Uncountability of the Rationals?
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From: William Hughes on 7 Jun 2010 12:01 You still have Let the number of elements in a set S be #(S) Set F be the set of all finite naturals. Consider #(F) Clearly #(F) cannot be a finite natural. But your claim is that #(F) cannot be an infinite natural. So the number of elements in F is not any type of natural. - William Hughes
From: David R Tribble on 7 Jun 2010 12:02 Transfer Principle wrote: >> Hold on a minute. Earlier, TO and Tribble were discussing >> something called the H-riffics. Now Virgil is referring >> to something called the T-riffics. > Tony Orlow wrote: > The T-riffic numbers were developed to numerically represent infinite > and infinitesimal numbers so that arithmetic could be performed on > them which may produce other infinities, infinitesimals, or even > finite numbers as a result. In addition to the finite unit One, there > is an infinite unit, Big'Un, and its multiplicative inverse, the > infinitesimal Lil'Un. Tony's ideas share a lot in common with those of Archimedes Plutonium (of which thousands of his posts can be found here in sci.math). I.e., the sequence of naturals (eventually) contains infinite naturals, infinite sets must contain at least one infinite member, all numbers must have digital representations, and a few other wacky ideas. To contrast Tony's ideas with mathematical ideas possessed of something resembling logical rigor (and to toot my own horn), have a look-see at my humble attempts to extend the reals: http://david.tribble.com/text/hnumbers.html My "suprareals" share some characteristics with Cantor normal form and also with the surreals. Turns out that they are actually a re-invention of sorts of the Levi-Civita field. See: http://en.wikipedia.org/wiki/Levi-Civita_field http://en.wikipedia.org/wiki/Cantor_normal_form -drt
From: Tony Orlow on 7 Jun 2010 12:31 On Jun 7, 12:02 pm, David R Tribble <da...(a)tribble.com> wrote: > Transfer Principle wrote: > >> Hold on a minute. Earlier, TO and Tribble were discussing > >> something called the H-riffics. Now Virgil is referring > >> to something called the T-riffics. > > Tony Orlow wrote: > > The T-riffic numbers were developed to numerically represent infinite > > and infinitesimal numbers so that arithmetic could be performed on > > them which may produce other infinities, infinitesimals, or even > > finite numbers as a result. In addition to the finite unit One, there > > is an infinite unit, Big'Un, and its multiplicative inverse, the > > infinitesimal Lil'Un. > > Tony's ideas share a lot in common with those of > Archimedes Plutonium (of which thousands of his posts > can be found here in sci.math). I.e., the sequence of > naturals (eventually) contains infinite naturals, infinite sets > must contain at least one infinite member, all numbers > must have digital representations, and a few other wacky > ideas. Oh, David, don't even go there. Archimedes thinks the universe is the nucleus of a plutonium atom in a bigger universe, for god knows what reason. I have never claimed the standard naturals include infinite values, nor denied that the set of points in [0,1] is infinite in size but contains no infinite values, nor claimed that there exists any finite digital representation of any transcendental number nor a last digit thereof, or any such wacky ideas. Your disingenuity is most unbecoming, and I would appreciate it if you not spin fabrications about what you believe to be my dysfunction. The splinter you perceive in my eye is but a reflection of the log in yours. > > To contrast Tony's ideas with mathematical ideas possessed > of something resembling logical rigor (and to toot my own horn), > have a look-see at my humble attempts to extend the reals: > http://david.tribble.com/text/hnumbers.html Indeed, humble, as you really do nothing particularly spectacular. > > My "suprareals" share some characteristics with Cantor > normal form and also with the surreals. Turns out that > they are actually a re-invention of sorts of the Levi-Civita field. > See: > http://en.wikipedia.org/wiki/Levi-Civita_field > http://en.wikipedia.org/wiki/Cantor_normal_form > > -drt I've reinvented lots of things. That's fun, but not ground-breaking. Have a good day. Tony
From: Brian Chandler on 7 Jun 2010 14:36 Tony Orlow wrote: > On Jun 6, 5:16 am, Brian Chandler <imaginator...(a)despammed.com> wrote: > > Transfer Principle wrote: > > > On Jun 5, 9:59 am, Virgil <Vir...(a)home.esc> wrote: > > > > In article > > > > <db5cbe4b-a8b8-4b6a-ae47-05fa05dd6...(a)i28g2000yqa.googlegroups.com>, > > > > Tony Orlow <t...(a)lightlink.com> wrote: > > > > > Yes, that's where I apply N=S^L. > > > > Which is a wrong now as when first dropped on an unsuspecting world.. > > > > > IIRC, TO's statement N=S^L means that the number of > > > strings of length L from a language of size S is > > > equal to N. > > > > > I disagree with Virgil that it's "wrong." > > > > No, of course the actual meaning of "N=S^L" is correct. The problem is > > that when Tony says "apply N=S^L" he refers to one of his "proofs" > > that an infinite set of natural numbers must include at least one > > number which is itself "infinite" (though he never really defines what > > this means). I will try to reconstruct the argument, which goes > > something like: > > Oh, here we go..... > > The argument that the truly infinite set of naturals must include an > infinite natural has nothing particular to do with N=S^L. That's a > simple matter of the nth element of any initial segment of N being > equal to n and existing for any segment of size n or greater. Don't know what that last sentence means, either. > > > > > (Tony 'knows' that the set P of all 'natural numbers' must include > > "infinite naturals", but can't just state it.) > > Been there done that. Your failure to comprehend is not necessairly > due to my inability to express myself. True, very true. My failure to comprehend might well be because your intellect is so much greater than mine. On the other hand, even if my failure to comprehend is in spite of your ability to express yourself perfectly, it doesn't necessarily imply that what you are saying is other than confused nonsense. After all, they laughed at Coco the Clown, didn't they? > > So, consider "N=S^L". The number of strings (over alphabet size S) of > > length L is N. > > Yes. > > > > > Well, the number of strings (over alphabet size S) of maximum length L > > is N. (Not exactly true, but close. Failiing to distinguish "finite > > strings of no fixed maximum length" from "possibly infinite strings" > > is at the heart of this argumentation technique.) > > Good imaginatorializing. Actually I thought it was good Orlovian salivation. Actually this formula is for the strings of length (all) exactly L. But a formula for strings of length *up to* L is similar. > > But for there to be an infinite number of strings ("set N = oo"), > > since S is constant, we have "L=oo". (Confusion and non-sequitur) > > (On your part, without a doubt) Yours, sunshine. "Infinite" isn't the "maximum length", let alone the length of all the strings. Your error is embedded in the step in which you glide over this little "detail". > Because N=S^L, the only way N can be infinite is if either S or L is > infinite. With an infinite alphabet we have an infinite number of > words of any given length, except 0 of course. Further, N can only be > uncountable, in my opinion, if either S or L or both are uncountable. > That's where we disagree, I believe. No, no, we don't disagree, at all. You opine this and that, and I agree that you may opine anything you like, really. The tunnel of love, the twilight zone, this is all excellent poetry. How can one "disagree" with poetry. Brian Chandler
From: Virgil on 7 Jun 2010 15:34
In article <934dae7c-c568-4cb2-9ec2-a8cb4d2ec605(a)t10g2000yqg.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > If you really think that I don't believe in the distinction between > the countable and uncountable then I don't know who you think you were > talking to or what you think I said. Additionally, it is not ZFC that > I object to, but the extension of the theory with the model or > cardinality as "set size" for the infinite case. Simple bijection does > not equate to "equinumerosity" in my book. That's "equicardinality" > for the mathematically rigorous, no? What is more natural than extending the notion that for finite sets "numerosity" is the same as "cardinality" to all sets? And all of TO's attempts to define some other useful measure(s) of numerosity for non-finite sets failed miserably! |