CANTOR DISPROOF <<<<<<<<<<<<<<<<
in [Logic]
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From: |-|ercules on 30 Jun 2010 21:25 "Transfer Principle" <lwalke3(a)lausd.net> wrote > > Of course, one might wonder what exactly an infinite rectangular > region is > suppsed to be. For example, we consider the coordinate plane. Is a > single > quadrant (say the first quadrant) an infinite rectangle, Yes that's what I had in mind. http://i721.photobucket.com/albums/ww214/ozdude7/fencingVSmowing.png Here's the story again that tells of the difference to Sylvia's finite sequence induction. You and I start a landscaping business Herc And Syl's Landscaping Ad Infinitum I handle all the mowing, and you do the fencing. We get a call from Mr Fenceme and Mrs Mowme Blockheads. We drive to the property which appears to be divided into 2 blocks, both infinite rectangular lawns. On one block, you start doing the fencing for Mr Fenceme, completing the perimeters of larger and larger concentric rectangular paddocks. I get to the mowing for Mrs Mowme, completing larger and larger rectangular mown lawn areas, each building upon the earlier smaller rectangular lawn area. I'm well on my way to mowing the whole lawn. You never ever come close to fencing the entire lawn! ;-) The limit of mown lawn area as mowing time->oo is infinity. However, infinitely many fence sizes all have finite perimeters. Herc
From: Transfer Principle on 2 Jul 2010 03:12 On Jun 30, 7:54 pm, herbzet <herb...(a)gmail.com> wrote: > Transfer Principle wrote: > > But now what? I want to be able to discuss theories other than > > ZFC, including those theories which might prove the negation of > > Cantor's Theorem. > It seems evident to me that such a theory would be talking about > objects somewhat different from what is being discussed in more > standard theories, even if such objects are called "sets" in > both theories. > Even in a not-too-distant from standard set theory like NF, it > seems fairly evident to me that objects which could conceivably > satisfy NF are somewhat different from objects that would satisfy, > say, ZFC. This raises another interesting question, namely at what point beyond which a theory differs from ZFC such that we should no longer call the objects which satisfy them sets? This question has come up in other threads as well. By this line of argument, one could even point out that there are objects satisfying _ZF_ that are different from those satisfying ZFC, namely those without choice functions, are infinite yet Dedekind finite, and so on. So I wonder, where is the line that when crossed we can no longer call them sets, but, to use MoeBlee's name, "zets"? > And so what? What's at stake? If there's a theory, say NF, which satisfies Herc's intuitions, then perhaps he might find it more reasonable than ZFC. If it's true that Herc has criticized ZFC and exhibited behaviors that cause him to be considered a "troll," then perhaps knowledge of a theory such as NF will convince him to be less of a "troll." > > Is it possible to do so _without_ inciting a > > highly emotional reaction? If so, then I'd _love_ to do so, so that > > I can discuss theories other than ZFC without fear that the > > "troll" label will come up. > Please clarify: do you want to discuss theories other than ZFC, or > do you wish to discuss why people are so mean to the trolls, cranks, > and loons who show up with their revolutionary breakthroughs and > poo-flinging? > If just the former, I don't think you'll have any problems. I want to discuss theories other than ZFC -- and hope that I can do so with neither the five-letter insults nor the behavior that inspires those words appearing.
From: Transfer Principle on 2 Jul 2010 03:24 On Jun 30, 5:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > I refuse to believe that the only way to discuss alternate theories > > is to incite emotional reaction and be a "troll," just as I refuse to > > believe that the only posters who oppose ZFC are those who > > don't understand ZFC. > (2) The only posters who oppose ZFC are those who don't understand ZFC. > If, on the contrary, they *did* understand mathematical theories and ZFC > in particular, why must they wait for a hero like you to make their > so-called theories rigorous? Surely, they would see the need and do the > work on their own. Was ZFC the work of a single set theorist? Sure, Cantor laid the foundation, but the fact that ZFC is named after _two_ set theorists, Zermelo and Frankel, speaks for itself. Similarly, if we work together, we might be able to come up with a rigorous set theory as well. Not only have I helped, but WM's writings have inspired Herc as well. As the saying goes, "two heads are better than one." > To give you an example: for a while, I did a little research in ZFA, > anti-well-founded set theory. I liked the theory. I suppose that I > found it preferable to ZFC at the time. I did not argue that ZFC was > wrong, or give tired, silly arguments that the axiom of regularity is > false because it's false in ZFA. The fact that ZFC was regarded as a > foundation of mathematics while ZFA was not did not bother me. I worked > in ZFA because it provided a nice setting for the things I was doing. > That's rather different than the behaviors we see here. As Hughes is allowed to use a theory which proves the negation of Regularity, likewise Herc should be allowed to use a theory which proves the negation of Cantor's Theorem. I wonder what combination of axioms and posting behavior will lead to Herc being granted that freedom.
From: Transfer Principle on 2 Jul 2010 03:36 On Jun 29, 11:20 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > "Transfer Principle" <lwal...(a)lausd.net> wrote > > I still believe that it's possible to find a workable schema that > > describe Herc's intuitions, but it won't be easy. > phi( <[1] 2 3 4...> ) & An ((phi ( <[1 2 ... n] n+1 n+2 ...>) -> phi( <[1 2 ... n n+1] n+2 n+3 ...> )) > -> > phi( <[1 2 3 4...]> ) An interesting schema. But of course, this raises the question as to what exactly the <> and [] symbols stand for. For example, some posters use <> to denote an n-tuple, so that <x y> would denote the (Kuratowski) ordered pair. But that would make most of the "n-tuples" infinite -- which would turn them into sequences. So under this interpretation: <[1 2 ... n] n+1 n+2 ...> is the sequence whose 0th entry is [1 2 ... n], whose 1st entry is n+1, whose 2nd entry is n+2, and whose mth entry is n+m. Reasonable enough. But now we must turn to the [1 2 ... n] notation which uses [] brackets instead of <>. A few earlier posters, including zuhair and tommy1729, used [] to denote objects under the flattened mereology. Hopefully, this isn't how Herc is using them, since this would make the theory a bit more complicated. What would help is to see more instances of this schema. We already know that Herc intends to apply it to lists such as: 0.100000000... 0.110000000... 0.111000000... 0.111100000... and conclude that the digit 1 appears in every position, or something like that. Let me think about this for a while.
From: |-|ercules on 2 Jul 2010 04:10
"Transfer Principle" <lwalke3(a)lausd.net> wrote > On Jun 29, 11:20 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> "Transfer Principle" <lwal...(a)lausd.net> wrote >> > I still believe that it's possible to find a workable schema that >> > describe Herc's intuitions, but it won't be easy. >> phi( <[1] 2 3 4...> ) & An ((phi ( <[1 2 ... n] n+1 n+2 ...>) -> phi( <[1 2 ... n n+1] n+2 n+3 ...> )) >> -> >> phi( <[1 2 3 4...]> ) > > An interesting schema. But of course, this raises the question > as to what exactly the <> and [] symbols stand for. > > For example, some posters use <> to denote an n-tuple, so that > <x y> would denote the (Kuratowski) ordered pair. But that > would make most of the "n-tuples" infinite -- which would turn > them into sequences. So under this interpretation: > > <[1 2 ... n] n+1 n+2 ...> > > is the sequence whose 0th entry is [1 2 ... n], whose 1st entry > is n+1, whose 2nd entry is n+2, and whose mth entry is n+m. Maybe <1* 2* 3* 4 5 6 ...> would be clearer and closer to standard induction, I thought it was clear < [1 2 3] 4 5 6...> is the 3 digit prefix in the sequence of N. > > Reasonable enough. But now we must turn to the [1 2 ... n] > notation which uses [] brackets instead of <>. A few earlier > posters, including zuhair and tommy1729, used [] to denote > objects under the flattened mereology. Hopefully, this isn't > how Herc is using them, since this would make the theory a bit > more complicated. > > What would help is to see more instances of this schema. We > already know that Herc intends to apply it to lists such as: > > 0.100000000... > 0.110000000... > 0.111000000... > 0.111100000... > > and conclude that the digit 1 appears in every position, or > something like that. Let me think about this for a while. What would be better is if the schema did NOT work on the above example. The lack of constraint on the suffix, the fact that there is no distinction where the finite prefix ends and the suffix starts blah blah.. Herc |