Uncountability of the Rationals?
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From: Tony Orlow on 8 Jun 2010 21:52 On Jun 8, 8:09 pm, David R Tribble <da...(a)tribble.com> wrote: > MoeBlee wrote: > >> WHAT exact ordering on sets in GENERAL do you mean '<' to stand for? > > Tony Orlow wrote: > > I was referring to sets of real (or hyperreal) quantities, specifically.. > > But then what about other ordered sets containing members > other than reals, hyperreals, naturals, or numbers at all? Does > your "bigulosity" still apply to them, in general? > > For example, consider the sets constructed from the "primitive" > element 'o'. Some examples: > T1 = { o } > T2 = { o, {o} } > T3 = { o, {o}, {{o}} } > > Likewise consider some infinite sets built in this same way: > R1 = { o, {o}, {{o}}, {{{o}}}, ... } > R2 = { o, {{o}}, {{{{o}}}}, {{{{{{o}}}}}}, ... } > > Now we'll define an order relation '<' for these particular sets. > We define o < {o}. So in general, > o < {o} < {{o}} < {{{o}}} < {{{{o}}}} < ... > > Now we can see that all of the sets above can be ordered > sets, using our new '<' relation. > > So the question is, given these ordered sets, can you calculate > the relative bigulosity for them? Specifically, does R1 have a > larger or smaller bigulosity than R2? > > -drt That depends how the strings are interpreted. If each bracket in one is equal to one bracket in the other, R2 is a subset of R1, including all the "even" members of the set. When dealing with the countably infinite, they can only be compared parametrically, and the correspondences need to be defined. Bigulosity is not as simple as cardinality, for sure. Not all situations are easily handled, and some may be viewed from a couple different perspectives, giving different results. But, since omega doesn't really exist as a number in my theory, that's not really a problem. TOny
From: Tim Little on 8 Jun 2010 22:54 On 2010-06-08, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 8, 2:00 am, Tim Little <t...(a)little-possums.net> wrote: >> What in particular do you think is unanswered about the continuum >> hypothesis? > > Whether aleph_1 is equal to c, or whether there exists an aleph > between aleph_0 and c. The answer is different for different models of ZFC. What is unclear about that? >> What is nonexistent about a well ordering of the reals? > > Ummm... none has ever be explicitly discovered or defined. So? There are plenty of natural numbers that have not been explicitly discovered or defined either. Does that mean that they don't exist too? >> > Remember my big challenge? What is the width of a countably infinite >> > complete list of digital numbers? >> >> In what salient way does your "big challenge" differ from "what is the >> width of a snarflek"? > > Do you think such a comment makes ME look bad? Do you not understand > English? I understand English, and mathematical English. I do not understand the private language in which your "big challenge" is phrased. What do you mean by "width of a ... list of digital numbers"? Since you have provided no definition of your private terms, they have no mathematical meaning: just like "snarflek". > Sorry, Tim, but I'm unlikely to respond to any more of your comments, I'm not surprised. Run along, now. - Tim
From: Aatu Koskensilta on 8 Jun 2010 22:55 Tim Little <tim(a)little-possums.net> writes: > On 2010-06-08, Tony Orlow <tony(a)lightlink.com> wrote: > >> Whether aleph_1 is equal to c, or whether there exists an aleph >> between aleph_0 and c. > > The answer is different for different models of ZFC. What is unclear > about that? Nothing. But this doesn't answer the question whether aleph_1 = c. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Virgil on 8 Jun 2010 23:18 In article <eda64f14-a9f9-449e-a875-bcb4fef979fa(a)r27g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Yes, it is directly related to CH, except that it addresses the > question of what lies between the finite and countably infinite. What deludes TO to suppose that there is enough "space" between the a collection of large finite sizes and a collection of small infinite sizes to accommodate anything? It is > generally accepted that a countably infinitely wide complete list of > digits contains all reals within the range of its greatest digit > (between 0 and 1 for digits from -1 down, for instance). A list of digits has no more digits in it that the base of that digital system, e.g., 10 for decimals. And "-1" is not just a digit.
From: Virgil on 8 Jun 2010 23:21
In article <233dbe8e-a3f4-4d2d-bd01-8b482656a1e8(a)w31g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 8, 8:09�pm, David R Tribble <da...(a)tribble.com> wrote: > > MoeBlee wrote: > > >> WHAT exact ordering on sets in GENERAL do you mean '<' to stand for? > > > > Tony Orlow wrote: > > > I was referring to sets of real (or hyperreal) quantities, specifically. > > > > But then what about other ordered sets containing members > > other than reals, hyperreals, naturals, or numbers at all? Does > > your "bigulosity" still apply to them, in general? > > > > For example, consider the sets constructed from the "primitive" > > element 'o'. Some examples: > > �T1 = { o } > > �T2 = { o, {o} } > > �T3 = { o, {o}, {{o}} } > > > > Likewise consider some infinite sets built in this same way: > > �R1 = { o, {o}, {{o}}, {{{o}}}, ... } > > �R2 = { o, {{o}}, {{{{o}}}}, {{{{{{o}}}}}}, ... } > > > > Now we'll define an order relation '<' for these particular sets. > > We define o < {o}. So in general, > > �o < {o} < {{o}} < {{{o}}} < {{{{o}}}} < ... > > > > Now we can see that all of the sets above can be ordered > > sets, using our new '<' relation. > > > > So the question is, given these ordered sets, can you calculate > > the relative bigulosity for them? Specifically, does R1 have a > > larger or smaller bigulosity than R2? > > > > -drt > > That depends how the strings are interpreted. That suggests that "biguosity" is subjective rather than objective, which rather spoils it for serious mathematical use. |