Uncountability of the Rationals?
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From: Transfer Principle on 5 Jun 2010 23:29 On Jun 5, 11:22 am, David R Tribble <da...(a)tribble.com> wrote: > Tony Orlow wrote: > >> There are always the H-Riffics. Remember "Well Ordering the Reals"? > David R Tribble wrote: > >> Yeah. Remember how several of us demonstrated that the H-riffics > >> is only a countably infinite set, and omits vast subsets of the reals > >> (e.g., all the multiples of powers of integers k, where k is not 2)? I don't recall what exactly TO's H-riffics are, but this post has piqued my interest. At first, by Tribble's comments here, I thought that the H-riffics were simply the elements of Z[1/2], the ring Z with 1/2 appended to it. It consists of all rationals with a finite binary expansion. Thus, for example, 1/3 is not an H-riffic. > Tony Orlow wrote: > > Sure when using 2 as a base, the numbers you mention are uncountably > > distant from the beginning of the uncountable sequence. But then, I am > > using "uncountable sequence" in a rather nonstandard way. > It's more basic than that. Your H-riffic set completely omits most of > the reals, such as any multiple of any integral power of 3 (e.g., 3, > 1/3, 27, etc., ad infinitum). > Besides, your set is only countable (which is obvious from its very > definition), so it can't possibly contain all the reals. It doesn't > even contain all the rationals. But now Tribble writes that the H-riffics not only exclude the value 1/3, but the value _3_ as well. I can see the ring Z[1/2] being worth defending, but a set excluding _3_ is indefensible. If 3 really isn't an H-riffic, then I wonder what 2+1 is in the H-riffics. But first, I'd like to see what exactly the H-riffics are in the first place.
From: Transfer Principle on 5 Jun 2010 23:36 On Jun 5, 10:04 am, Virgil <Vir...(a)home.esc> wrote: > In article > <796c4157-f47d-4aba-b056-8ddc38d46...(a)c10g2000yqi.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > Surely you remember the T-Riffics? > Does Tony Orlow really want to maintain that ANY part of his idiotic > "T-Riffics" was ->generally accepted<- ? Hold on a minute. Earlier, TO and Tribble were discussing something called the H-riffics. Now Virgil is referring to something called the T-riffics. If by "T-riffics" Virgil is actually referring to the "H-riffics" as mentioned by Tribble, then for once, I actually agree with Virgil. For according to Tribble, the H-riffics lack a value for 3. Even _I_ can't accept a theory in which one can't even prove the existence of 3 (especially if it does prove the existence of 4, which, being a power of two, does exist in this theory). If the T-riffics are distinct from the H-riffics, then I would like to learn more about the T-riffics before I attempt to pass judgment. I don't mind learning more about sets other than the classical real numbers (i.e., standard R) and standard set theories.
From: Transfer Principle on 5 Jun 2010 23:56 On Jun 4, 4:07 pm, r...(a)trash.whim.org (Rob Johnson) wrote: > In article <f4287888-3e95-4f17-b16b-e96af8cfd...(a)s41g2000vba.googlegroups..com>, > David R Tribble <da...(a)tribble.com> wrote: > >If the number of true (constructible) statements proved by PA and > >the number of true statements proved by PA + Con(PA) are both > >countable, does it not follow that those two sets have the same > >cardinality? > >(Or can we not place all provable statements into sets?) > I don't think it is the number of provable statements that one > considers important, but what those statements are. Although we may > be able put the statements proved by PA and the statements proved by > PA + Con(PA) into a 1-1 correspondence, there are probably statements > provable by PA + Con(PA) that are not provable by PA that are useful. Of course there are only countably many statements provable in either PA or PA+Con(PA) -- simply because there are only countably many _statements_, period. It's easy to show that there are only countably many wellformed formulas (wff's) of finite length over a finite language. Thus, even if PA (and hence PA+Con(PA), of course) were _inconsistent_ (so that _every_ statement would be a theorem of PA) there would still be only countably many theorems of PA. (Of course, this all assumes that the base theory in which we are counting the theorems is ZFC and cardinality. In TO's Bigulosity theory, this isn't the case.)
From: Transfer Principle on 6 Jun 2010 00:34 On Jun 4, 3:16 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 4, 3:24 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > I used "Bigulosity", remember? Surely you do, you rascal. > You could use "jiggywiggywosity" or whatever you want. It doesn't > address the point I've made. That YOU have some neologism concerning > your own mathematical thought-cloud doesn't address that, as far as > mere formal ZFC is concerned, it is not substantive whether we say > 'size' or 'zize'. But the fact that TO uses the word "bigulosity" strongly implies that he's using a theory _other_than_ ZFC. Thus, any objections to TO's theory just because it proves a theorem whose negation is provable in ZFC is irrelevant. > > "The" formal theory. I like that. You should really become a Hare > > Krishna devotee. > Oh please, Swami Sri Nondevotee, of course, I mean whatever particular > theory is under discussion at the time. That doesn't imply that I > don't recognize that there are INFINITELY many theories. MoeBlee likes to claim that he recognizes theories other than ZFC, but he really isn't as open-minded as he purports to be. I don't believe that the Z-based theories have a monopoly on the theories that are worthy of consideration (in terms of representing math for the sciences and so on). On the contrary, I believe that there can exist a TO-based theory that can be just as powerful as ZFC wrt math for the sciences. If only MoeBlee would let us look for such a theory. > Good question. Perhaps Cantor and Dedekind in modern times, though > Galileo certainly explored bijection long before, and decided there > was no real answer, and Aristotle first made the distinction between > potential (countable) and actual (uncountable) infinities. This is > some of the ancestry of transfinitology. As far as I know, neither Galileo nor Aristotle proposed what could fairly be called a set theory. But it does show the desirability of a pre-Cantorian set theory (among _several_ posters, not just TO), perhaps based not just on Galileo and Aristotle's notions, but also on those of another ancient Greek mathematician, Euclid, who once stated that "the whole is greater than the part" -- which contradicts ZFC's cardinality, but agrees with TO's Bigulosity. > Anyway, so everything Galileo and/or Aristotle said was true and wise? So _nothing_ Galileo and/or Aristotle said was true and wise? > > ToeKnee > BoerEeng. "Boring." This, of course, represents the reaction that MoeBlee would have if TO really were to post a rigorous theory applicable to sciences based on his own non-ZFC notions. MoeBlee would be "bored."
From: Transfer Principle on 6 Jun 2010 00:42
On Jun 5, 9:59 am, Virgil <Vir...(a)home.esc> wrote: > In article > <db5cbe4b-a8b8-4b6a-ae47-05fa05dd6...(a)i28g2000yqa.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > Yes, that's where I apply N=S^L. > Which is a wrong now as when first dropped on an unsuspecting world. IIRC, TO's statement N=S^L means that the number of strings of length L from a language of size S is equal to N. I disagree with Virgil that it's "wrong." On the contrary, since a string is essentially a function with domain L and codomain S, N=S^L is simply the _definition_ of cardinal exponentiation (as in the exponentiation of standard _cardinals_ in ZFC). So TO is only telling us that the definition of exponentiation of bigulosities is identical to that of standard ZFC cardinals. So it astonishes me that for the one point which TO agrees with ZFC, Virgil tells TO that he's still wrong. |