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From: Eckard Blumschein on 8 Apr 2005 11:47 On 4/8/2005 4:24 PM, Dave Rusin wrote: > Eckard Blumschein wrote: >> >>> "There are neither more nor less nor equally many real numbers >>> as compared to the rational ones." > > Did you mean to say the _number of_ real numbers is the same as > the _number of_ rational numbers? There are some fellows over in > comp.ai.philosophy and sci.philosophy.meta you should talk to. Thank you for the hint. I was not aware of there groups. No. I argue that the number of rational numbers exceeds any limit as does the number of real numbers. The operation oo-oo is not reasonable because oo is a quality , not a quantity. Sincerely, Eckard
From: Matt Gutting on 8 Apr 2005 12:07 Eckard Blumschein wrote: > On 4/8/2005 4:24 PM, Dave Rusin wrote: > >>Eckard Blumschein wrote: >> >>>>"There are neither more nor less nor equally many real numbers >>>>as compared to the rational ones." >> >>Did you mean to say the _number of_ real numbers is the same as >>the _number of_ rational numbers? There are some fellows over in >>comp.ai.philosophy and sci.philosophy.meta you should talk to. > > > Thank you for the hint. I was not aware of there groups. > No. I argue that the number of rational numbers exceeds any limit as > does the number of real numbers. The operation oo-oo is not reasonable > because oo is a quality , not a quantity. I don't see what subtraction has to do with it. I agree that oo is not a *number* in the same sense that, e.g., pi is a number. I don't think one can talk meaningfully about the number of numbers in either of these sets - that is, I don't think one can answer the question "How big is this set?" for either set. I do think, however, that one can find a measure of comparison between the two sets - that it is possible to answer the question "Is this set bigger (in some sense) than that one?" By the measures generally adopted by mathematicians we decide that the reals are "bigger" in a very specific sense. But "bigger" has nothing to do with a numerical answer to the question "How many?" for either set - since this question cannot, in fact, be answered. Matt > > Sincerely, > Eckard > >
From: fishfry on 8 Apr 2005 20:10 In article <4256A7A6.8050305(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 4/8/2005 4:24 PM, Dave Rusin wrote: > > Eckard Blumschein wrote: > >> > >>> "There are neither more nor less nor equally many real numbers > >>> as compared to the rational ones." > > > > Did you mean to say the _number of_ real numbers is the same as > > the _number of_ rational numbers? There are some fellows over in > > comp.ai.philosophy and sci.philosophy.meta you should talk to. > > Thank you for the hint. I was not aware of there groups. > No. I argue that the number of rational numbers exceeds any limit as > does the number of real numbers. The operation oo-oo is not reasonable > because oo is a quality , not a quantity. > " oo is a quality , not a quantity" seems like a statement of philosophy, not mathematics. In your M280 paper I believe you said that you agree that there is no bijection from the natural numbers to the reals. That's the DEFINITION of "more" in this context. So by definition, there are more reals than naturals. Clearly as a matter of physical reality, that is somewhat of a meaningless statement. There are no infinite sets in the physical universe. The real numbers do not exist in the physical universe. The statement "there are more reals than rationals" is not a statement of physics; it's a statement about mathematics. It depends for its truth on the specific technical DEFINITION of "more," which is taken to mean that there's no bijection.
From: Eckard Blumschein on 9 Apr 2005 06:41 On 4/8/2005 6:07 PM, Matt Gutting wrote: >>>>>"There are neither more nor less nor equally many real numbers >>>>>as compared to the rational ones." >>> >>>Did you mean to say the _number of_ real numbers is the same as >>>the _number of_ rational numbers? There are some fellows over in >>>comp.ai.philosophy and sci.philosophy.meta you should talk to. >> >> >> Thank you for the hint. I was not aware of there groups. >> No. I argue that the number of rational numbers exceeds any limit as >> does the number of real numbers. The operation oo-oo is not reasonable >> because oo is a quality , not a quantity. > > I don't see what subtraction has to do with it. It depends on the result of subtraction whether a number is larger, equal or smaller than the other one. I agree that oo is not > a *number* in the same sense that, e.g., pi is a number. oo is just a quality, not a number at all. > > I don't think one can talk meaningfully about the number of numbers Well, In German language I would rather say the Anzahl (amount) of numbers. See above: Originally I wrote "more numbers". > in either of these sets - that is, I don't think one can answer the > question "How big is this set?" for either set. That is indeed the point. > I do think, however, that > one can find a measure of comparison between the two sets - that it is > possible to answer the question "Is this set bigger (in some sense) than > that one?" I see this the basic fallacy. Without a quantitative measure there is no possibility of comparison. The quality is the same: Just infinite. > By the measures generally adopted by mathematicians we decide > that the reals are "bigger" in a very specific sense. Do you really decide? I rather found out that you are following Cantor's split way of thinking/believing. > But "bigger" has > nothing to do with a numerical answer to the question "How many?" for > either set - since this question cannot, in fact, be answered. "Bigger" is the result of a quantitative comparison. But it has nothing to do with countable or not. Eckard
From: Eckard Blumschein on 9 Apr 2005 07:09
On 4/9/2005 2:10 AM, fishfry wrote: > " oo is a quality , not a quantity" seems like a statement of > philosophy, not mathematics. I was told that children early learn oo+1=oo. I think this is good so. Mathematics needs a sound understanding of the quality oo. > > In your M280 paper I believe you said that you agree that there is no > bijection from the natural numbers to the reals. I do not doubt in that. > That's the DEFINITION of "more" in this context. Yes, this was Cantors fallacious ambition from the very beginning. > So by definition, there are more reals than naturals. This definition is not reasonable. Cantor perhaps deliberately violates the still even in mathematics valid meaning of infinity: oo+1=oo. > Clearly as a matter of physical reality, that is somewhat of a > meaningless statement. Let's focus on mathematics. Cantor's phasmagoras relative to physics proved even worse. > There are no infinite sets in the physical > universe. The real numbers do not exist in the physical universe. Nonetheless, physics needs idealized models. A peculiarity of the reals is that they are not even mathematically real in the sense they cannot be represented by a finite series of numerals. > > The statement "there are more reals than rationals" is not a statement > of physics; No it is a mathematically wrong statement. > it's a statement about mathematics. It depends for its truth > on the specific technical DEFINITION of "more," which is taken to mean > that there's no bijection. I vote for a mathematics that is self-consistent. Cantor's detour only caused at least questionable notions like infinite numbers and paradoxa. It hampered the effort to find appropriate axioms. It gave rise to the illusion that real numbers are really numbers, and I looked in vain for any benefit except for income of those who edited his schizophrene and mystical rather than logically retraceable theory, wrote pertaining books on set theory and got lucratice posts. Please notice that I can compellingly explain all of Cantor's "evidence". Also I was surprized that the puzzle of open questions driving me got suddenly and completely resolved on that correct basis. On the other hand, the difficulties with Cantors's set theory and his infamous continuum hypothesis are not even partially resolverd after more than a century of effort by the most prominent mathematicians. |