Uncountability of the Rationals?
in [Math]
|
Prev: Collatz conjecture
Next: Beginner-ish question
From: Tim Little on 25 Jun 2010 01:59 On 2010-06-24, Transfer Principle <lwalke3(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
From: Tony Orlow on 25 Jun 2010 10:18 On Jun 20, 3:40 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <f4ac19cf-f3be-44ab-90b7-62cc17275...(a)u26g2000yqu.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > On Jun 18, 10:37 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > On Jun 17, 12:31 pm, Virgil <Vir...(a)home.esc> wrote: > > > > > > > David R Tribble wrote: > > > > > > >> So the question becomes, what axiom justifies your logical > > > > > > >> leap, applying a property of finite sets to infinite sets without > > > > > > >> largest members? > > > > > > Tony Orlow wrote: > > > > > > > There is no leap. > > > > Then there can be no difference between finite sets and infinite sets, > > > > thus everything true of finite sets must be true of infinite sets, thus > > > > all infinite sets must be finite since that is true of finite sets. > > > > At least following TO's logic. > > > > This is, incidentally, a problem that occurs with many posters > > > who want the properties of all finite sets to extend to all > > > infinite sets. In general, such posters want more properties > > > to extend from finite sets to infinite sets than standard > > > theory allows, but they certainly don't want the property of > > > _finiteness_ to extend to infinite sets! > > > > So we see how any schema of the form: > > > > (Ax (x finite -> phi(x)) -> Ax phi(x) > > > > fails if we let phi(x) be "x finite" (and, of course, at least > > > one infinite set exists). > > > > TO attempts to prevent this by preventing phi from being just > > > any function, but instead limiting the schema to _algebraic_ > > > functions using a few chosen operations. > > > Hi Transfer - > > > Actually, I avoid this conundrum by resticting my properties to > > statements of inequality among formulaically expressed quantities. By > > focusing on inequalities I am establishing quantitative order among > > different expressions, and by extending those expressions to the > > infinite case, I am concocting a method of distinguishing among > > different countable infinities. Whether they are technically algebraic > > or not is of little consequence. For the purposes of IFR it is only > > required that they be monotonic bijections between N+ and any set. > > TO allow a monotonic function between N+ and any other set, that other > set must be ordered and ORDER ISOMORPHIC TO N+, so that Bigulosity > cannot be applied either to unordered sets or to ordered sets whose > ordering differs from that of N+. Thus, for example, if one reorders N+ > to make odds less than evens then bigulosity does not apply to the > reordered set. > > It cannot even be applied to N+ as an unordered set. > > Thus bigulosity applies to, at most, only infinite well ordered sets > with only one non-successor.- Hide quoted text - > > - Show quoted text - In cases where you have the exact same set in different order, it is the exact same set with the exact same Bigulosity, as the Axiom of Extensionality entails. TOny
From: Tony Orlow on 25 Jun 2010 10:32 On Jun 21, 10:53 pm, Transfer Principle <lwal...(a)lausd.net> 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. > > Notice that mathematicians do not agree whether N should be > {1,2,3,4,5,6,7,8,9,...} or {0,1,2,3,4,5,6,7,8,9,...}. If the > former, then N+ = N and the "+" is redundant. But in either > case, N+ = {1,2,3,4,5,6,7,8,9,...} regardless of which > convention we're using, which is why we include it. > > This raises another question. I've noticed that those who > are ZFC (Herc-)religionists have a tendency to identify N > with omega, and thus they consider 0eN, since 0eomega is > uncontroversial (since omega\{0} wouldn't even be an > ordinal at all.) But those whom they call using five-letter > insults tend to consider ~0eN. In particular, ~0eN to WM, and > regardless of whether TO considers 0eN, he clearly prefers > working with omega\{0} to omega itself. For the purposes of IFR, N+, the positive naturals, is the standard countably infinite set. I don't mind 0. It's a wonderful number, without which we would have no infinity or useable science. But, the equations work properly using N+ without 0. We don't generally talk about the "0th element", now, do we? Peace, Tony
From: Tony Orlow on 25 Jun 2010 10:51 On Jun 22, 12:03 am, Brian Chandler <imaginator...(a)despammed.com> wrote: > 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. Incorrect. When I speak of N+ I mean the positive "pofnats". I have introduced the T-riffics as a representation of infinite naturals in *N, but N+ is N+. > > 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"!) It really means the relationship can't be formulaically quantified in any way that leads to quantitiative ordering using IFR. In order to state that one diverging formula is greater than another in the limit, one needs to extend inductive proof in a way that orders such formulas, much like Big O notation in computer and information science. > > 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". Now, you are bringing N=S^L into the picture. What you don't understand is that the set must be bijected with N+ in some manner. If you are going to allow for tav elements in set A, then you are talking about log2(tav) bit positionss (still countably infinite, so that shouldn't bother you). If you want to say there are tav bit positions possible, then you have 2^tav bit strings in your list (still countably infinite, as far as Bigluosity is concerned, which might bother you). Set B *might* be represented in digital format and thus become strings of characaters rather than quantities, and if so, then we parameterize according to string length and digital number base. As pure quantities, N is N having Bigulosity tav+1, and not some list of strings. When it comes to set C, it *looks* like set A, however, if there are tav+1 naturals (starting at 0) in decimal, there are log_10(tav+1) bit positions. Given N=S^L, with S=2 and L=log_10(tav+1), we have 2^log_10(tav+1)=log_5(tav+1) elements. Yes, different rules work with languages than with pure quantities. That doesn't make the system inconsistent. You rmember enough to make fun and disparage, but not to actually make sense? Motivation guides thought and action. Tony
From: Jesse F. Hughes on 25 Jun 2010 10:48
Tony Orlow <tony(a)lightlink.com> writes: > For the purposes of IFR, N+, the positive naturals, is the standard > countably infinite set. Can you tell us exactly how you define N+? This question may seem silly, but it really is important. For comparison, I would define N+ to be the least inductive_1 set, where ind_1(x) <-> 1 in x & (Ay)(y in x -> y+1 in x). Thus, N+ has the property ind_1(N+) & (Ax)(ind_1(x) -> N+ c x). Is this also your definition? -- Jesse F. Hughes "I have written many words to sci.math, some of them are not even meaningless." --Ross Finlayson |