Uncountability of the Rationals?
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From: David R Tribble on 26 Jun 2010 22:56 Tony Orlow wrote: > 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. Granted, you claim that omega can't exist as a natural, counting number, set size, or bigulosity. But how does all that keep it from existing as some other kind of number? Perhaps you could clarify what you mean by "number".
From: Tim Little on 26 Jun 2010 22:59 On 2010-06-26, Tony Orlow <tony(a)lightlink.com> wrote: > There is some proof left to do, but the fact is that taking log(|x|) > repeatedly tends around 0, and so the probability of the root of any > number becoming 0 through this recursive process eventually > approaches 1. False. The probability is actually zero. - Tim
From: David R Tribble on 26 Jun 2010 23:12 Tony Orlow wrote: > N+ is the standard countably infinite set of Bigulosity "tav". All > other countably infinite sets are measured therewith. Well, then how about showing us how to measure the bigulosity of a set larger[*] than N+: G = { 1, {1}, 2, {1,2}, {2,3}, 3, {1,2,3}, {2,3,4}, {4,5,6}, 4, {1,2,3,4}, {2,3,4,5}, {3,4,5,6}, {4,5,6,7}, ... } You can see how G is built up from members of N+, where the members are displayed in rows above for clarity. Each row starts with the next natural k from N+, followed by k more elements, each one being a set of k elements itself. Thus each row contains k+1 elements. So it would appear that the size[*] of G is something like: (1+1) + (2+1) + (3+1) + (4+1) + ... = 2 + 3 + 4 + 5 + ... It's obvious that G is a countable[*] set. So what would the bijection between N+ and G be, in order to compute the bigulosity[*] of G? [*] Terms in Tony's BO Theory.
From: Tony Orlow on 27 Jun 2010 01:26 On Jun 26, 10:19 pm, David R Tribble <da...(a)tribble.com> wrote: > Jesse F. Hughes wrote: > >> Is it your opinion that N+ has a last element? > > Tony Orlow wrote: > > No, and that's beside the point. If there are an infinite number of > > unit increments between any two values then there is an infinite > > difference between them. If not, then it's a countable sequence. > > In R (the reals), there is no infinite "number of unit increments" > and no infinite difference" between any two elements in R. There are no infinite quantitative differences between any two standard real values, however, between any two distinct real numbers do there not exist an uncountable number of others? But, that's not the point, still. Please pay attention. The reals do not have any minimum finite difference between successive elements. The naturals do. There is at most (or exactly) one natural per unit interval. That's how the unit interval is defined. > Yet it is not a countable set. No, R has internal uncountability in any finite segment, whereas N has 1 element per unit segment, a finite range. > > In contrast, the ordered set: > S = N U { w } = { 0, 1, 2, 3, ..., w } > has an "infinite distance" between w and all of its other > members, yet it is a countable set. Only if you consider w, or 'tav', or whatever, to be some actual infinite number, um, after all the other ones ended, even though they have no end. Sorry. x-1<x. It's a primordial fact. Tav is not a specific value, and neither is aleph_0. Aleph_1 is another story.... > > > N+ is countable. Some subsets of R *might* be measurable even though not > > continuous (perhaps the Cantor set?), > > Consider the discountinuous subset of R: > D = { x | 0 < x < 1 or 2 < x < 3, for x in R }, > i.e., the union of the real intervals (0,1) and (2,3). > Unless I'm mistaken, it is "measurable". Yes, and the measure of (x,x+1] = 1 zillion. Therefore, we have 2 zillion minus the two endpoints for 2zillion-2. Tony
From: Virgil on 27 Jun 2010 01:40
In article <f7318a3f-31c2-414e-8620-3325b7e3176b(a)x27g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 26, 10:19�pm, David R Tribble <da...(a)tribble.com> wrote: > > Jesse F. Hughes wrote: > > >> Is it your opinion that N+ has a last element? > > > > Tony Orlow wrote: > > > No, and that's beside the point. If there are an infinite number of > > > unit increments between any two values then there is an infinite > > > difference between them. If not, then �it's a countable sequence. > > > > In R (the reals), there is no infinite "number of unit increments" > > and no infinite difference" between any two elements in R. > > There are no infinite quantitative differences between any two > standard real values, however, between any two distinct real numbers > do there not exist an uncountable number of others? > > But, that's not the point, still. Please pay attention. The reals do > not have any minimum finite difference between successive elements. > The naturals do. There is at most (or exactly) one natural per unit > interval. That's how the unit interval is defined. > > > Yet it is not a countable set. > > No, R has internal uncountability in any finite segment, whereas N has > 1 element per unit segment, a finite range. > > > > > In contrast, the ordered set: > > � S = N U { w } = { 0, 1, 2, 3, ..., w } > > has an "infinite distance" between w and all of its other > > members, yet it is a countable set. > > Only if you consider w, or 'tav', or whatever, to be some actual > infinite number, um, after all the other ones ended, even though they > have no end. Since in FOL+ZFC all those 'numbers' are sets, there is no problem in having other sets as numbers, and having the set omega, of all finite ordinal numbers, as an ordinal number is no problem. |