CANTOR DISPROOF <<<<<<<<<<<<<<<<
in [Logic]
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From: Transfer Principle on 25 Jun 2010 22:08 On Jun 23, 9:17 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > On Jun 24, 2:01 pm, Sylvia Else <syl...(a)not.here.invalid> wrote: > > If you follow that algorithm, you cannot assign a finite element number > > to 1/9 because 1/9, which is an infinitely recurring decimal, must be > > prededed by infinitely many finite sequences. Since 1/9 is not assigned > > a finite element number, the algorithm does not list all the reals. > This sounds like George Greene's argument that the > permutations are only finite length. > It's not something I'm likely to sway you on. But > 11 > could just as well be 111... And the induction holds. I believe that someone else already commented on this, either here or in one of the other Herc threads, but since these threads are so long, I'll just comment again here. Herc states that some form of "induction" holds. Apparently, this induction is of the form: (phi(0.1) & phi(n 1's) -> (phi(n+1 1's))) -> phi(0.111...) in other words, it's an induction intended to extrapolate from finite(ly many 1's) to infinite(ly many 1's). Many other posters have suggested similar types of induction schemata, including TO, who discusses what he calls his "infinite case induction" schema in another active thread. Of course, Herc's and TO's induction schemata are incompatible, since TO proclaims himself to be a "Post-Cantorian" while Herc is considered "Anti-Cantorian." But these schemata are extremely unpopular among the ZFC Herc-"religionists." Let me explain why, without explicitly mentioning the axioms of ZFC (lest I be labeled a ZFC "religionist" myself). Let's say, as Herc states, we have a list -- which we might as well call List X -- containing 0.1, 0.11, 0.111, and so on. So according to Herc's induction schema, 0.111... must also appear on the list -- i.e., we must be able to assign a finite element number to 0.111... according to the schema. Now let us form a new list, List Y. This list will be formed by taking List X and crossing out 0.111... from the list. So now we ask, does this list contain 0.111...? One might say, of course not, since we just crossed out 0.111... from the list, but List Y still contains 0.1, 0.11, 0.111, and so on, then by Herc's induction schema, it must contain 0.111... after all. So List Y both contains and doesn't contain 1/9, which is a blatant contradiction. So now we can see why schemata like Herc's are usually unpalatable to Herc-"religionists." The induction forces every list that contains 0.1, 0.11, 0.111, etc. to contain 1/9, including lists that were specifically constructed to contain 0.1, 0.11, 0.111, etc. and _not_ 1/9, like List Y above. The schema forces additional structure onto a list (such as imposing that any convergent sequence of list entries must have a limit that is also a list entry), when the ZFC "religionists" prefer to have the freedom to write lists without this restrictive property. So now what? I myself have nothing against induction schemata like those of Herc or TO, but these posters can't expect the majority of sci.math posters to go along with them.
From: Transfer Principle on 25 Jun 2010 22:37 On Jun 24, 1:33 pm, "Mike Terry" <news.dead.person.sto...(a)darjeeling.plus.com> wrote: > "Sylvia Else" <syl...(a)not.here.invalid> wrote in message > > A permutation of a list of computable reals is another list of > > computable reals. The elements have the same width they always did. > Yes, but that's not what Herc means by permutations. The feeling I get is > that he just means the following: > "Permutations" of n digits: means all strings of length n digits. I admit that I was the one who introduced the word "permutation" to the Herc threads. Originally, Herc's arguments involved taking a list of ternary reals and _shuffling_ them to produce almost any real on the diagonal of the shuffled list. In this case, we really are talking about a permutation in the case of a bijection between a set and itself, since if F : omega -> R is a list, and g : omega -> omega is a permutation (of omega), then the nth entry of the original list is F(n) and the nth entry of the shuffled list is F(g(n)). But now Herc's arguments no longer involve shuffling a list around, and so use of the word "permutation" is awkward. Perhaps I shouldn't have used that word in the first place. > I imagine this applies to finite n, and for omega (first infinite ordingal), > so we can talk of "permutations of omega digits", just meaning all infinite > digit strings. > I also get the feeling that his argument is essentially: > 1) We can list all "permutations" of 1 digits in finite number of steps > 2) then add all "permutations" of 2 digits > 3) then 3 digits > 4) ... then n digits > 5) ... > [Herc has provided instructions for going from the n-digit step to the > (n+1)-digit step. In this way all created "permutations" have a definite > index position. So Herc has defined a bona-fide list of finite > "permutations".] > 6) this list contains permutations of "width" n, for all n > 7) therefore the "width of the sequence" is omega (1st infinite ordingal) > 8) therefore the list "contains" all permutations of length omega > 9) since all real numbers have digit sequences of length omega, > all real numbers are "contained" in the sequence. > (Specifically, all anti-diagonals are "contained" in the sequence) Notice that Cooper states that he is using a sort of induction schema, possibly of the form: (phi(1) & An (phi(n) -> phi(n+1))) -> phi(omega) This schema is invalid in standard ZFC, of course. But Herc rejects ZFC as a "religion," and so we shouldn't use ZFC to prove that Herc is wrong. > Obviously this is just confusion induced by his own terminology for > "contains", which doesn't mean the list contains entries in the usual sense > (of the entries actually being somewhere in the list!). > If we could clarify what "contains" means for Herc, we could maybe pin down > the error to (8), or possibly there is no error with a suitable > interpretation of "contains" - unless there is a mysterious step (10) which > comes next. (Maybe using (9) to refute Cantor's proof in some way?) Even among ZFC Herc-"religionists," some people use the phrase "x contains y" to denote "yex," while others use the same phrase to denote "y subset x" (i.e., "Az (zey -> zex)"). (Another poster, tommy1729, actually disliked the fact that "is an element of" and "is a subset of" aren't identical. So he sought a new theory, to be called TST, based on a mereology in which these two relationships are unified." So it might be interesting to define Herc-"contains" in terms of the primitive "e" of ZFC. But once again, the problem is that Herc rejects ZFC, and indeed, he probably refers to "contains" and "lists" in order to avoid the "is an element of" and "is a set" of ZFC. Terry points out that a natural reading of "a list contains _____" sounds as if it should mean "_____ is a list entry." But this has already been refuted by Herc, since the list "3, 3.1, 3.14, ..." is said to "contain" pi, even though pi isn't an entry. (Then again, one wonders whether one can use Herc's induction schema to prove that the list has pi as an entry after all.) An explanation from Herc as to what "contains" means might be forthcoming, but it is unreasonable for Terry and Else to expect him to give his definition in terms of the primitive "e" of ZFC, since he rejects ZFC.
From: Sylvia Else on 25 Jun 2010 23:18 On 26/06/2010 3:40 AM, Graham Cooper wrote: > On Jun 25, 8:23 pm, Sylvia Else<syl...(a)not.here.invalid> wrote: >> On 25/06/2010 7:07 PM, Graham Cooper wrote: >> >> >> >> >> >>> On Jun 25, 6:54 pm, Sylvia Else<syl...(a)not.here.invalid> wrote: >>>> OK. >> >>>> "1 start with an infinite list of all computable reals". >> >>>> That is any list of all the computable reals, howsoever constructed. >> >>>> "2 let w = the maximum width of complete permutation sets" >> >>>> Where a complete permutation set is all the possible combinations of >>>> some finite number of digits. So this step doesn't involve doing >>>> anything with the list described in step 1? It's a completely >>>> independent step? >> >>>> Sylvia. >> >>> Hmmm. Did you consider that the CPS found in the list of >>> step 1 was what I meant. Step 1 - consider this list... >> >> I'm reluctant to assume you mean anything unless it's stated. It seems >> to cause difficulties. However, apparently CPS is an abbreviation for >> "complete permutation set". >> >> So the list of all computable reals contains as a subset complete >> permutation sets whose width is unbounded. Slightly rewording 2, gives us: >> >> "2 let w = the maximum width of those complete permutation sets" >> >> and the next step is >> >> "3 contradict 2" >> >> How is it to be contradicted? >> >> Sylvia. > > There is no (finite) maximum. So there is no finite maximum. How does that advance your proof? Sylvia.
From: Sylvia Else on 26 Jun 2010 06:25 On 26/06/2010 1:18 PM, Sylvia Else wrote: > On 26/06/2010 3:40 AM, Graham Cooper wrote: >> There is no (finite) maximum. > > So there is no finite maximum. How does that advance your proof? > > Sylvia. By which I meant - what is the next step? Sylvia.
From: |-|ercules on 26 Jun 2010 06:45
"Sylvia Else" <sylvia(a)not.here.invalid> wrote > On 26/06/2010 3:40 AM, Graham Cooper wrote: >> On Jun 25, 8:23 pm, Sylvia Else<syl...(a)not.here.invalid> wrote: >>> On 25/06/2010 7:07 PM, Graham Cooper wrote: >>> >>> >>> >>> >>> >>>> On Jun 25, 6:54 pm, Sylvia Else<syl...(a)not.here.invalid> wrote: >>>>> OK. >>> >>>>> "1 start with an infinite list of all computable reals". >>> >>>>> That is any list of all the computable reals, howsoever constructed. >>> >>>>> "2 let w = the maximum width of complete permutation sets" >>> >>>>> Where a complete permutation set is all the possible combinations of >>>>> some finite number of digits. So this step doesn't involve doing >>>>> anything with the list described in step 1? It's a completely >>>>> independent step? >>> >>>>> Sylvia. >>> >>>> Hmmm. Did you consider that the CPS found in the list of >>>> step 1 was what I meant. Step 1 - consider this list... >>> >>> I'm reluctant to assume you mean anything unless it's stated. It seems >>> to cause difficulties. However, apparently CPS is an abbreviation for >>> "complete permutation set". >>> >>> So the list of all computable reals contains as a subset complete >>> permutation sets whose width is unbounded. Slightly rewording 2, gives us: >>> >>> "2 let w = the maximum width of those complete permutation sets" >>> >>> and the next step is >>> >>> "3 contradict 2" >>> >>> How is it to be contradicted? >>> >>> Sylvia. >> >> There is no (finite) maximum. > > So there is no finite maximum. How does that advance your proof? There seems to be 2 possibilities. 1 All finite digit permutations occur to infinite length. 2 It would be very difficult to come up with a new sequence of digits that wasn't on the computable reals list Herc |