Uncountability of the Rationals?
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From: David R Tribble on 15 Jun 2010 21:36 David R Tribble wrote: >> I'm still confused, because it looks for all the world as if the sets >> N = { 1, 2, 3, ... } >> and >> R = { sqrt(1), sqrt(2), sqrt(3), ... } >> have exactly the same number of members. For a given k in N+, >> there is a sqrt(k) in R at exactly the same position. (Forget >> bijections, I'm applying your BO set ordering and member indexes.) > Tony Orlow wrote: > You're still thinking within the bax, David. You know AneN (n>1 -> > E(reR: r=sqrt(n)), and that it's a countable set that occurs more > often than the naturals, as you move at some measurable speed upward > along the number line from that 1. > > So, you must admit, there is some kind of "moreness" that occurs with > the square roots of naturals, and the same measure of "lessness" that > occurs with the squares of naturals (of course with a natural density > of 0). I know you think so, but I don't see how. If every natural k in N+ has a square root sqrt(k) in S, how are there more roots than naturals? Does each natural have only a single (positive) square root, or are there more roots than that? I'll ask the question again, since you didn't bother answering it before: Are there square roots r in S such that r*r is not a natural in N+? If so, can you provide an example r? If not, does this mean that there are more members in S than in N+?
From: Jesse F. Hughes on 15 Jun 2010 21:35 Transfer Principle <lwalke3(a)lausd.net> writes: > This, of course, goes back to the question that comes up often in > these threads. Is it possible for one to accept all of the axioms of a > theory, without accepting all of the consequents of those axioms? And > if one does reject theorems proved from axioms that he finds > unobjectionable, does he deserve to be called by a five-letter insult? What fine questions! Of course one should not be required to accept the logical consequences of the axioms he accepts. Why should he? What sort of fascist makes these rules up? Everyone should have the right to accept whatever theorems they want and to reject those they don't want. And no one else should be allowed to criticize them for their personal choices. That's what freedom is all about. -- "When you go to class today, if your professor talks about algebraic number theory, or misuses Galois Theory[,] I want you to carefully notice how you feel. Hold on to that feeling so that you never forget it." --James S. Harris, on channeling rage via Galois theory.
From: David R Tribble on 15 Jun 2010 21:45 Transfer Principle wrote: > Of course, on sci.math less than a year ago there was an > adherent of ZFC+AD who refused to accept that his chosen > theory is inconsistent. In other words, he accepts all of > the axioms of ZFC+AD, but not the theorem "1=0" which is > provable in ZFC+AD. This poster defies classification. Not at all. "Crank" works just fine.
From: Brian Chandler on 16 Jun 2010 00:18 Jesse F. Hughes wrote: > Tony Orlow <tony(a)lightlink.com> writes: > > > On Jun 15, 2:17 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Tony Orlow <t...(a)lightlink.com> writes: <snip> > >> That little bit of math is utterly unclear. > >> > >> For instance, you claim that sqrt(w) * sqrt(w) = w, but that does *not* > >> follow from the above principle, *even if* we assume that the above > >> principle entails that sqrt(w) is defined. > >> > >> The above principle just allows us to conclude that sqrt(w) < w. > > > > True. So? It doesn't mean there isn't an easy way to envision omega^2 > > or sqrt(omega). > > Envision? Look, we're supposed to be able to prove whatever properties > you think are true of sqrt(w) and w^2. We shouldn't have to magically > "envision" them. Thus, if sqrt(w) * sqrt(w) = w, there should surely be > a proof of this fact. > > How do you suggest we do this, aside from our magical envisioning > powers? Jesse, again I think you are being a bit short on imagination here. In Conway's ONAG, at the bottom of page 13 there is a paragraph: "It should not take the reader too long to verify that z = w/2.When he has done this, ... [he can] verify our assertion that {0, 1, 2, 3, ... | w, w/2, w/4, w/8, ...} is a square root of w." Now if you or I were reading this we might just take Conway's word for it that this assertion holds, or we might decide to do the grunt, and check. Either way, we see that he has defined something in terms of his primitives (all numbers are constructed as {A|B} where A and B are sets) which can be called the root of omega ('w'). Now imagine Tony reading ONAG (this was suggested to him, and perhaps he did look at it). He can only read it through the Fido filter (if you doubt this, reread the first post in this thread to remember just how confused he is). You know, "!!!! !!! !!!! !!!! !!!! !!! FIDO !!!! !!! !!!!! !! !!!! !! !!! BALL !!!! !!! !!!!!! FIDO". I believe Tony is working on a book, which is why suggesting he reads an existing one is not likely to work. No doubt once published his book will be (in his own estimation anyway) the ultimate book on Set Theory and Why It Is Wrong. Can he write it through the inverse Fido filter? Don't see why not. I do speak Japanese, and I sometimes dream things in Japanese, which make sense at least as far as dreams ever make sense. But I don't speak German, other than highly fragmentary bits of youth hostel stuff: however, I recall a memorable dream in which I was with some German speakers, who were indeed speaking German, and just like in real life I could sort of semi-understand parts of it. How did my brain generate this German for me to semi- understand? Well, wait for Tony's book, and you will have some idea. Brian Chandler
From: Transfer Principle on 16 Jun 2010 01:48
On Jun 15, 9:18 pm, Brian Chandler <imaginator...(a)despammed.com> wrote: > Jesse F. Hughes wrote: > > Envision? Look, we're supposed to be able to prove whatever properties > > you think are true of sqrt(w) and w^2. We shouldn't have to magically > > "envision" them. Thus, if sqrt(w) * sqrt(w) = w, there should surely be > > a proof of this fact. > > How do you suggest we do this, aside from our magical envisioning > > powers? > Jesse, again I think you are being a bit short on imagination here. In > Conway's ONAG, at the bottom of page 13 there is a paragraph: > "It should not take the reader too long to verify that z = w/2.When he > has done this, ... [he can] verify our assertion that > {0, 1, 2, 3, ... | w, w/2, w/4, w/8, ...} > is a square root of w." Ah, yes, Conway's surreal numbers. These have come up in several threads similar to this one. So yes, sqrt(omega) is a surreal. But the big difference between surreals and TO's numbers is that the latter is supposed to be a sort of set _size_ (Bigulosity). It makes no sense to state that the set of perfect squares has the surreal sqrt(omega) as its set size. |