Uncountability of the Rationals?
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From: Tony Orlow on 25 Jun 2010 11:35 On Jun 24, 3:55 am, Transfer Principle <lwal...(a)lausd.net> 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. 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. Hi Transfer - I didn't realize this thread was continuing. Almost sorry for twisting your neurons a bit, but I think you are enjoying it. :) What my proof about N+ indicates is not that its Bigulosity, tav, is either finite or infinite, but that it cannot be either, and therefore can't actually exist. Its use in Bigulosity is as a virtual, or parametric, size. Since we know it is at least as large as any finite, we can apply ICI to this variable, and order countably infinite sets according to Bigulosity. > > > 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. Really, I meant "actually" infinite, or uncountable. Countably infinite sets have no absolute size in my theory, either finite or infinite. 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)? There is no last element of N+, hence no absolute size. > > 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. All "pofnats" (plain old finite naturals) neN+ share the property that 1/n>0. This cannot be considered true in any standard sense for infinite n. > > 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. I consider the statement, "Ax xen->xem" to imply "|n|<|m|", but I don't particularly consider "nem" to be equivalent to "n<m". > > 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. Okay. > > 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. Then this schema does not represent Bigulosity. N+ has no largest member, and tav is not an absolute size. If it were, then N+ would contain tav, but it cannot, because then tav would be finite, and tav +1 would be a finite natural not included in N+. See? Tav cannot exist as an absolute size, but only as a standard countable infinity for comparison and ordering purposes. > > 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. Well, it is Dedekind infinite, in that such a bijection can surely exist. It just doesn't prove equibigulosity. > > 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). Okay, I should clear something up, here, as a misconception seems to be spreading. Bigulosity does not just measure subsets of N+. It measures any set formulaically bijected with N+, which may include elements outside of N+. Secondly, any uncountable set, when ordered linearly (or with a finite number of child nodes each), must contain an element which is infinitely distant from the foundation of the set. A countably infinite set never does, which is why it's not "actually" infinite. > > 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. Eh... tav-1 is not an element of N+. There is simply the matter of an extra element, 0, in one of the sets. IFR provides a means to calculate this difference i the finite case, and ICI extends the method to the infinite (or at least as large as any finite, such as tav). > > 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). Only if I concede that omega is in fact infinite, which I don't. > > 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. Professor Plutonium did not invent the adic numbers. Neither did I. :) > > > 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. No it doesn't. There are no elements unmapped. There is a difference in frequency of elements as the range increases. That is all IFR detects, but it does it accurately. > > 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: No, tav is the Bigulosity of N+, not a set. > > 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. That's why I invented the T-riffics, with two ends and an uncountable string of digits in between, capable of including infinite bit positions, both positive and negative, and therefore capable of expressing infinite and infinitesimal values. > > 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. If the strings of B are bijected string-wise with those of C, they may be considered equibigulous, but when the strings of C are specified to be binary naturals, then one needs to decide how one is comparing the two sets, and decimal notation comes into play with N=S^L. > > 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. One may conjecture as to whether tav is even or odd, or divisible by any given natural. However, tav doesn't exist anywhere on the *N number line, so those questions are unanswerable and ultimately irrelevant. Thanks for your comments. Tony
From: Tony Orlow on 25 Jun 2010 11:55 On Jun 24, 9:22 am, Brian Chandler <imaginator...(a)despammed.com> wrote: > 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'. Okay, Brain. 'Tav' is most surely elastic, and not a tight right end. It would be helpful if you would stop misrepresenting things. > > <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. I've been pretty good lately, don't you think? You're up to your old tricks, and you have to dredge up five-year-old posts to make yourself look better. Lame. > > > 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?) > ICI in the making, Baby. Why don't you go find my baby pictures, too. I bet there's something embarrassing there. > > > 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. I explained "pofnat" for you. > > <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. It's probably logically equivalent to my statement that tav is not finite, but not infinite, and therefore is self-contradictory or non- existent. Sounds spot-on 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? Okay, Brain, you're right here. There are no unmapped elements. But, Transfer is thinking. That's a good thing. I corrected him in an earlier post today about that. > > > 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? Actually, I like that term. It sounds Hobbitlike to me, O Bilbo. "Yes, the Bagginses has my preciousssss" - Gollum > > > 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. Brainiac - Personal attacks are a waste of time. They only make you look belligerent. Transfer is trying to understand what I'm trying to explain, and I appreciate that. He is having a little trouble, since my theory, as much as it is formalized, is at odds in ways with standard theory which are hard to pinpoint. In our recent interactions, he's gleaned more than you have in years. You remember what you can laugh at, for that purpose alone, but laughter is no replacement for understanding. So, laugh away, enjoy yourself, and see where that takes you. > > Brian Chandler- Hide quoted text - > Peace, Tony
From: Tony Orlow on 25 Jun 2010 11:57 On Jun 24, 10:01 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Brian Chandler <imaginator...(a)despammed.com> writes: > >> 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. > > In Walker's defense, he said that the formula phi should not contain the > symbol tav. Good point. Maybe phi should not include "finite" either, but at least, specifying an inequality without a limit of 0 is sufficient to avoid supposed contradictions. Tony > > -- > Jesse F. Hughes > > "[Iota]'s the smallest infinitesimal, Russell, there are smaller > infinitesimals." -- Ross Finlayson
From: Tony Orlow on 25 Jun 2010 12:09 On Jun 24, 11:30 am, David R Tribble <da...(a)tribble.com> wrote: > Transfer Principle (Walker) quoted: > > >> Tony Orlow wrote, crica 2005: > >> 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? > > Brian Chandler wrote: > > Well, there you go. It's the loop of string argument. But this > > argument is totally bogus. (Surely Walker can see that?) > > You left out the part where Tony decides that there are > "finite but unbounded" sets, like N. That is, they contain > only finite elements (numbers), but since they have no > "identifiable" largest element, they are "unbounded". Foul: Misrepresentation I never said any of that. There are uncountable sets of only finite numbers such as {xeR: 0<x<1}. There are countably infinite sets which are bounded such as {xeQ,neN: q=1/n}, with a largest element. Where you have a minimum positive difference between successive elements (like 1), then any infinite number of them means an infinite difference between the first and last. > > The situation is, and has always been since Tony started > posting, that he wants everything to act like "numbers" > including set "sizes", and specifically that they can be mapped > to points on the real number line and obey the usual rules > of (finite) arithmetic. Thus his constant mention of properties > "without regard to finiteness", "infinite induction", "formulaic" > infinities, infinite set members, "unit infinities", "bogus" > bijections, > and so forth. Excuse me for trying to make math use "numbers". My bad. > > Look, Tony has demonstrated that he's a Crank many times > over, with all of the usual symptoms: unwillingness to change his > ideas even in the face of blatant illogic, abusive replies to > disbelievers in his faith, proofs by declaration, assumed authority > on terminology and topics he has no clue about, expectation of > the impact of his ideas on the world, etc. This isn't an "abusive reply" or obnoxious personal attack, is it? The sliver you perceive in my eye is but a reflection of the log in yours. > > I admit to finding the last few years occasionally entertaining, > but lately it's become, well, boring. Nothing new is being offered, > and more importantly, I'm not learning much new from the > posters who actually know something about set theory. I left for a few years. Maybe that's why you were bored. I think the fact that you're boring to most besides yourself probably goes back quite a number of years before that. Probably the tuba lessons with the stick.... TOny
From: Tony Orlow on 25 Jun 2010 12:11
On Jun 25, 1:59 am, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-24, Transfer Principle <lwal...(a)lausd.net> wrote: > > > In other words, if all but finitely many naturals share a property > > phi, then so does tav. > > In particular "is a finite ordinal" is true of tav. By the defining > property of ordinals, it thus contains *only* finite ordinals. > > > So tav appears to contain all the pofnats. But it also contains > > elements which aren't pofnats. > > And so we have a contradiction. Gee, that didn't take long. > > - Tim I am the one who pointed out the contradiction with tav. It doesn't exist any more than omega. It's a limit, not a number. That does not invalidate the theory whatsoever. Tony |