Uncountability of the Rationals?
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From: Brian Chandler on 22 Jun 2010 00:03 Transfer Principle wrote: > On Jun 20, 9:29 am, Brian Chandler <imaginator...(a)despammed.com> > wrote: > > Jesse F. Hughes wrote: > > > ...but I'm pretty sure that you haven't > > > said anything that implies the conclusion you've just drawn. > > Not in what you are talking about Jesse. But the fundamental problem > > is obvious: Tony is talking about something else entirely. You're not > > sure -- I'm not sure -- what the "+" is in "N+" > > It refers to the positive naturals, just as "Q+" denotes the > positive rationals, "R+" denotes the positive reals, etc. OK. That is a very minor point, though, compared with the real issue: when Jesse talks about N+ he will mean the set {1,2,3,4,5,6,7,8,9,...} of Pofnats (normal, mathematical, plain-old-finite naturals). But Tony is not really talking about the same set at all -- we have called his the tnats, but tN+ would also do just fine. tN+ includes at least a bunch of "infinite naturals", perhaps "tav" itself, perhaps some "worms", a "twilight zone", and who know, perhaps some old farm implements. Perhaps if you're going to be channelling Tony you can clear up exactly what tN+ is? You could start by looking out some of his "proofs" that the (any?) set of all "naturals" must include some "infinite" ones, and let us know whether you feel you are going to have to defend these proofs. Incidentally, you managed to write quite a long screed about "strong bijections", without as far as I can see capturing in any sense at all the distinction from normal bijections. (In practice "non-strong" just seems to mean "Tony doesn't like it"!) Here's a start for you: consider the 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) I hope you can immediately see canonical bijections A <-> B and A <-> C. Tony claims that B and C have different bigulosities, so your job is to say which of the bijections (or both!) is not "strong". Brian Chandler
From: Transfer Principle on 24 Jun 2010 03:55 On Jun 21, 9:03 pm, Brian Chandler <imaginator...(a)despammed.com> wrote: > Transfer Principle wrote: > > On Jun 20, 9:29 am, Brian Chandler <imaginator...(a)despammed.com> > > wrote: > > > Not in what you are talking about Jesse. But the fundamental problem > > > is obvious: Tony is talking about something else entirely. You're not > > > sure -- I'm not sure -- what the "+" is in "N+" > > It refers to the positive naturals, just as "Q+" denotes the > > positive rationals, "R+" denotes the positive reals, etc. > OK. That is a very minor point, though, compared with the real issue: > when Jesse talks about N+ he will mean the set {1,2,3,4,5,6,7,8,9,...} > of Pofnats (normal, mathematical, plain-old-finite naturals). But Tony > is not really talking about the same set at all -- we have called his > the tnats, but tN+ would also do just fine. tN+ includes at least a > bunch of "infinite naturals", perhaps "tav" itself, perhaps some > "worms", a "twilight zone", and who know, perhaps some old farm > implements. This is something that I've been wondering myself -- whether tav can be an element of a set whose Bigulosity TO is trying to find. Some of my posts have been directed at finding the Bigulosity of such sets, while others assume that TO is only finding the Bigulosity of subsets of standard N+ (i.e, sets whose elements are what Chandler calls "Pofnats"). So TO can prove that his N+ contains "tnats"? I wonder how, since his ICI applies only to algebraic functions of _naturals_, not _sets_ of naturals. > Perhaps if you're going to be channelling Tony you can > clear up exactly what tN+ is? You could start by looking out some of > his "proofs" that the (any?) set of all "naturals" must include some > "infinite" ones, and let us know whether you feel you are going to > have to defend these proofs. Chandler suggests that I go look up some old posts of TO's, and sure enough, by using the search them "tnat," I found a discussion from 2005 between the two of them in which TO discusses these "tnats": TO, 2005: Gee you sure like to hear yourself talk, as opposed to listening to what others are saying. Randy also just accused me of saying that infinite sets require infinite members. You would pretend I am just being stupid, when you folks can't even follow basic logical proof. Each pair of naturals has a difference of 1. The largest member of a set of n distinct naturals will be at least n. Therefore, a set of an infinite number of natural numbers will contain infinite values. If there is no largest finite number, is there a largest finite set of naturals? Why do we consider the naturals to be finite and the set to be infinite, when the members are a finite constant quantity apart frome each other? So according to TO, not just the full set of naturals, but any infinite set of naturals must contain infinite values. But then Chandler responds to TO's post as follows: Chandler, 2005: > Now consider the subset of Tnats that are in fact finite: call them the > FTnats. How big is the set of FTnats? Is it finite? If so, do you not > agree that we can count through the elements - the FTnats, that is - > and know that we will STOP at some point in our ditty. Call the number > at which we stop FTLast. Well, what is FTLast+1 ? (Notice that Chandler used the name "FTnats" in 2005 for the standard finite naturals. Now he uses the term "pofnats.") In other words, TO tells us that his infinite set of tnats must contain infinite values, so Chandler asks us to apply the Separation Schema to this set to obtain a set of pofnats. And so Chandler asks, is the resulting set infinite as in standard theory (even though according to TO, all infinite sets of naturals contain naturals which aren't pofnats), or is the set finite (in which case it would be missing a pofnat, FTLast+1)? So what gives? Earlier in this thread, I mentioned a schema which might fit TO's theory: If phi doesn't contain the symbol "tav," then all closures of: ((En fin. nat. (phi(n)) & Ax (phi(x) -> phi(xu{x}))) -> phi(tav) are axioms. In other words, if all but finitely many naturals share a property phi, then so does tav. From this schema, we can prove that tav contains 0 as an element (since all but finitely many naturals contain 0 -- recall that for naturals m and n, "nem" is equivalent to "n<m"), tav contains 1 as an element (since all but finitely many naturals contain 1), tav contains 2 as an element (since all but finitely many naturals contain 2), and so on. Indeed, tav contains all the pofnats. We can let phi(x) be the formula "x < x+1" in order to prove that "tav < tav+1," and similarly let phi(x) be the formula "x > x-1" to prove that "tav > tav-1." So many instances of ICI are in fact instances of this schema. So tav appears to contain all the pofnats. But it also contains elements which aren't pofnats. For example, the schema tells us that tav contains a maximum element (since all but finitely many naturals contain a maximum element). Furthermore, since that maximum element for all but finitely many naturals n is n-1, we prove that the maximum element of tav is tav-1. Earlier, I was puzzled by the fact that the schema proves that tav is "finite," despite containing all the pofnats. But after thinking about it for a while, I realize that this "finite" isn't the same as "plain old finite." It could refer to Dedekind finite, in which we prove that no bijection between tav and any of its proper subsets exists. In this way, none of tav's proper subsets can have the same Bigulosity as tav. But then, what if we try to do what Chandler does and use the Separation Schema to form the set: w = {ne(tav) | n pofnat} Wouldn't w be a set which can be bijected to one of its proper subsets (via the successor function, of course), which could then be extended to a bijection between tav and its proper subset via: f(x) = x+1, xew = x otherwise As it turns out, we can't do this in this theory -- precisely because there is _no_ first-order formula phi such that: An (phi(n) <-> n pofnat) Believe it or not, in this theory, every subset of tav formed via Separation must either include at least one tnat that isn't a pofnat, or exclude at least one pofnat, whether we like it or not! And any formula that seemingly defines pofnat will end up including elements that aren't pofnats (i.e., "n is a finite natural," written using only the primitive "e" in a standard manner, ends up including objects that aren't pofnats, including tav-1, for example). In other words, the theory differs from ZF, which of course does prove that there is a set containing all and only pofnats. Such a theory is often unpalatable to most sci.math posters, since it imposes additional structure on arbitrary sets. (A similar discussion is going on in one of the Herc threads.) And so, we find out that if we attempt to find a bijection between a set and its proper subset, it turns out that the set contains additional infinite elements which aren't mapped to unique elements under the alleged bijection. Say we have: T = {0,1,2,3,4,5,...} S = {1,2,3,4,5,6,...} and we wish to map T to S by f(t) = t+1. As it turns out, f isn't a bijection, because we really have something like: T = {0,1,2,3,4,5,...,tav-3,tav-2,tav-1} S = {1,2,3,4,5,6...,tav-4,tav-3,tav-2,tav-1} and T contains an element tav-1 such that f(t) isn't in S. These infinite elements automatically appear in this theory, whether we like it or not. There remains a problem with this theory. Doesn't the Axiom of Infinity prove the existence of omega, which is exactly the set of all pofnats? This is why, when I first stated the theory, I stated that the schema should be added to ZF-Infinity rather than ZF, because the schema is inconsistent with omega. In this thread, TO states that he isn't opposed to the Axiom of Infinity, and Virgil agreed that Bigulosity depends on Infinity. But as it turns out, the existence of omega is inconsistent with TO's 2005 statement that any infinite set of naturals must contain at least one infinite natural (i.e., one tnat that isn't a pofnat). TO wrote another interesting post in 2005: TO, 2005: In another post I made the point that we cannot pinpoint the largest finite or the smallest infinite, but that doesn't mean that there aren't infinite members in the set. We can count backwards from ....999999 as easily as counting forwards from 0. We just can't find the dividing line between these "subsets". My point stands unchanged. Note that TO's reference to "...999999" sounds very much like the AP-adics of Archimedes Plutonium. > Here's a start for you: consider the 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) > I hope you can immediately see canonical bijections A <-> B and A <-> > C. Tony claims that B and C have different bigulosities, so your job > is to say which of the bijections (or both!) is not "strong". As I mentioned before, any alleged bijection between a set and one of its proper subsets indicates that the set contains extra elements which aren't mapped to unique elements. In particular, we know that B contains extra elements. There is no set N of pofnats in this theory, but instead the set tav of tnats: tav = {0,1,2,3,4,5,...,tav-3,tav-2,tav-1} What about the binary part? Since TO refers to "...999999," I see no reason why we can't have infinite binary representations for the tnats as well. I'm not sure whether we can identify tav-1 with ....999999 in decimal, or ...111111 in binary (a la AP-adics). We need more info from TO to learn more. Thus the canonical bijection between B and C in ZFC isn't a bijection in this theory at all. Even if tav-1 = ...111111 (decimal), it can't be the same as ...111111 (binary) as one of them would have more digits than the other. I remember TO referring to log_10(tav) and log_2(tav), so these might apply here. And so whether a bijection is "strong" or not really depends on what extra elements are added. My question about even and odd integers really asks which extra elements need to be added to the odd integers. So should it be: {1,3,5,7,9,...,tav-7,tav-5,tav-3,tav-1} or {1,3,5,7,9,...,tav-8,tav-6,tav-4,tav-2} Based on TO's comments, it should be the former.
From: Jesse F. Hughes on 24 Jun 2010 08:15 Transfer Principle <lwalke3(a)lausd.net> writes: > Wouldn't w be a set which can be bijected to one of its > proper subsets (via the successor function, of course), which > could then be extended to a bijection between tav and its > proper subset via: > > f(x) = x+1, xew > = x otherwise > > As it turns out, we can't do this in this theory -- precisely because > there is _no_ first-order formula phi such that: > > An (phi(n) <-> n pofnat) > > Believe it or not, in this theory, every subset of tav formed via > Separation must either include at least one tnat that isn't a pofnat, > or exclude at least one pofnat, whether we like it or not! What you're saying *might* be true, but you have not proved it. And no wonder, since you don't have an explicit theory, as far as I recall. Actually, on reflection, I think you're wrong. Why not define w in the usual way? Namely, w = Intersection{ x c Tav | x is inductive }. I don't see where this will go wrong, but no surprises there. It's still not clear to me how Tony's theory is supposed to differ from ZFC. I know that there are these two odd axioms, ICI and IFR, that implicitly grant extensions to functions, but surely he wants to also take away some axioms of ZFC or amend them in some way? If not, the usual construction of w suffices to define your set of pofnats. -- One these mornings gonna wake | Ain't nobody's doggone business how up crazy, | my baby treats me, Gonna grab my gun, kill my baby. | Nobody's business but mine. Nobody's business but mine. | -- Mississippi John Hurt
From: Jesse F. Hughes on 24 Jun 2010 09:07 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > Transfer Principle <lwalke3(a)lausd.net> writes: > >> Wouldn't w be a set which can be bijected to one of its >> proper subsets (via the successor function, of course), which >> could then be extended to a bijection between tav and its >> proper subset via: >> >> f(x) = x+1, xew >> = x otherwise >> >> As it turns out, we can't do this in this theory -- precisely because >> there is _no_ first-order formula phi such that: >> >> An (phi(n) <-> n pofnat) >> >> Believe it or not, in this theory, every subset of tav formed via >> Separation must either include at least one tnat that isn't a pofnat, >> or exclude at least one pofnat, whether we like it or not! > > What you're saying *might* be true, but you have not proved it. > > And no wonder, since you don't have an explicit theory, as far as I > recall. > > Actually, on reflection, I think you're wrong. Why not define w in the > usual way? Namely, > > w = Intersection{ x c Tav | x is inductive }. > > I don't see where this will go wrong, but no surprises there. It's > still not clear to me how Tony's theory is supposed to differ from ZFC. > I know that there are these two odd axioms, ICI and IFR, that implicitly > grant extensions to functions, but surely he wants to also take away > some axioms of ZFC or amend them in some way? > > If not, the usual construction of w suffices to define your set of > pofnats. Well, this last claim of mine is not so clear, since it's possible that w defined this way still has infinite elements (because, perhaps, every inductive set in this theory has infinite elements). Note, however, (1) That this has not been proved, but merely asserted by Walker, because (2) there is no clearly defined theory here as far as I can see. Finally, (3) even if w has infinite elements, w serves as a canonical choice for Tav, since it is the least inductive set. Why doesn't Tony just use w? -- "[In the movie, Tom Green] delivers a child, severs the umbilicus with his teeth and then swings the baby over his head before tenderly handing it to the stunned, blood-spattered mother[...] This was, I have to say, a bit much." -- New York Times movie reviewer A. O. Scott
From: Brian Chandler on 24 Jun 2010 09:22
Transfer Principle wrote: > On Jun 21, 9:03 pm, Brian Chandler <imaginator...(a)despammed.com> > wrote: > > Transfer Principle wrote: > > > On Jun 20, 9:29 am, Brian Chandler <imaginator...(a)despammed.com> > > > wrote: > > > > Not in what you are talking about Jesse. But the fundamental problem > > > > is obvious: Tony is talking about something else entirely. You're not > > > > sure -- I'm not sure -- what the "+" is in "N+" > > > It refers to the positive naturals, just as "Q+" denotes the > > > positive rationals, "R+" denotes the positive reals, etc. > > OK. That is a very minor point, though, compared with the real issue: > > when Jesse talks about N+ he will mean the set {1,2,3,4,5,6,7,8,9,...} > > of Pofnats (normal, mathematical, plain-old-finite naturals). But Tony > > is not really talking about the same set at all -- we have called his > > the tnats, but tN+ would also do just fine. tN+ includes at least a > > bunch of "infinite naturals", perhaps "tav" itself, perhaps some > > "worms", a "twilight zone", and who know, perhaps some old farm > > implements. > > This is something that I've been wondering myself -- whether > tav can be an element of a set whose Bigulosity TO is trying > to find. I think that one of Tony's working principles is that anything he can see with his razor-sharp intuition to be true about finite sets is also true of "infinite sets" (at least unless his RSI makes him aware of a contradiction). Now an obvious intuition about a finite set of adjacent naturals strung out along a number line is that we could measure the "size" of the set by the following method: Take a piece of string, wrap it completely around the set of naturals in a loop, and pull tight; then remove, measure the length of the loop and divide by 2. Now the pulling tight step ensures that the string is a minimal loop, held in place by the smallest natural on the left and the largest natural on the right. Tony is Quite Sure that exactly the same is true of "infinite sets". Therefore, for the loop of string to be tight, there must be something at the right end around which it is tightly wound. Fast-forward to infinite set of adjacent naturals. Tony knows the string must still be tightly wound around something at the right end, and it must (in some sense!) be a "number", but Tony _can_ see that it really can't be a perfectly ordinary natural ("pofnat"), so hypothesizes that it any of various miasmas, "infinite naturals", "unidentifiable naturals" and so on. When talking about the bigulosity of a set of adjacent naturals, I think it is fairly clear that there must always be this mysterious entity at the right end, keeping the string taut. (Among the properties of this mystery object, I'm fairly sure we have sighted "non-existence" at least once!) But more usually, a property of the thing at the right end keeping the string taut is that it is "infinite". This is basically how Tony derives his necessary existence of "infinite naturals" within any set of naturals which we call "infinite". So anyway, Tony deals with "numbers", and elastic concept that surely includes 'tav'. <snip, for the narrative is long, and dawn not infinitely far off> > Chandler suggests that I go look up some old posts of TO's, > and sure enough, by using the search them "tnat," I found a > discussion from 2005 between the two of them in which TO > discusses these "tnats": > > TO, 2005: > Gee you sure like to hear yourself talk, as opposed to listening to > what others > are saying. Randy also just accused me of saying that infinite sets > require > infinite members. You would pretend I am just being stupid, when you > folks > can't even follow basic logical proof. Right, Tony starts off by slinging a few (moderate) insults. > Each pair of naturals has a > difference > of 1. The largest member of a set of n distinct naturals will be at > least n. > Therefore, a set of an infinite number of natural numbers will contain > infinite > values. If there is no largest finite number, is there a largest > finite set of > naturals? Why do we consider the naturals to be finite and the set to > be > infinite, when the members are a finite constant quantity apart frome > each > other? Well, there you go. It's the loop of string argument. But this argument is totally bogus. (Surely Walker can see that?) > So according to TO, not just the full set of naturals, but any > infinite set > of naturals must contain infinite values. But then Chandler responds > to > TO's post as follows: > > Chandler, 2005: > > Now consider the subset of Tnats that are in fact finite: call them the > > FTnats. How big is the set of FTnats? Is it finite? If so, do you not > > agree that we can count through the elements - the FTnats, that is - > > and know that we will STOP at some point in our ditty. Call the number > > at which we stop FTLast. Well, what is FTLast+1 ? > > > (Notice that Chandler used the name "FTnats" in 2005 for the standard > finite naturals. Now he uses the term "pofnats.") The narrative is not only long, but verges on the tedious. Yes, this is how mathematics is done -- I can call things anything I like, as long as I explain what the words mean. <snip a bit more> > So what gives? Earlier in this thread, I mentioned a schema which > might fit TO's theory: > > If phi doesn't contain the symbol "tav," then all closures of: > > ((En fin. nat. (phi(n)) & Ax (phi(x) -> phi(xu{x}))) -> phi(tav) > are axioms. In other words, if all but finitely many naturals share a > property phi, then so does tav. OK, so if phi(n) is the property "n is not tav", this means that given an infinite set of naturals, none of which is 'tav', this implies that tav is not tav. Doesn't sound likely to me. <skip lots more> > > Here's a start for you: consider the 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) > > I hope you can immediately see canonical bijections A <-> B and A <-> > > C. Tony claims that B and C have different bigulosities, so your job > > is to say which of the bijections (or both!) is not "strong". > > As I mentioned before, any alleged bijection between a set and one > of its proper subsets indicates that the set contains extra elements > which aren't mapped to unique elements. Huh? Consider the mapping f: n -> 2n (over the (proper, mathematical) naturals) This is a map from N to the subset being the evens. Can you explain what elements are not mapped to "unique??" elements? > In particular, we know that B contains extra elements. Huh? You are trying to use "Bagulosity theory"? Instead of sets being like boxes, they are string bags, so even after you have put an element in the set it might leak out through a hole, or someone might slip something else in while you're not looking? > There is no > set N of pofnats in this theory, but instead the set tav of tnats: > > tav = {0,1,2,3,4,5,...,tav-3,tav-2,tav-1} > > What about the binary part? Since TO refers to "...999999," I see > no reason why we can't have infinite binary representations for the > tnats as well. I'm not sure whether we can identify tav-1 with > ...999999 in decimal, or ...111111 in binary (a la AP-adics). We > need more info from TO to learn more. > > Thus the canonical bijection between B and C in ZFC isn't a > bijection in this theory at all. Even if tav-1 = ...111111 (decimal), > it can't be the same as ...111111 (binary) as one of them would > have more digits than the other. I remember TO referring to > log_10(tav) and log_2(tav), so these might apply here. > > And so whether a bijection is "strong" or not really depends on > what extra elements are added. It also might depend on what day of the week it is, whether the moon is in Saturn, or Saturn in the moon. Or not. I confess, Transfer, that you really mystify me. I can understand people like Tony, who are very confused, very confident, yet somehow not quite entirely stupid. Tony has ideas, and unfortunately lacks the skills to see that these ideas are No Good. But you don't seem to come up with any ideas at all, except this hopeless quest to discover the "alternative theory" in which people like Tony are "working". I just can't understand the motivation for that. Brian Chandler |