An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 28 Oct 2006 15:39 In article <4543543d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > C is simply the set of all integers, which we can consider twice the > size of N. There's really nothing to formulate about that. TO can consider the set of all integers any way he likes, but that does not validate his considerations. The set of all integers is precisely the same "size" as the set of all naturals when "size" is measured by cardinality. Measured by TO's standards, it should be slightly larger or slightly smaller than twice the size of the naturals, depending on whether 0 is counted as a natural or not.
From: Lester Zick on 28 Oct 2006 15:40 On 28 Oct 2006 11:00:52 -0700, imaginatorium(a)despammed.com wrote: > >Tony Orlow wrote: [. . .] >Do you think you are an "original thinker"? I certainly consider myself an original thinker in general scientific, mechanical, and mathematical terms, Brian. ~v~~
From: Virgil on 28 Oct 2006 15:41 In article <45435501(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <454286e8(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> stephen(a)nomail.com wrote: > >>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >>>> Tony Orlow wrote: > >>>>> David Marcus wrote: > >>>>>> Your question "Is there a smallest infinite number?" lacks context. > >>>>>> You > >>>>>> need to state what "numbers" you are considering. Lots of things can > >>>>>> be > >>>>>> constructed/defined that people refer to as "numbers". However, these > >>>>>> "numbers" differ in many details. If you assume that all subjects that > >>>>>> use the word "number" are talking about the same thing, then it is > >>>>>> hardly surprising that you would become confused. > >>>>> I don't consider transfinite "numbers" to be real numbers at all. I'm > >>>>> not interested in that nonsense, to be honest. I see it as a dead end. > >>>>> > >>>>> If there is a definition for "number" in general, and for "infinite", > >>>>> then there cannot both be a smallest infinite number and not be. > >>>> A moot point, since there is no definition for "'number' in general", as > >>>> I just said. > >>>> -- > >>>> David Marcus > >>> 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. If someone were to ask "does there exist a smallest > >>> positive non-zero number?", the answer depends on what sort > >>> of "numbers" you are talking about. > >>> > >>> Stephen > >> Like, perhaps, the Finlayson Numbers? :) > > > > Any set of numbers whose properties are known. Are the properties of > > "Finlayson Numbers" known to anyone except Ross himself? > > Uh, yeah, I think I understand what his numbers are. Perhaps you've seen > our recent exchange on the matter? They are discrete infinitesimals such > that the sequence of them within the unit interval maps to the naturals > or integers on the real line. Is that about right, Ross? Then no one can understand them because those alleged properties are self contradictory.
From: David Marcus on 28 Oct 2006 15:41 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> stephen(a)nomail.com wrote: > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> stephen(a)nomail.com wrote: > >>>>> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >>>>>> Tony Orlow wrote: > >>>>>>> David Marcus wrote: > >>>>>>>> Your question "Is there a smallest infinite number?" lacks context. You > >>>>>>>> need to state what "numbers" you are considering. Lots of things can be > >>>>>>>> constructed/defined that people refer to as "numbers". However, these > >>>>>>>> "numbers" differ in many details. If you assume that all subjects that > >>>>>>>> use the word "number" are talking about the same thing, then it is > >>>>>>>> hardly surprising that you would become confused. > >>>>>>> I don't consider transfinite "numbers" to be real numbers at all. I'm > >>>>>>> not interested in that nonsense, to be honest. I see it as a dead end. > >>>>>>> > >>>>>>> If there is a definition for "number" in general, and for "infinite", > >>>>>>> then there cannot both be a smallest infinite number and not be. > >>>>>> A moot point, since there is no definition for "'number' in general", as > >>>>>> I just said. > >>>>>> -- > >>>>>> David Marcus > >>>>> 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. If someone were to ask "does there exist a smallest > >>>>> positive non-zero number?", the answer depends on what sort > >>>>> of "numbers" you are talking about. > >>>>> > >>>>> Stephen > >>>> Like, perhaps, the Finlayson Numbers? :) > >>> If they were sensibly defined then sure you could talk about them. > >>> Nothing Ross has ever said has made any sense to me, and > >>> I severely doubt there is any sense to it, but I could be wrong. > >>> The point is, there are different types of numbers, and statements > >>> that are true of one type of number need not be true of other > >>> types of numbers. > >>> > >>> Stephen > >> Well, then, you must be of the opinion that set theory is NOT the > >> foundation for all mathematics, but only some particular system of > >> numbers and ideas: a theory. That's good. > > > > That's rather an amazing leap from what Stephen actually said. > > > > Do you agree that there is no logical contradiction between the fact > > that there is a least positive integer and the fact that there is no > > least positive real number? > > Yes, of course, the reals being continuous. What makes you think I would > disagree with that? Because we thought you had been saying that there can't be a smallest X because there is no smallest Y. Case 1: X = "infinite ordinal" Y = "infinite non-standard real" Case 2: X = "positive integer" Y = "positive real" The logic seems identical in the two cases. All that changes are the values of X and Y. > > Do you agree that there is no logical contradiction between the fact > > that there is a least infinite ordinal and the fact that there is no > > least non-standard real number? > > I think that the concept of a least infinite number in any real sense > violates the fact that one can always remove 1 from it and it will still > be infinite. Robinson directly uses this idea. Limit ordinals directly > violate it. Is it true or not? Does removal of a nonzero quantity always > result in a smaller value, or not? That's the issue. > > So, yes, there is no deductive contradiction between the two, because > they have a difference of axiomatic assumptions to begin with. That > difference causes a contradiction BETWEEN the two. > > > Feel free to give your reasoning, but please also answer each question > > with either "Agree" or "Disagree". > > agreed I think you are saying there is no (logical) contradiction. However, I'm afraid I don't know what you mean by "That difference causes a contradiction BETWEEN the two". Perhaps you are saying the two concepts are different? And, you don't think that infinite ordinals should be called "infinite numbers"? -- David Marcus
From: Lester Zick on 28 Oct 2006 15:42
On Sat, 28 Oct 2006 13:31:51 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >imaginatorium(a)despammed.com wrote: >> Anyway, I'm getting a giggle from hearing about you "reading" Robinson; >> makes me wonder if that's how you can find anything of merit in >> Lester's endless drivel - you just cruise through looking for an >> attractive sentence here or there? > >If you don't realize that the words are supposed to convey rigorous >mathematics, you can read a math book the same way that you do a novel. >I fear that most undergraduates who are not math majors read their math >books this way. And most modern math professors write them the same way. ~v~~ |