An uncountable countable set
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From: Lester Zick on 27 Oct 2006 14:31 On Thu, 26 Oct 2006 23:28:04 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >MoeBlee wrote: >> Tony Orlow wrote: [. . .] >> We JUST agreed that 'smallest infinity' means two different things when >> referring to ordinals and when referring to certain kinds of other >> orderings! It is AMAZING to me that even though I took special care to >> make sure this was clear, and then you agreeed, you NOW come back to >> conflate the two ANYWAY! > >Ahem. I said that Robinson's analysis seems to have nothing to do with >transfinitology. They appear to be unrelated. However, they cme to two >very different conclusions regarding a basic question: is there a >smallest infinite number? It seems clear to me there is not, for the >very same reason that Robinson uses: if there is an infinite number, you >can subtract 1 and get a different, smaller infinite number. It's the >same logic y'all use to argue that there's no largest finite. It's >correct. The Twilight Zone between finite and infinite CANNOT really be >pinpointed that way. Hey, Tony. You know this is an interesting problem but I think you're wasting your time here arguing the issue with the Holy Order of Self Righteous Mathematikers. Let me outline my own thinking for you. I think there is a smallest infinity but that subtracting finites from infinites isn't the way to get at the problem because as far as I can tell arithmetic operations cannot be defined between infinites and finites any more than finite division by zero can be. Instead you need a different approach altogether and I suspect the way to get at the problem is to assess the kind of infinity according to the number of infinitesimals in various intervals. And in this manner I suspect you'll find the infinity associated with straight line segments is the smallest and various kinds of curves larger. ~v~~
From: Lester Zick on 27 Oct 2006 14:35 On Fri, 27 Oct 2006 01:37:04 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: [. . .] >It is interesting that when we try to ask Tony a question that doesn't >mention balls or vases or time, his answer involves balls, vases, and >time. I'm afraid to ask what 1 + 1 is because the answer might be "noon >doesn't exist". So if the definition for "1+1" entails "1(x)+1(x)" "balls" don't lie in the "domain of discourse" for "1+1"? Curious to say the least. ~v~~
From: imaginatorium on 27 Oct 2006 14:38 David Marcus wrote: > Lester Zick wrote: > > On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > > >A very simple example is that there exists a smallest positive > > >non-zero integer, but there does not exist a smallest positive > > >non-zero real. > > > > So non zero integers are not real? > > That's a pretty impressive leap of illogic. Gosh, you obviously haven't seen Lester when he's in full swing. (Have _you_ searched sci.math for "Zick transcendental"?) Brian Chandler http://imaginatorium.org
From: Lester Zick on 27 Oct 2006 14:39 On Fri, 27 Oct 2006 12:01:19 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Tony Orlow wrote: >> stephen(a)nomail.com wrote: >> > What are you talking about? I defined two sets. There are no >> > balls or vases. There are simply the two sets >> > >> > IN = { n | -1/(2^floor(n/10)) < 0 } >> > OUT = { n | -1/(2^n) < 0 } >> >> For each n e N, IN(n)=10*OUT(n). > >Stephen defined sets IN and OUT. He didn't define sets "IN(n)" and "OUT >(n)". So, you seem to be answering a question he didn't ask. Given >Stephen's definitions of IN and OUT, is IN = OUT? According to MoeBlee's recent lectures on the subject of exhaustive mathematical definitions one cannot simply define IN and OUT, one must use a placeholder such as IN(x) and OUT(x) to establish the domain of discourse. Not exactly to my taste but there it is. And I'm sure Moe must be right because he says he is. ~v~~
From: MoeBlee on 27 Oct 2006 14:43
Lester Zick wrote: > >By having read a proof. > > A proof that there can be "no consistent theory . . .". Truly > fascinating. Do tell us more about this proof. I said no such thing as that there is a proof that there is no consistent theory. I say something, and you come back to tell me I said something completely different. You are the winner! > Arithmetic: 1, 2, 3 . . . The calculus: disintegration and integration > of definite integrals. Infinity as the number of infinitesimals. Do I attempt a telepathic tap into your brain for all of that mathematics, or do you recommend a book I can read that has all that mathematics (especially 'infinity as the number of infinitesimals') but avoids all the nasty "neo-mathematiker" propaganda? Or should I just "think it for myself" and carry it around privately in my brain the way you do? MoeBlee |