Obections to Cantor's Theory (Wikipedia article)
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From: malbrain on 28 Jul 2005 11:52 Martin Shobe wrote: > On 26 Jul 2005 22:07:39 -0700, malbrain(a)yahoo.com wrote: > > >Martin Shobe wrote: > >> On 26 Jul 2005 17:15:10 -0700, malbrain(a)yahoo.com wrote: > >> > >> >Chris Menzel wrote: > >> >> On Tue, 26 Jul 2005 16:39:58 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > >> >> > ... > >> >> > then that function needs to be taken into account. This nonsense > >> >> > about an infinite set of finite whole numbers is pretty bad too, but > >> >> > probably without any real consequences. > >> >> > >> >> You seem to agree that the set of whole numbers is infinite. But there > >> >> was an inductive argument a few posts back that all the whole numbers > >> >> are finite, and hence that the set of finite whole numbers is infinite. > >> >> There was some real mathematics there. > >> > > >> >How does it follow that the count of finite whole numbers is infinite? > >> >How is this established by the Peano axioms? > >> > >> A set, A, is infinite if, and only if, there exists a one-to-one > >> function, f:A -> A, such that f(A) is a proper subset of A. > >> > >> Or equivalently, > >> > >> A set, A, is infinite if, and only if, there exists a function, f:A -> > >> A, such that > >> 1) for all x,y in A, f(y)=f(x) => x=y. > >> 2) there exists an x in A such that for all y in A, f(y) =/= x. > >> > >> Since there are no sets, and we are interested only in the domain of > >> PA, we have > >> > >> The domain of PA is infinite if, and only if, there exists a function, > >> f, such that > >> 1) for all x,y f(y)=f(x) => x=y. > >> 2) there exists an x such that for all y, f(y) =/= x. > >> > >> The successor function meets those criteria. Therefore, the domain of > >> PA is infinite. > >> > >> The only problem that I can see with this is that it's a theorem about > >> PA instead of a theorem of PA. > > > >It's ABOUT PA because you back-filled a definition for infinite to PA? > > It's about PA because the question isn't even expressable in PA. > There are no functionn in the domain of PA, just numbers. > > >> > You have Tony agreeing to > >> >the axiom of infinity apriori, when this is not indicated. > >> > >> The axiom of infinity is not needed to prove that a set is infinite. > >> The axiom of infinity is needed to prove that infinite sets exist. > > > >This KOAN is going to be a tough sell. karl m > > What do you think the axiom of infinity says? I'm not looking for the > logical consequences, just what the axiom is. Well, perhaps I need to define KOAN first before I answer your question. Etymology: Japanese kOan, from kO public + an proposition : a paradox to be meditated upon that is used to train Zen Buddhist monks to abandon ultimate dependence on reason and to force them into gaining sudden intuitive enlightenment. The axiom of infinity imports the set of natural numbers into set theory. karl m
From: malbrain on 28 Jul 2005 12:01 stephen(a)nomail.com wrote: > In sci.math Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > Just as a matter of prevention, shouldn't we at least _try_ to find > > a way of living together, then? > > > Han de Bruijn > > You are the one who seems to have a problem living together > with mathematicians. What is preventing you from simply > living and let live? Plenty. It's called life. karl m
From: Victor Eijkhout on 28 Jul 2005 11:50 Virgil <ITSnetNOTcom#virgil(a)COMCAST.com> wrote: > > You're forgetting that S and L in those series go to infinity. > > How is it that S, L and S^L though all finite, can be of different sizes? > > Whenever S^L is larger that either S or L, why not take a new S and L > each equal to the old S^L and recalculate a new S^L. > > Does this process ever have to stop? If at some step S^L ever reached an > infinite value, for which finite S and L does S^L become infinite? I think it's sufficiently established that TO basically doesn't understand that the order of quantifiers is relevant. Any person with the capacity for logical though would have read this post of yours or many like it, contemplated it for a second, then slapped his/her forehead and exclaimed "of course, how could I have been so silly". TO is clearly not such a person. V. "but hey, electrons are cheap" -- email: lastname at cs utk edu homepage: www cs utk edu tilde lastname
From: malbrain on 28 Jul 2005 12:14 Dik T. Winter wrote: > In article <1122503371.218414.268340(a)g47g2000cwa.googlegroups.com> malbrain(a)yahoo.com writes: > > Dik T. Winter wrote: > ... > > > > > > 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. > > > > > > That is not an answer. > > > > Well, the OBVIOUS answer to your question is, "I'm talking about C" > > However, I'm not that vulgar. I tend to translate discussions into C > > because I find it to be more universally understood than java. karl m > > So what? When someone talks about java with "unbounded" standard types, > what is the point stating that C does not have "unbounded" standard types? How is the length of an integer determined by a java program running on the virtual-java-machine? karl m
From: malbrain on 28 Jul 2005 12:19
Robert Kolker wrote: > Virgil wrote: > > > > > TO has yet to produce any axiom system of his own, and, judging by the > > quality of his arguments, is extremely unlikely ever to be able produce > > one of any value. > > Orlow is a mathematical incompetent. In addition to this, his arrogant > ignorance of mathematics has indicated stupidity. Not knowing something > is no shame, as we are all born ignorant. Refusal to learn is the mark > of the Truly Stupid. In horse training you have to lead a horse to water AND make it drink, BOTH. karl m |