Uncountability of the Rationals?
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From: K_h on 17 Jun 2010 21:10 "Tony Orlow" <tony(a)lightlink.com> wrote in message news:7dbc2694-4152-42df-9ff6-2b709ad58dda(a)e35g2000vbl.googlegroups.com... On Jun 16, 7:59 pm, "K_h" <KHol...(a)SX729.com> wrote: > "Tony Orlow" <t...(a)lightlink.com> wrote in message > > news:f6d161b0-150a-4c47-bea4-bc898e5a0f87(a)i31g2000yqm.googlegroups.com... > On Jun 15, 8:22 pm, Transfer Principle <lwal...(a)lausd.net> > wrote: > > > > > necessary. I understand that that provides the > set-theoretic basis for N, but to view quantities > as sets might not capture all the qualities of > quantities. That may sound like gibberish, but my Not sure what you have in mind here. We can think of the quantity of distance of the interval [0,2] as being twice as big as the quantity of distance of [0,1] but the sets for the real intervals [0,1] and [0,2] both have the same cardinal size of 2^[ALEPH_0]. So we need to be clear in our terminology and not use quantity for cardinal size when we're talking about a quantity of distance (for example). I hope that is clear. > > Definition: > > For an ordinal S, the cardinal of S, notated |S|, is the > > least ALEPH_K equinumerous to S (if S is infinite) or is > > S > > itself (if S is finite) where K is an ordinal. > > Right, that definition is based on the previous one, > which is what is in question, not in terms of consistency > with ZFC, but in terms of whether the axioms imply the > definition. If you don't like a given definition then it is best for you to say what you don't like about it. _
From: Virgil on 17 Jun 2010 21:58 "Tony Orlow" <tony(a)lightlink.com> wrote in message news:7dbc2694-4152-42df-9ff6-2b709ad58dda(a)e35g2000vbl.googlegroups.com... > Right, that definition is based on the previous one, > which is what is in question, not in terms of consistency > with ZFC, but in terms of whether the axioms imply the > definition. Axioms do not imply definitions.
From: Jesse F. Hughes on 17 Jun 2010 22:01 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > > Right. Thanks for that correction. > > In case others had trouble (like me) finding the correction, here it is: > > Rotwang <sg552(a)hotmail.co.uk> writes: > >> ITYM (Ay,z in x)(y in z or z in y) & >> (Aw,y,z in x)((w in y & y in z) -> w in z) & >> (A z)(((Ay)( y in z -> y in x ) & (Ey)( y in z )) -> >> (Ey in z)(Aw in z)(y in w or y = w) ) & > ^^^^^^^^ >> (A y in x)(A z in y)( z in x ) > > My expression of the well-order condition omitted "or y = w". Oops. One more correction. In the first clause, we need as well "or y = z", and so it should be: x is an ordinal iff (Ay,z in x)(y in z or z in y or y = z) & ^^^^^^^^ (Aw,y,z in x)((w in y & y in z) -> w in z) & (A z)(((Ay)( y in z -> y in x ) & (Ey)( y in z )) -> (Ey in z)(Aw in z)(y in w or y = w) ) & ^^^^^^^^ (A y in x)(A z in y)( z in x ) -- "The papers are currently at journals. [When published,] make no mistake, there will be no place on this planet where you can hide. Remember, I'm not talking about something vague here. I'm talking about publication in journals." James S. Harris. Wow. Journals.
From: Tim Little on 18 Jun 2010 01:10 On 2010-06-17, Jesse F. Hughes <jesse(a)phiwumbda.org> wrote: > I don't really see that we can guarantee to get as near to any > arbitrary real as we want by traversing this tree. Shouldn't it really have a +/- at the front as well, or are we talking only of nonnegative reals? With either adjustment, it is true. Such a sequence can be obtained by taking successive logarithms of absolute values, noting the signs at each step. For example, 3 is approximated to eight decimal digits by 2^2^2^-2^-2^-2^-2^2^-2^2^-2^2^-2^-2^2^2^-2^2^-2^2^2^-2^2^-2^2^-2^2^-2^-2^2^0 It is fairly easy to prove that a slightly modified lexicographic order on distinct sign sequences agrees with the usual ordering on the reals. Proving that distinct reals yield distinct sign sequences requires somewhat more, but can be done by analysing stability under iteration of log_2|x|. - Tim
From: Jesse F. Hughes on 18 Jun 2010 08:12
Tim Little <tim(a)little-possums.net> writes: > On 2010-06-17, Jesse F. Hughes <jesse(a)phiwumbda.org> wrote: >> I don't really see that we can guarantee to get as near to any >> arbitrary real as we want by traversing this tree. > > Shouldn't it really have a +/- at the front as well, or are we talking > only of nonnegative reals? With either adjustment, it is true. If you say so. > Such a sequence can be obtained by taking successive logarithms of > absolute values, noting the signs at each step. For example, 3 is > approximated to eight decimal digits by > > 2^2^2^-2^-2^-2^-2^2^-2^2^-2^2^-2^-2^2^2^-2^2^-2^2^2^-2^2^-2^2^-2^2^-2^-2^2^0 > > It is fairly easy to prove that a slightly modified lexicographic > order on distinct sign sequences agrees with the usual ordering on the > reals. Proving that distinct reals yield distinct sign sequences > requires somewhat more, but can be done by analysing stability under > iteration of log_2|x|. It's still not at all obvious to me, even with your example of 3 to eight digits, nor do I see easily how to modify lexicographic ordering to make it work out in the tree, but I am not quite curious enough to dig in and see. If you say that finite and infinite paths in Tony's tree correspond in a natural way to reals, and every real is thus represented, then I'll accept it. -- Jesse F. Hughes "As you can see, I am unanimous in my opinion." -- Anthony A. Aiya-Oba (Poeter/Philosopher) |