Why can no one in sci.math understand my simple point?
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From: Tim Little on 20 Jun 2010 23:12 On 2010-06-21, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > The same algorithm does work for all n. Look at the first n terms to > n places of accuracy. Provide computer code, please. Here's the required function signature: int getDigit(BigInteger n). You get to fill in the function body. >> For some fixed M, you could embed M digits of the diagonal into the >> algorithm, and the algorithm would work for all n < M. But that's not >> enough: it has to work for all n. > > I can in fact compute the nth digit of the anti-diagonal for any n > by hand. "By hand" is irrelevant to the definition of a computable real, especially when your "hand" is connected to "eyes" that have the list as input. The list is *not* input to the algorithm. > The same objection could be raised to Cantor's proof; after all, he > accepts a purportedly infinite list as input to his algorithn as > well. Cantor does not claim to be proving existence of any finite algorithm. You are. > Or are you saying I have to do this without using the infinite list > itself? You may use the infinite list to determine which finite algorithm to use, but the algorithm must be self-contained and not refer to the list. The *only* input to the algorithm you select will be a natural number, which denotes n. > I note that Cantor's original proof also requires an infinite list > of Reals, but you seem to have no problem with that. Indeed I don't. Cantor is not claiming that the antidiagonal is a computable real, and so finite algorithms are irrelevant. You *are* claiming that the antidiagonal is a computable real, and so you must prove the existence of a suitable finite algorithm for the antidiagonal. - Tim
From: Tim Little on 20 Jun 2010 23:19 On 2010-06-21, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > "Tim Little" <tim(a)little-possums.net> wrote in message >> On 2010-06-21, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: >>> No, I only need a finite algorithm which will calculate it to n >>> decimal places with finite input. >> >> You have previously stated that you can do this with "about three >> lines of Java code". >> >> Here's your Java method signature: int antidiagonalDigit(BigInteger n). >> >> Here are the first twenty digits of the first twenty members of my >> list: >> >> 93525532854532512838 >> 32127313472944276266 >> 70595916184994935423 >> 20733652719572401688 >> 30472031767118774150 >> 47190325821263633948 >> 74236814853458351851 >> 58521903865615844550 >> 91701104659863267390 >> 39510921669610680229 >> 19656091025330568974 >> 49591084533660072011 >> 81683520683673830124 >> 93720622611168810054 >> 50284245443806050152 >> 11702670934156383526 >> 58534679962278312978 >> 68478827933494896765 >> 83579375050000862329 >> 50241229144880453593 >> >> Now let's see your Java code. > > Here is my p-code. As I suspected, your claim that you could actually do it was pure puffery, and you probably know as little about programming as you do about mathematics. > Take the nth digit of the nth item. The n'th item of *what*? The list is not a parameter, it is not a global variable, it is completely out of scope. You fail already. - Tim
From: Tim Little on 20 Jun 2010 23:37 On 2010-06-21, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > OK, I have a list of all Reals. Which real is missing? Denoting your list by L, antidiag(L) is missing. If you *tell* me L, I can verify that it is missing. Otherwise you'll have to verify it yourself. Either way, it is missing. >>> But you have made an excellent point in your example. When Cantor is >>> confronted with some digit as poorly defined your Chaitan's number >>> example >> >> It's not poorly defined at all; it is precisely defined. > > No, its not precisley defined, you can't tell me its value to arbitrary > accuracy. It looks like the concept of mathematical definition is also an area of incompetence for you. Failing that, there really in no point in continuing any further. Goodbye. - Tim
From: Newberry on 20 Jun 2010 23:39 On Jun 20, 6:43 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <b1d23660-1b7a-4682-affa-952cf2d9e...(a)e34g2000pra.googlegroups.com>, > > > > > > Newberry <newberr...(a)gmail.com> wrote: > > On Jun 20, 1:18 pm, Virgil <Vir...(a)home.esc> wrote: > > > In article > > > <f92c169d-ee85-40c2-aa82-c8bdf06f7...(a)j4g2000yqh.googlegroups.com>, > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 20 Jun., 17:51, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > > > > Cantor was this utterly insane freak who chose not to accept Newberry's > > > > > word for it, and instead *prove* that there was no list of all real > > > > > numbers. Obviously, his proof is nonsense, because, after all, Newberry > > > > > said there was no list. > > > > > His proof is nonsense because it proves that a countable set, namely > > > > the set of all reals of a Cantor-list and all diagonal numbers to be > > > > constructed from it by a given rule an to be added to this list, > > > > cannot be listed, hence, that this indisputably countable set is > > > > uncountable. > > > > That is a deliberate misrepresentation of the so called "diagonal proof". > > > > What that proof (not actually by by Cantor himself) ays is that any > > > listing of reals is necessarily incomplete because given such a listing > > > one can always construct a real not listed in it. > > > Please CONSTRUCT the anti-diagonal in the space provided below. > > > Virgil's anti-diagonal construction > > BEGIN > > It is not mine but Cantor's, though the following notation is mine. > > N represents the set of natural numbers. > > NxN is the Cartesian product of N with itself with members (m,n) where m > and n are members of N. > > F: NxN --> {0,1} is a list of functions, F(m,--), from N to {0,1} > with, for each m in N, F(m,--): n --> F(m,n), being such a function. > > Then the anti-diagonal function g:N --> {0,1} is defined by > g(n) = 1-F(n,n) in {0,1}, > so that g(n) <> F(n,n) for all n, > and thus g(--) <> F(n,--) for all n.. > > Note the Cantor himself never published an anti-diagonal argument for > the reals themselves, though some of his followers did. > > > END Not sure why you think you had to tell us how the anti-diagonal is defined. You claimed you could CONSTRUCT it. Please go ahead and do so. > > > What Cantor himself proved was that for any list of infinite binary > > > sequences there are binary sequences not listed in that list. > > > > Cantor had previously proved by quite a different method, which WM seems > > > unable or unwilling to criticize, that the set of reals was not > > > countable. > > > > > Even completely blinded matheologicians should be able, perhaps after > > > > some contemplation, to recognize that. > > > > What completely blinded matheologicians are able, perhaps after > > > some contemplation,to recognize is that WM is not working in the same > > > world as mathematicians do.- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -
From: Newberry on 20 Jun 2010 23:49
On Jun 20, 7:02 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > On Jun 20, 8:51 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Newberry <newberr...(a)gmail.com> writes: > >> > That's right. There is no formula or algorithm to construct the list.. > >> > It means that you would have to flip each and every digit one by one.. > >> > And that is impossible. > >> Of course, it is perfectly reasonable to accept that there is no such > >> list, because Newberry says so. > > > Ah, so it's nice to see that Newberry has joined Herc and WM in > > this ever-growing thread. > > Yes, how nice! Er, how come? Why do you think this is a nice thing? > > Honestly? It's not that I think it's a bad thing --- though, to be > honest, I think Newberry's better than the argument he gives here --- > but why do you think it's a positive development? > > > > >> Cantor was this utterly insane freak who chose not to accept Newberry's > >> word for it, and instead *prove* that there was no list of all real > >> numbers. > > > But this proof that there is no list of all real numbers must use > > the axioms of some _theory_ (such as ZFC) -- a theory which > > Newberry isn't required to accept just because Hughes says so. > > Of course he's not required to accept it. So? > > Nonetheless, it is a *theorem*, namely a theorem of ZF. > > > > > So Hughes continues to be a Herc-religionist (i.e., a religionist > > according to Herc), appealing to ZFC to prove Newberry wrong even > > though there's no reason to assume that Newberry is working in ZFC (or > > any other theory proving Cantor's Theorem). > > If Newberry means that Cantor's Theorem is not a Theorem in some other > (specified) theory, then, of course, I'd have no opinion at all, until I > saw the theory. > > So, let's ask Newberry! > > Newberry, what theory did you have in mind? What I had in mind is that if you drop the axiom of extent you can either conclude that there is no such list or that the anti-diagonal does not exist analogously to the conclusion that the set R = {x | ~(x in x} does not exist. BTW, Cantor did not work in ZFC and the argument given by McCullough appeared to be on the intuitive level rather than in any particular formal theory. I will grant you that Zermelo figured how to construct a system where the diagonal argument goes through and his system is probably consistent. It is a nonsense nevertheless. All the transfinite set theory is a completely superfluous metaphysics. > -- > Jesse F. Hughes > "I'm not going to forget what I've seen. I understand the devastation > requires more than one day's attention." > -- G. W. Bush reassures Hurricane Katrina victims. Two days, minimum. |