappear. People think (and IMHO in error) that the definition > captures correctly the concept "size". Increase your set size! This all-natural new formula will give you the true superset you always dreamt of! No injections required! Now you can fill all pigeonholes and still have your member available for more! Hear the testimonials: WM: It actually has infinite potential. Certainly the largest finite possible. TO: It goes beyond finite. I did things on the Peano that others claimed impossible. Gives a whole new meaning to the word. You are probably right. The interest of the resulting propositions would diminish. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum |