Uncountability of the Rationals?
in [Math]
|
Prev: Collatz conjecture
Next: Beginner-ish question
From: David R Tribble on 9 Jun 2010 14:19 David R Tribble wrote: >> You completely avoided answering my question, which was >> meant as a concrete example to answer the bigger question: >> "is bigulosity general enough for to be useful for all sets?" > Tony Orlow wrote: > In some cases Bigulosity cannot make a greater distinction than > cardinality, ... You mean that in *most* cases it cannot. Based on your description, bigulosity only applies to ordered sets of monotonically increasing numeric members. That's a very tiny subclass of all interesting sets. > ... and may have to simply classify a set as countably infinite. What about all the sets with cardinality greater than Aleph_0? Are their bigulosities all the same? If so, how do you show this? > It is not as simple as cardinality, and therefore not as > universal, but includes a collection of notions which lead to > solutions to a great many situations which better staisfy most > intuitions. "Notions" and "intuitions" are not mathematically definable things. The "notion" that "2 + 2 = 4" is fine for the "satisfying the intuition" that 2 apples plus 2 apples is 4 apples, but completely breaks down when you're dealing with modular arithmetic, finite fields, non-commutative operators, and all sorts of other areas of math. Indeed, it even fails for unions of sets, a "notion" that is highly "intuitive". More to the point, "notions" and "intuition" are subjective weasel words that mean you don't have a precise coherent definition that is expressible in mathematical notation. As a concrete example, the "notion" that a bijection proves that an infinite set has the same number of members as any of its infinite proper subsets is quite "intuitive" to me (and most of the other posters here), regardless of whether it is to you. David R Tribble wrote: >> Because, you see, cardinality applies to every set. If bigulosity >> only applies to a tiny fraction of sets, well, its usefulness is >> equally tiny. More to the point, it can't be a replacement for >> cardinality, since there are so many sets it can't be used for, >> and cardinality works for every set. But since you'd like it >> to be just such a replacement, and it just plain can't, that >> makes it, well, uninteresting. > Tony Orlow wrote: > Well, when you come up with a tool that replaces the screwdriver, > hammer and wrench, I will admire your rock with great disinterest as > well. I expect nothing more than that from you. David R Tribble wrote: >> Your bigulosity kind of resembles Natural Density (or Asymptotic >> Density) (for sets) or perhaps Schnirelmann density (for sequences), >> both of which have been around for some time: >> http://en.wikipedia.org/wiki/Natural_density >> http://en.wikipedia.org/wiki/Schnirelmann_density > Tony Orlow wrote: > I'm rather amazed that folks may be willing to accept simpler, less > general ideas such as these, but object to infinite-case induction and > the inverse function rule as being too restrictive because it's not as > general as cardinality. It does not compute. Yes, it's amazing that people accept concepts firmly grounded in mathematical logic and having precise and coherent definitions and meanings, but object to vaguely constructed notions that are not. Once you offer some logical rigor for your notions, people will be more open to discussing them. For now, all we have is imprecise descriptions and your assurances that they work and are truly wonderful. > IFR goes far beyond either of those already. (sigh) So you claim. Could you show us an example or two where Natural density fails but your "IFR" works? Precisely defining what exactly "IFR" is would help, too. If it really does give multiple results for the same sets (as you've said), then you can't go around claiming that it is mathematically sound.
From: Virgil on 9 Jun 2010 14:53 In article <54092252-9627-4f1d-a441-50971c85f813(a)d37g2000yqm.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 8, 11:43�pm, David R Tribble <da...(a)tribble.com> wrote: > > David R Tribble wrote: > > >> But then what about other ordered sets containing members > > >> other than reals, hyperreals, naturals, or numbers at all? Does > > >> your "bigulosity" still apply to them, in general? > > > > >> For example, consider the sets constructed from the "primitive" > > >> element 'o'. Some examples: > > >> �T1 = { o } > > >> �T2 = { o, {o} } > > >> �T3 = { o, {o}, {{o}} } > > > > >> Likewise consider some infinite sets built in this same way: > > >> �R1 = { o, {o}, {{o}}, {{{o}}}, ... } > > >> �R2 = { o, {{o}}, {{{{o}}}}, {{{{{{o}}}}}}, ... } > > > > >> Now we'll define an order relation '<' for these particular sets. > > >> We define o < {o}. So in general, > > >> �o < {o} < {{o}} < {{{o}}} < {{{{o}}}} < ... > > > > >> Now we can see that all of the sets above can be ordered > > >> sets, using our new '<' relation. > > > > >> So the question is, given these ordered sets, can you calculate > > >> the relative bigulosity for them? Specifically, does R1 have a > > >> larger or smaller bigulosity than R2? > > > > Tony Orlow wrote: > > > That depends how the strings are interpreted. > > > > Standard set notation (obviously). > > > > > If each bracket in one > > > is equal to one bracket in the other, R2 is a subset of R1, including > > > all the "even" members of the set. > > > > So far, so good. > > > > > When dealing with the countably > > > infinite, they can only be compared parametrically, and the > > > correspondences need to be defined. Bigulosity is not as simple as > > > cardinality, for sure. Not all situations are easily handled, and some > > > may be viewed from a couple different perspectives, giving different > > > results. But, since omega doesn't really exist as a number in my > > > theory, that's not really a problem. > > > > Oh, there you dropped the ball. > > > > You completely avoided answering my question, which was > > meant as a concrete example to answer the bigger question: > > �"is bigulosity general enough for to be useful for all sets?" > > In some cases Bigulosity cannot make a greater distinction than > cardinality, and may have to simply classify a set as countably > infinite. It is not as simple as cardinality, and therefore not as > universal, but includes a collection of notions which lead to > solutions to a great many situations which better staisfy most > intuitions. For example? I have yet to be presented with any situation in which "bigulosity" had any advantage whatsoever over standard set theory. And it certainly does not "staisfy" me. > > > > > Because, you see, cardinality applies to every set. If bigulosity > > only applies to a tiny fraction of sets, well, its usefulness is > > equally tiny. More to the point, it can't be a replacement for > > cardinality, since there are so many sets it can't be used for, > > and cardinality works for every set. But since you'd like it > > to be just such a replacement, and it just plain can't, that > > makes it, well, uninteresting. > > Well, when you come up with a tool that replaces the screwdriver, > hammer and wrench, I will admire your rock with great disinterest as > well. Your "tool" does not replace anything. > I'm rather amazed that folks may be willing to accept simpler, less > general ideas such as these, but object to infinite-case induction and > the inverse function rule as being too restrictive because it's not as > general as cardinality. It does not compute. "It does not compute" is OUR line!
From: Transfer Principle on 9 Jun 2010 22:10 On Jun 7, 7:30 am, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 5, 11:36 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > If the T-riffics are distinct from the H-riffics, then I > > would like to learn more about the T-riffics before I > > attempt to pass judgment. I don't mind learning more > > about sets other than the classical real numbers (i.e., > > standard R) and standard set theories. > The T-riffic numbers are much like the adic numbers in that they > express number strings of infinite length in a finite representation, > and therefore are only capable of expressing "rational" numbers with > respect to the scale we are addressing. The digital point of normal > digital systems is retained to the right of the 1's digit at location > 0, but in addition we can insert other digital points uncountably to > the left or right of this middle point. Also, like classical digital > systems, one can use any natural base above 1, so we may have binary, > octal, decimal, or hexidecimal (or whatever) T-riffic number systems. > The choice of extra digital points to the right or left of the > classical digital point is based on the formulaic infinitude one want > to express. In order to be consistent with normal digital systems, > with the classical digital point at location 0, we can place other > digital points to the right or left by specifying the digit location > relative to that point. In a T-riffic of base x, for instance, one > zillion would be expressed with a 1 to the left of a point at > logx(zillion). Thus, where z is a zillion, one zillion in base two > would be: > 1.(log2(z))000...000.0. > A zillionth can be expressed as: > 0.000...001.(-log2(z))0 Thanks. So essentially it is a system of infinitesimals. Later on, Tribble points out that it is similar in some ways to the AP-reals. A key difference between the AP-reals and the T-riffics is that the former is decimal, while the latter is binary. In searching for a way to make the T-riffics more rigorous, we can start by letting z, the symbol for "zillion," be a primitive and go from there. > Now, central to my theory is infinite-case induction. You may dredge > up a thread I started some years ago called "Infinite Induction and > the Limits of Curves", in response to a challenge regarding infinite- > case induction from Chas Brown. In any case, infinite-case induction > is simply an extension of finite inductive proof to the infinite case, > without reference to any limit ordinals or transfinite concepts. > Thanks to all the critics and naysayers I was able to refine the rule > so that it was consistent. Simply stated, any inequality which may be > inductively proved to be true for any value greater than some > particular finite value can be considered true for any positive > infinite value, provided that the difference between the two > expressions upon which the inequality is based does not have a limit > of zero as the variable approaches infinity. Thus x+2<x*2<x^2<2^x<x^x > for any x greater than 2. Ah, infinite-case induction. Once again, TO is hardly the first person to desire that induction be extended to infinite as well as finite cases. TO claims that his induction rule is consistent, but I suspect that no one is convinced of its consistency. I've tried to find a way to make something similar consistent, but so far it hasn't worked. We need something roughly like this: Infinite-Case Induction Schema: If phi is a formula that doesn't mention the primitive symbol z, then as many closures of: (AxAy ((x<y & phi(x)) -> phi(y))) -> phi(z) as possible to avoid inconsistency are axioms. To make this rigorous, we need to replace "as many closures as possible" with a rigorous rule to determine which closures we are discussing. Otherwise, the schema is called "oracular" and will be rejected.
From: Tony Orlow on 10 Jun 2010 08:17 On Jun 9, 2:19 pm, David R Tribble <da...(a)tribble.com> wrote: > David R Tribble wrote: > >> You completely avoided answering my question, which was > >> meant as a concrete example to answer the bigger question: > >> "is bigulosity general enough for to be useful for all sets?" > > Tony Orlow wrote: > > In some cases Bigulosity cannot make a greater distinction than > > cardinality, ... > > You mean that in *most* cases it cannot. Based on your description, > bigulosity only applies to ordered sets of monotonically increasing > numeric members. That's a very tiny subclass of all interesting > sets. No, That's IFR in particular which is used on countable sets of reals mapped from some segment of N. > > > ... and may have to simply classify a set as countably infinite. > > What about all the sets with cardinality greater than Aleph_0? > Are their bigulosities all the same? If so, how do you show this? > > > It is not as simple as cardinality, and therefore not as > > universal, but includes a collection of notions which lead to > > solutions to a great many situations which better staisfy most > > intuitions. > > "Notions" and "intuitions" are not mathematically definable > things. The "notion" that "2 + 2 = 4" is fine for the "satisfying the > intuition" that 2 apples plus 2 apples is 4 apples, but completely > breaks down when you're dealing with modular arithmetic, > finite fields, non-commutative operators, and all sorts of > other areas of math. Indeed, it even fails for unions of sets, > a "notion" that is highly "intuitive". I wasn't speaking about any of those areas, so again, you think changing the subject proves some point, when it just makes you look like you have adhd. > > More to the point, "notions" and "intuition" are subjective weasel > words that mean you don't have a precise coherent definition > that is expressible in mathematical notation. You like to hear yourself talk, don't you? > > As a concrete example, the "notion" that a bijection proves > that an infinite set has the same number of members as any > of its infinite proper subsets is quite "intuitive" to me (and most > of the other posters here), regardless of whether it is to you. How nice for you. I'm guessing it only took a couple years to "develop" that intuition. > > David R Tribble wrote: > >> Because, you see, cardinality applies to every set. If bigulosity > >> only applies to a tiny fraction of sets, well, its usefulness is > >> equally tiny. More to the point, it can't be a replacement for > >> cardinality, since there are so many sets it can't be used for, > >> and cardinality works for every set. But since you'd like it > >> to be just such a replacement, and it just plain can't, that > >> makes it, well, uninteresting. > > Tony Orlow wrote: > > Well, when you come up with a tool that replaces the screwdriver, > > hammer and wrench, I will admire your rock with great disinterest as > > well. > > I expect nothing more than that from you. I present a box of tools. You present a rock. > > David R Tribble wrote: > >> Your bigulosity kind of resembles Natural Density (or Asymptotic > >> Density) (for sets) or perhaps Schnirelmann density (for sequences), > >> both of which have been around for some time: > >> http://en.wikipedia.org/wiki/Natural_density > >> http://en.wikipedia.org/wiki/Schnirelmann_density > > Tony Orlow wrote: > > I'm rather amazed that folks may be willing to accept simpler, less > > general ideas such as these, but object to infinite-case induction and > > the inverse function rule as being too restrictive because it's not as > > general as cardinality. It does not compute. > > Yes, it's amazing that people accept concepts firmly grounded > in mathematical logic and having precise and coherent > definitions and meanings, but object to vaguely constructed > notions that are not. I have precisely stated what IFR and ICI are. If you cannot understand, it's only due to deliberate ignorance. > > Once you offer some logical rigor for your notions, people > will be more open to discussing them. For now, all we have > is imprecise descriptions and your assurances that they work > and are truly wonderful. And I'll I get from you are defensive insults. > > > IFR goes far beyond either of those already. (sigh) > > So you claim. Could you show us an example or two where > Natural density fails but your "IFR" works? I just did. Natural density gives a measure of zero for any set mapped from the naturals which does not use a linear mapping formula. Any higher power mapping formula, such as the set of squares, results in a measure of 0. In IFR, results agree with natural density for linear formulas. For any other invertible mapping formula, which includes all bijections, IFR gives a formulaic result in terms of omega. Granted, the formulas don't make sense to you unless you accept the concept of infinite-case induction, which has not been refuted as inconsistent in any manner. > > Precisely defining what exactly "IFR" is would help, too. > If it really does give multiple results for the same sets (as > you've said), then you can't go around claiming that it is > mathematically sound. I didn't say that. Where IFR is applicable it gives precise results, but that is only for linearly ordered sets of real quantities. TOny
From: Michael Stemper on 10 Jun 2010 08:44
In article <c36fedbe-da79-4372-a21e-6e6fb14be725(a)h13g2000yqm.googlegroups.com>, MoeBlee <jazzmobe(a)hotmail.com> writes: >On Jun 8, 5:03=A0pm, Transfer Principle <lwal...(a)lausd.net> wrote: >> So the best we can do is choose one of these two equally >> good notions to preserve. Standard Cantorian cardinality >> rejects the second in favor of the first. But I see no >> reason that we can't reject the first in favor of the >> second -- and this would give us TO's Bigulosity. > >Fine. Except "bigulosity" awaits a coherent definition. It's a perfectly cromulent term! -- Michael F. Stemper #include <Standard_Disclaimer> This message contains at least 95% recycled bytes. |