Obections to Cantor's Theory (Wikipedia article)
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From: Martin Shobe on 27 Jul 2005 19:40 On Wed, 27 Jul 2005 15:45:58 -0400, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >Martin Shobe said: >> On Tue, 26 Jul 2005 14:46:42 -0400, Tony Orlow (aeo6) >> <aeo6(a)cornell.edu> wrote: >> >> >Martin Shobe said: >> >> On Mon, 25 Jul 2005 15:52:08 -0400, Tony Orlow (aeo6) >> >> <aeo6(a)cornell.edu> wrote: >> >> >> >> >Martin Shobe said: >> >> >> On Wed, 20 Jul 2005 10:57:58 -0400, Tony Orlow (aeo6) >> >> >> <aeo6(a)cornell.edu> wrote: >> >> >> >> >> >> >Barb Knox said: >> >> >> >> In article <MPG.1d4726e11766660c989f2f(a)newsstand.cit.cornell.edu>, >> >> >> >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> >> >> >> [snip] >> >> >> >> >> >> >> >> >Infinite whole numbers are required for an infinite set of whole numbers. >> >> >> >> >> >> >> >> Good grief -- shake the anti-Cantorian tree a little and out drops a >> >> >> >> Phillite. Here's a clue: ALL whole numbers are finite. Here's a >> >> >> >> (2nd-order) proof outline, using mathematical induction (which I >> >> >> >> assume/hope you accept): >> >> >> >> 0 is finite. >> >> >> >> If k is finite then k+1 is finite. >> >> >> >> Therefore all natural numbers are finite. >> >> >> >> >> >> >> >> >> >> >> >That's the standard inductive proof that is always used, and in fact, the ONLY >> >> >> >proof I have ever seen of this "fact". Is there any other? I have three proofs >> >> >> >that contradict this one. Do you have any others that support it? >> >> >> > >> >> >> >Inductive proof proves properties true for the entire set of naturals, right? >> >> >> >> >> >> Yep. >> >> >> >> >> >> >That entire set is infinite right? >> >> >> >> >> >> Yep. >> >> >> >> >> >> >Therfore, the number of times you are adding >> >> >> >1 and saying, "yep, still finite", is infinite, right? >> >> >> >> >> >> Yep. But be careful here, at *every* stage of this process, we have >> >> >> still only done it a finite number of times. >> >> >Uhhh.... Look at what you just agreed to. The number of times you are adding 1 >> >> >is infinite. But, now you say it is always a finite number of times? make up >> >> >your mind. >> >> >> >> It's quote made up. I have chosen to follow the route you so >> >> cavalierly ignored. >> >?????? Huh? >> >> There is no finite upper bound on how many times I can apply it. >> However, each application occurs after only a finite number of >> previous applications. >Do you, or do you not, have an infinite set? Yes. >Do you, or do you not, egenrate >one member per iteration. Yes. > How many iterations are requires to produce an >infinite set, one at a time? (sigh) Never happens. >> >> >> > So, you have some way of >> >> >> >adding an infinite number of 1's and getting a finite result? >> >> >> >> >> >> Nope. You weren't careful. >> >> >You contradicted yourself. Do you apply successor and increment the value a >> >> >finite number of times, or an infinite number of times? Be careful. >> >> >> >> There is no end to how much I can apply successor. However, I can >> >> only apply it a finite number of times. No contradiction here. >> >Except for the fact that somehow you got an infinite set ina finite number of >> >steps, producing 1 element at a time. How does that work? >> >> Do you remember that proof of mine about the existence of a smallest >> cardinal? The first few lines of the proof get you one of the more >> common constructions of the naturals. I get it in finitely many steps >> by showing that there is a set at least that big, and throwing out the >> things I don't want. >I don't recall that in detail. By smallest cardinal, I assume you mean smallest >infinite cardinal? That doesn't exist, unless you allow bullshit like omega-1 >=omega. Sorry Yes I meant smallest infinite cardinal. And I gave you a proof of it. You responded by saying you had printed it out and would get back to me. It doesn't rely on omega - 1 = omega. See http://groups-beta.google.com/group/sci.math/browse_frm/thread/b31134b5d940005d/6f258e835695ad03?lnk=st&q=smallest+infinite+cardinal+Shobe&rnum=2&hl=en#6f258e835695ad03 Martin
From: malbrain on 27 Jul 2005 20:00 Tony Orlow (aeo6) wrote: > Chris Menzel said: > > On Tue, 26 Jul 2005 16:39:58 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > > > ... > > > The big problem in transfinite cardinality is the assumption that a > > > bijection necessarily indicates equal sizes for infinite sets, as it > > > does for finite sets. > > > > It's not an assumption, it's a definition, one that you reject. That's > > fine, but until you provide equally rigorous alternatives, it's the only > > game in town. And frankly, you've got no business rejecting it just > > because some of the consequences conflict with your intuitions. The > > mathematics of the infinite is occasionally surprising, especially when > > one doesn't have a complete understanding of the subject matter. > > > > > When the only way to form a bijection is with a mapping function, > > > > How else? > > > > > 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. Why have you not responded to a > > mathematical proof that all the members of the infinite set of whole > > numbers are finite? It would be your chance to show everyone where the > > error in the argument lies. > > > > Chris Menzel > > > > > I have repsonded to that proof over and over. Where were you when we were > discussing the nature of inductive proof, and the implicit infinite loop in the > construction that no one seems to have considered, and the fact that adding 1 > an infinite number of times, as is done in the generation of the infinite set > of naturals, produces an infinite sum? Unfortunately, I'm afraid you wore out his patience, or perhaps you missed his last post which he directed only to sci.math. > I tell you what. You refute my inductive proof that the set of naturals is > finite, without refuting your own proof, and then we'll talk. Good luck. I > don't expect any response, since you folks tend to ignore any proof you can't > throw "largest finite" at. Your bag of tricks is really rather finite, and more > and more transparent. Look at Barb's java program. It clearly generates a stream of numbers that has no limit. karl m
From: Martin Shobe on 27 Jul 2005 20:17 On 27 Jul 2005 09:34:46 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: >And your set description of chords says nothing whatsoever >about the physics of music. Can you tell me why {C,E,G} >is a chord and {C,C#,D,D#} is not? {C,C#,D,D#} is a chord. It's called a tone cluster. I agree with the rest though. Martin
From: Daryl McCullough on 27 Jul 2005 21:25 stephen(a)nomail.com says... >Here is another inconsistency for Tony to mull over. >Apparently there is a largest finite natural number (of course >Tony denies this in every other post, but uses it >in every "proof"). This base 10 representation >of this number requires L digits. The base 2 representation >of this number, which presumably exists because all natural >numbers can be representd in any base, requires roughly 3L digits. > >Why do no 3L digit long base 10 numbers exists? Or do >no 3L digit long numbers exist in any base, so that really >larger numbers cannot be represented in small bases? >The mysteries of Orlovian math. Basically, for any particular statement S, Tony will sometimes claim S, and sometimes claim the negation. Whichever one is most convenient for his argument. So sometimes there is a biggest finite natural, sometimes there isn't. Sometimes mathematical induction is valid, sometimes it isn't. Sometimes you use bijections to prove that sets are the same size. Sometimes you don't. -- Daryl McCullough Ithaca, NY
From: Martin Shobe on 27 Jul 2005 21:54
On Wed, 27 Jul 2005 13:27:07 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >MoeBlee wrote: > >> Meanwhile, I'm still fascinated by your inconsistent theory, posted at >> your web site, which is: >> >> Z, without axiom of infinity, bu with your axiom: Ax x = {x}. >> >> Would you say what we are to gain from this inconsistent theory? > >No. But first we repeat the mantra: > > A little bit of Physics would be NO Idleness in Mathematics > >It is physically correct that every member of a set is at the same time >a _part_ of the set, meaning that a e A ==> a c A , where e stands for >membership and c for being a subset. The above axiom that a member of a >set cannot physically distinguished from a set which contains only the >member - the envelope {} means nothing, physically - is only a weaker >form of this idea. Since sets aren't physical, it is not physically correct to assert that a e A ==> a c A. From what I understand of the history of set theory, physics wasn't the inspiration, and it certainly isn't its purpose at the current time. That doesn't mean that you can't explore theories where a e A ==> a c A. Just don't confuse what you are doing with standard set theory. (And it would be nice if you indicate that you aren't doing standard set theory, if for no other reason than to avoid some confusion.) >There has been a thread on this topic in 'sci.math' as well, called >"Set inclusion and membership". It is noted that a "set theory without >the membership" actually exists. Google it up as "mereology" and you >will find quite some clues. > >Now I simply add this axiom to ZFC and see what happens. Nothing else. >If the consequence may be that ZFC scrumbles into nothingness, then, >unfortunately, we are not going to gain anything. But I consider that >not to be my problem, because I'm just doing .. the mantra. >No kidding: it IS a problem, but I find that first things come first. If you are going to use that as an axiom, then the set theory you add it to should include sets that aren't well founded. Make that very few well-founded sets. Martin |