CANTOR DISPROOF <<<<<<<<<<<<<<<<
in [Logic]
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From: Transfer Principle on 30 Jun 2010 02:00 On Jun 28, 9:58 pm, herbzet <herb...(a)gmail.com> wrote: > Tim Little wrote: > > In short, just another crank. > Yeh, well, I actually defended him from the beginning when he > showed up in sci.logic with his project of providing formal > systems in which the assertions of various cranks can be > demonstrated. I think that is a valid intellectual exercise, > at least, and could provoke some actually interesting > discussions about fundamental assumptions we routinely make > in logic/math. I actually attempted to do this a few times in this thread, but when Herc stated that he was trying to use some form of induction, all I could muster was a schema of the form: (phi(.1) & An (phi(n 1's) -> phi(n+1 1's))) -> phi(.111...) Of course, such schemata are invalid in standard theory, and I even attempted to warn Herc that the majority of posters in this thread are likely to reject such schemata. I still believe that it's possible to find a workable schema that describe Herc's intuitions, but it won't be easy.
From: Transfer Principle on 30 Jun 2010 02:03 On Jun 28, 9:58 pm, herbzet <herb...(a)gmail.com> wrote: [Oops! Reposting because I forgot to remove aus.tv per herbzet's request] > Tim Little wrote: > > In short, just another crank. > Yeh, well, I actually defended him from the beginning when he > showed up in sci.logic with his project of providing formal > systems in which the assertions of various cranks can be > demonstrated. I think that is a valid intellectual exercise, > at least, and could provoke some actually interesting > discussions about fundamental assumptions we routinely make > in logic/math. I actually attempted to do this a few times in this thread, but when Herc stated that he was trying to use some form of induction, all I could muster was a schema of the form: (phi(.1) & An (phi(n 1's) -> phi(n+1 1's))) -> phi(.111...) Of course, such schemata are invalid in standard theory, and I even attempted to warn Herc that the majority of posters in this thread are likely to reject such schemata. I still believe that it's possible to find a workable schema that describe Herc's intuitions, but it won't be easy.
From: |-|ercules on 30 Jun 2010 02:20 "Transfer Principle" <lwalke3(a)lausd.net> wrote > On Jun 28, 9:58 pm, herbzet <herb...(a)gmail.com> wrote: >> Tim Little wrote: >> > In short, just another crank. >> Yeh, well, I actually defended him from the beginning when he >> showed up in sci.logic with his project of providing formal >> systems in which the assertions of various cranks can be >> demonstrated. I think that is a valid intellectual exercise, >> at least, and could provoke some actually interesting >> discussions about fundamental assumptions we routinely make >> in logic/math. > > I actually attempted to do this a few times in this thread, but > when Herc stated that he was trying to use some form of > induction, all I could muster was a schema of the form: > > (phi(.1) & An (phi(n 1's) -> phi(n+1 1's))) -> phi(.111...) > > Of course, such schemata are invalid in standard theory, and I > even attempted to warn Herc that the majority of posters in this > thread are likely to reject such schemata. > > 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...]> ) Herc
From: Transfer Principle on 30 Jun 2010 02:30 On Jun 28, 8:57 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > "Sylvia Else" <syl...(a)not.here.invalid> wrote > > I have questioned whether what you're doing is induction. P(n) -> P(n+1) > > is the result of an inductive proof, not the proof itself. > > Sylvia. > You and I start a landscaping business Herc And Syl's Landscaping Ad Infinitium > 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. Interesting analogy. 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, or is a half- plane (say the union of the first two quadrants) an infinite rectangle? > On one block, you start doing the fencing for Mr Fenceme, completing the perimeters of larger and larger concentric > rectangular paddocks. But then we notice that neither a quadrant nor a half-plane is the union of any number of concentric rectangular regions. Indeed, the union of infinitely many concentric rectangular regions is either a (finite) rectangular region (if the dimensions are increasing but bounded), a strip (if the lengths are increasing to infinity but not the widths), or the entire plane. Since a finite rectangle can hardly be described as "infinite," and since the entire plane leaves no room for Mrs. Mowme's lawn, I conclude that an "infinite rectangle" is actually an infinite strip. > 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. > Being a man, I'm well on my way to mowing the whole lawn. OK. I assume that on the first step, Herc mows a small finite segment of the strip (whose width is the same as that of the entire strip), then on subsequent steps, mows rectangles on either side of the starting rectangle so that after each step, the center of the mowed rectangle remains constant. If each step takes, say, half as long as the previous step, then the entire lawn can be mowed in finite time. > Because you're a woman [snip sexism]
From: Jesse F. Hughes on 30 Jun 2010 08:54
Transfer Principle <lwalke3(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. Let's take these one at a time. (1) The only way to discuss alternate theories is to incite emotional reaction and be a "troll". No one believes this is so. (2) The only posters who oppose ZFC are those who don't understand ZFC. This is a matter of fact, not necessity. I imagine there are perfectly intelligent philosophers of mathematics who understand ZFC yet believe that it is a "bad" theory for various philosophical reasons. Those people aren't posting here on the group. It is obvious that the people who post about the evils of ZFC on this group are, indeed, a confused lot that don't understand mathematics. 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. > It's possible to be an _expert_ of the proof of Cantor's Theorem in > ZFC and _still_ prefer to work in a theory in which its negation is a > theorem, and it's possible to discuss such a theory without an > emotional "trolling" reaction. Of course. But the cranks here do not simply claim to "prefer" to work in an alternate theory. First, they do not have an alternate theory that they claim to prefer. Second, their criticisms do not come off as mere preference. 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. -- "Another factor one has got to look at is the amount of liquidity in the system. In other words, is there enough liquidity to enable markets to be able to correct? And I am told there is enough liquidity in the system to enable markets to correct." -- Guess who. |