Uncountability of the Rationals?
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From: David R Tribble on 27 Jun 2010 17:43 David R Tribble wrote: >> In contrast, the ordered set: >> S = N U { w } = { 0, 1, 2, 3, ..., w } >> has an "infinite distance" between w and all of its other >> members, yet it is a countable set. > Tony Orlow wrote: > Only if you consider w, or 'tav', or whatever, to be some actual > infinite number, um, after all the other ones ended, even though they > have no end. Sorry. x-1<x. It's a primordial fact. Tav is not a > specific value, and neither is aleph_0. Aleph_1 is another story.... The context we're talking in here is Set Theory, as has been pointed out several times. Therefore every entity we talk about is, by definition, a set, or something equivalent to a set. This include numbers, which are just sets having certain properties. [At this point, feel free to reject that view. It won't change anything.] Therefore omega (w) is a set. It's an infinite set. It's also an ordinal, possessing all the properties of ordinals. [You will doubtless reject that as well.] Likewise, N (which is omega) is a set. N+ is a set, and could, with a little finessing, be considered a number as well.
From: David R Tribble on 27 Jun 2010 17:58 David R Tribble wrote: >> Perhaps you could clarify what you mean by "number". > Tony Orlow wrote: > A number is a location in an n-dimensional mathematical space, such > that every number is denoted by a unique n-tuple. There is no location > called omega or tav. Ah, now we're making some progress. For you, a "number" must be a point in some geometric (Euclidean?) n-space. Your notion, though, appears to omit numbers like the hyperreals and surreals, and spaces like the projective plane (aka the Riemann sphere). You would also seem to have problems with disconnected and noncontinuous metric spaces, and infinite-dimension manifolds. Obviously, ordinals and cardinals are right out. Of course, most of your discussion here has been within the context of Set Theory. Perhaps you could explain what a "number" is within that context (any set theory will do, even your BO theory), without bringing in the whole geometrical "number line" in the process.
From: Transfer Principle on 28 Jun 2010 02:13 On Jun 27, 12:23 am, Brian Chandler <imaginator...(a)despammed.com> wrote: > MoeBlee wrote: > > I thought Orlow doesn't accept limit ordinals. Are there new updates > > released on that matter this week? > Under Transfer's new regime, it's possible we are only supposed to > remember the last thing our Dear Leader said on any particular topic Suppose that back when Chandler was a fifth grader, his teacher decided to grade him not on his current work, but by going back to his work from kindergarten and grade it according to a fifth grade rubric, and therefore fails him (unless, of course, Chandler was an especially precocious kindergartner). Then, when Chandler protests that this is unfair, the teacher states that only an oppressive "regime" would deny her the right to use kindergarten work to grade him. The point I'm trying to make is not that we are only supposed to remember the last thing that TO said, but instead, if he made a claim and was convinced that he was wrong, give him credit for having learned that he was wrong. > Definition schmefinition. Faced with my example sets: > > > A = { "0", "10", "11", "100", ... } of (two-ended!) strings over > > > alphabet {0,1} starting with 1 > > > B = N ... the set of naturals (including 0), which we might represent > > > in binary > > > C = { 0, 10, 11, 100, ... } of integers whose decimal representation > > > only includes digits 0 and 1 (no sign) The reason that Chandler brings up sets A, B, and C is in hopes that TO would see that his Bigulosity can't assign a sensible set size to these three sets, and so cardinality is a much more sensible set size. But suppose TO were to see (what Chandler considers to be) the error of his ways, and learns that cardinality really is a more sensible set size -- in short, Chandler convinces that TO is "wrong" and so TO corrects himself. Would this mean that, say in 2015, TO would write something about set size, and then someone would then write, "Well, according to this post from 2010, TO believes that B and C have different sizes," etc., rather than give him credit for having learned something in 2010? And if this is the case, then why should TO be convinced of anything that Chandler tells him at all? Why should TO learn that everything he's written is "wrong," if after learning this, posters will still quote his "wrong" statements from earlier? And therefore, why should Chandler even bother attempting to convince TO that he's "wrong" if he plans on quoting the "wrong" statements forever anyway? If someone who used to think that 2+2=5 is convinced that 2+2=4 then give him credit for learning that 2+2=4. Don't forever quote him when he used to believe that 2+2=5.
From: Transfer Principle on 28 Jun 2010 03:47 On Jun 27, 12:23 am, Brian Chandler <imaginator...(a)despammed.com> wrote: > Faced with my example sets: > > > A = { "0", "10", "11", "100", ... } of (two-ended!) strings over > > > alphabet {0,1} starting with 1 > > > B = N ... the set of naturals (including 0), which we might represent > > > in binary > > > C = { 0, 10, 11, 100, ... } of integers whose decimal representation > > > only includes digits 0 and 1 (no sign) > ... Transfer decided that the way to avoid awkward consequences of the > obvious bijections A<->B and A<->C, was to strike off bijections that > are not "strong". (Seems a terrible choice of word to me -- I would > have used something like Tconsistent.) So a "strong bijection" is one > which doesn't upset [TO]. OK then, "Tconsistent" it is. > But anyway, we've also been told that the number of naturals varies > anyway, depending on what base they're written in. As I wrote earlier, Chandler's purpose of his sets A,B,C is to convince TO that this is silly. The set of naturals ought to have the same size no matter what base they're written in, but that the only way to accomplish this is to allow sets to have the same size as some of their proper subsets. But this does raise an interesting question that I've asked before, but not received a sufficient answer for. Suppose we wanted to define a set size in ZF with certain reasonable properties (that I'll delineate later in this post). Can one prove that therefore, there must exist a set with the same size as one of its proper subsets? Let's state this formally. We start with ZF and add to its language a new primitive symbol -- but not "tav" or "size" or anything like that, but the symbol "<=" a two-place infix predicate symbol. We also define another symbol "~=" also a two-place infix predicate symbol: x~=y <->def (x<=y & y<=x) Here are some axioms for <= and ~=: 1. <= is a preorder. (Recall that in earlier posts, I've explained that I'm not going to take the time to write phrases such as "linear order," "equivalence relation," since I expect posters to know what these mean. 2. Axy (x<=y v y<=x) 3. Axy ((card(x) = card(y) & card(x) < omega) -> x~=y) (That is, x~=y agrees with cardinality for finite sets.) 4. Axy ((card(x) < card(y) & card(x) < omega) -> (x<=y) & ~(y<=x)) (That is, x<=y agrees with cardinality for finite sets.) 5. Axyz ((x,y,z mutually disjoint & x<=y) -> xuz<=yuz) (Here "u" denotes Boolean union.) So far, this theory is consistent (assuming ZF is), since standard cardinality has all of the properies ascribed to the relations <= and ~=. (Notice that if "x<=y" denotes "there exists an injection from x to y," then "x~=y" denotes "there exists a bijection from x to y" because of the Schroeder-Bernstein Theorem.) But we ask, is there another set size (such as Bigulosity) that also has these properties as well as guaranteeing that sets don't have the same set size as any of their proper subsets, or can we prove from these properties that there _must_ exist a set and its proper subset with the same size. In other words, can we prove the following: Eab (b proper subset a & a<=b) Notice that <= mod ~= is a total order. If we strengthened this to a wellorder, then I believe that we can prove that a set does have the same size as its proper subset. To do this, we let a be a <=-minimal infinite set, which must exist by the wellordering. Thus, for any set c, if c<=a then either a<=c or c is finite. In particular, let b be an infinite proper subset of a. We can prove that b<=a using 4. and 5. (First let x be the empty set and y be a\b in 4. to prove that 0<=a\b. Then let x be 0, y be a\b, and z be b in 5. to prove that 0ub<=(a\b)ub -- in other words, b<=a.) Thus from above, we have either a<=b or b is finite, and since b was chosen to be infinite, we conclude a<=b. QED But since <= mod ~= is only a _total_ order rather than a wellorder, there's no guarantee that any <=-minimal infinite set even exists! Thus, there could be an infinitely descending <=-chain of infinite sets, which is exactly what TO wants, since he has: {1,2,3,4,5,6,7,8,9,...} {2,3,4,5,6,7,8,9,...} {3,4,5,6,7,8,9,...} {4,5,6,7,8,9,...} .... with each set having smaller Bigulosity than the set immediately above it. So now we ask, can we nonetheless complete the proof of: Eab (b proper subset a & a<=b) If so, then there is no set size satisfying all of the desiderata given by TO. If not, then add the negation as an axiom: 6. Aab (b proper subset a -> ~(a<=b)) Then this set size satisfies TO's desiderata. But notice that these axioms define <= and ~= relations, but don't actually define tav or the Bigulosity of any particular set. I do this because so far, I want to focus on the <= and ~= relations themselves and not get bogged down by the names of any of the equivalence classes mod ~=. Once one has figured out <= and ~=, then one can start declaring any set x such that x~=N+ to have the Bigulosity tav. At this point, we can answer Chandler's and MoeBlee's question about the definition of a "strong" (or "Tconsistent") bijection. A "Tconsistent" bijection is a bijection between two sets x and y such that x~=y. Although this definition is actually eliminable down to the primitives (including "<="), it doesn't actually determine which bijections are Tconsistent. In other words, all we can show (assuming that we haven't already proved the negation of 6. above) is that there _exists_ a way to declare certain bijections Tconsistent, but not how to find a way of doing so. Then again, some people might point out that in ZFC, we can assign cardinalities to infinite sets using alephs, but in ZF+~AC, we can't prove that every set is equinumerous with an aleph (assuming consistency), and one wonders what is the canonical cardinality of a set not equinumerous to an aleph. We can prove that cardinalities exist for such sets (at least, we can tell whether the cardinalities are equal via bijections), but not what those cardinalities are. So, we can prove that at least one of the bijections, either A<->B or A<->C, isn't Tconsistent, but we can't tell which one it is -- at least not yet. But I'd like to know more about the provability of the negation of 6. using ZF+1.-5. before proceeding along this line any further.
From: Transfer Principle on 28 Jun 2010 03:55
On Jun 27, 12:18 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > Your sarcasm is ill-directed. That one detects a self-contradiction in > > a definition does not mean that the theory one is using is > > inconsistent. It means the definition is not acceptable, which in this > > case means tav is not a number, being neither finite nor infinite. > I'm not sure about the above contradiction, but just today, you've said > both that tav is an element of N+ and that tav is not an element of N+. > That's an inconsistency. > Most of us regard inconsistency as a bad thing in a theory. To me, the assumptions that lead to inconsistency were those such as "tav is a nonstandard natural number" and "infinite sets must contain at least one infinite element" that lead to inconsistency. These are based on TO's 2005 beliefs. In 2010, TO admits that these lead to inconsistency, since they can lead to tav being both finite and infinite or both in N+ and not in N+. And so, we avoid the inconsistency by noting that tav is _not_ a number (natural, nonstandard, or otherwise). And since it's not a number, it's not a finite number, nor can it be an infinite number. So this inconsistency is avoided. |