Uncountability of the Rationals?
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From: cbrown on 27 Jun 2010 02:12 On Jun 26, 6:47 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 26, 8:53 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > Nonetheless, I'm fascinated to learn that tav is in N+. N+ contains > > nothing but finite numbers, as you've said previously. Thus, tav is > > a finite number. > > > Agreed? > > Absolutely not!!! > > As already proven, tav is neither finite nor infinite, and therefore > exists only virtually, or "potentially". So when you talk about tav, it's from the same point of view as when I talk about samekh. Samekh is the unique integer which is greater than 5 and is not greater than 5, and therefore exists only virtually or "potentially". Agreed that samekh less than or equal to 5? Absolutely not! Explicitly, samekh is greater than 5, /by definition/!! The thing I love best about samekh is that it is provable that samekh tastes of raspberries and lavender with a hint of cabbage. Cheer - Chas
From: Brian Chandler on 27 Jun 2010 03:23 MoeBlee wrote: > On Jun 25, 9:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > On Jun 24, 10:59 pm, Tim Little <t...(a)little-possums.net> wrote: > > > > > On 2010-06-24, Transfer Principle <lwal...(a)lausd.net> wrote: > > > > 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. > > > > But now TO is posting again, so we can fix this error based > > on TO's latest comments. > > > > Let's summarize what TO has stated. > > > > 1. The set omega is exactly what we expect it to be. > > 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, but I believe ordinals are still schlock, and "omega" can be called "tav" if desired, and tav is (simultneously): an ordinal, an element of omega, a finite natural, non-existent (now _that_ must be a predicate!), a limit, contradictory, and probably some things I've forgotten, even though I do remember that I coined "tav" as a name when I wished to try "declaring a unit infinity". > > ... we can just go back to ZF and assign Bigulosities to the sets > > of ZF, as I originally intended to do. This means that we can > > go back to determining which bijections are "strong." > > I haven't been reading all the ruminations here. Please, what is the > definition of a 'strong bijection'. 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) .... 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 DL. But anyway, we've also been told that the number of naturals varies anyway, depending on what base they're written in. > > 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". Then what is the definition of > 'Bigulosity of x' such that every x ('x' ranging over all objects, I > presume?) has a bigulosity assigned to it.
From: Tony Orlow on 27 Jun 2010 10:16 On Jun 26, 10:36 pm, David R Tribble <da...(a)tribble.com> wrote: > Tony Orlow wrote: > >> 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. > > Jesse F. Hughes wrote: > >> Is it your opinion that N+ has a last element? > > Tony Orlow wrote: > >> No, and that's beside the point. > > You entirely missed the point of Jesse's question. > > You claim that for an infinite ordered set with a minimum difference > between members, the set must have an infinite difference between > the first and last elements. If it is truly infinite, that is, uncountable. > > N+ is an infinite set with a natural order, and has a minimum > difference between its members. So by your claim, the set must > have an infinite difference between its first and last members. There is no last, and the extent of N+ is not infinite. N+ is unboundedly finite, or potentially infinite. > > So Jesse asked the obvious question: does N+ therefore have a > last element? > > If not, then perhaps you meant something other than what you > actually wrote? We've been over this a thousand times. TOny
From: Tony Orlow on 27 Jun 2010 10:19 On Jun 26, 10:37 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > On Jun 26, 8:57 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Tony Orlow <t...(a)lightlink.com> writes: > >> > On Jun 26, 12:01 am, Transfer Principle <lwal...(a)lausd.net> wrote: > >> >> On Jun 24, 10:59 pm, Tim Little <t...(a)little-possums.net> wrote: > > >> >> > On 2010-06-24, Transfer Principle <lwal...(a)lausd.net> wrote: > >> >> > > 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. > > >> >> But now TO is posting again, so we can fix this error based > >> >> on TO's latest comments. > > >> > There is no error. The contradiction concerning omega or tav is > >> > deliberate. It doesn't exist as a number, as evidenced by its own > >> > self- contradiction. > > >> Well, *that*'s encouraging! > > >> Your system is purposely, not accidentally, inconsistent. > > >> -- > >> Jesse F. Hughes > > >> "I think the Iraqi people owe the American people a huge debt of > >> gratitude." -- G.W. Bush in January, 2007 > > > Disproof by contradiction is a well-established method in logic, > > dingaling. > > Yes, it is. > > So? > > It doesn't mean that people find inconsistent theories in classical > logic very interesting. > > (Though, I really do appreciate your patient tutelage in logic, kind > master. Thanks!) > > 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. Tony
From: Tony Orlow on 27 Jun 2010 10:23
On Jun 26, 10:32 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > >> > Rather, > >> > |N+|=tav > >> > Thus taveN+ > >> > tav=|N+| and |N+|=tav. > >> > True. > > >> That's not what I proved and you didn't point out any mistake. > > >> Nonetheless, I'm fascinated to learn that tav is in N+. N+ contains > >> nothing but finite numbers, as you've said previously. Thus, tav is > >> a finite number. > > >> Agreed? > > > Absolutely not!!! > > > As already proven, tav is neither finite nor infinite, and therefore > > exists only virtually, or "potentially". If we assign this size "tav" > > to that set, then it is not defined as belonging to either group of > > quantities. > > I don't understand any of those words. Let's try again. Do you agree > with each of the following? > > (1) Every element of N+ is a finite natural number. Yes. > > (2) Tav is an element of N+. No, tav is not a number, and therefore cannot be a member of this set of numbers. Tav is a variable Bigulosity, not any kind of actual count that would qualify as even an extended natural. > > You said (2) *just now* and, unless I'm much mistaken, you've agreed > with (1) quite recently. Now, bear with me here and see if you can't > understand how (1) and (2) imply > > (3) Tav is a finite natural number. If tav were a number, and there were exactly tav elements in N+, then tav would be an element of N+. Tav, however, is not a number or a real set size. It's a countably infinite variable. Tony |