Why can no one in sci.math understand my simple point?
in [Logic]
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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.
From: Owen Jacobson on 21 Jun 2010 00:32 On 2010-06-19 22:31:56 -0400, Peter Webb said: > Cantor's original proof requires the list to be provided in advance No, it does not. It's an existence proof showing that: IF a list of reals L(x) (where L : N -> R) exists, THEN a real A_L exists such that for every natural number n, L(n) ≠ A_L. Or, equivalently, showing that: For every function L from N to R, there exists a real r not in the image of L. Being a constructive proof, it's usually *presented* as if it were an algorithm or a procedure, but it's only for ease of comprehension. Unfortunately, that seems to have backfired on you. -o Incidentally, "countable" and "listable" mean the same thing in conventional set theories: an infinite set S is countable if and only if there exists an surjective function f from N to S. Since a "list" of elements of S is most easily formalized as a surjective function from N to S, denying that S is listable is equivalent to denying that it is countable.
From: Peter Webb on 21 Jun 2010 02:08 "Tim Little" <tim(a)little-possums.net> wrote in message news:slrni1tkli.jrj.tim(a)soprano.little-possums.net... > On 2010-06-21, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: >> "Tim Little" <tim(a)little-possums.net> wrote in message >> I do understand that. But I don't actually ask for a "finite algorithm" >> in >> my proof, I only require that I am provided a purported list of all >> comptuable Reals. > > Later in the proof, you claim that the antidiagonal real is > computable. In other words, that there exists a finite algorithm for > computing it. > > >> Yes, in practice, this means that each item is specified by a finite >> algorithm, but so what? I ask for a purported list of all computable >> Reals >> and show there is at least one missing. > > At least one *what* missing? A real? Fine - Cantor's proof shows > that there is a missing real. And if the nth digit of the nth term is computable, then so is the nth term of the anti-diagonal, as there is an explicit construction of the nth digit of the anti-diagonal based upon the nth digit of the nth term. > What's that? You want the antidiagonal > to be a *computable* real? Well, then - you are required to show that > there always exists a finite algorithm to compute it, and that is not > present in Cantor's proof. > Well, I don't actually have to write out all "infinite" decimals in its expansion to prove it is computable. All I really have to be able to do is to compute it to arbitrary accuracy. And Cantor's proof constructs the number to arbitrary accuracy. > So your modified "proof" is either invalid, or you must include > significant amounts of material not present in Cantor's proof. > Well, really only one thing. I have to produce an algorithm to estimates the anti-diagonal to arbitrary accuracy. Fortunately, the digit chnage rule of Cantor uniquely specifies the nth decimal place of the anti-diagonal in a completely finite closed procedure. > > As is well-known to competent mathematicians and computer scientists, > no amount of material will make it valid, as the modified "theorem" is > simply false. It is *not true* that any list of computable reals has > a computable antidiagonal. You give me a list of purported computable Reals, I can compute the anti-diagonal to arbitrary precision. Just like Cantor can compute the value of the missing Real to arbitrary precision. > > > - Tim
From: Peter Webb on 21 Jun 2010 02:14 "Tim Little" <tim(a)little-possums.net> wrote in message news:slrni1tnji.jrj.tim(a)soprano.little-possums.net... > 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. > OK, you tell me your list of computable Reals, and I will verify which one 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. > Funny, rather than address my proof you now seem to just want to be rude. Very crank like behaviour on your part. If this really is goodbye, then cheerio and thanks for the (mathematical) input. > > - Tim
From: Peter Webb on 21 Jun 2010 02:20
"Tim Little" <tim(a)little-possums.net> wrote in message news:slrni1tmi9.jrj.tim(a)soprano.little-possums.net... > 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. Of computable Reals. That is supposed to include all Computable Reals. > The list is not a parameter, it is not a > global variable, it is completely out of scope. You fail already. > > Actually, Cantor's proof starts with a purported list of all Reals, and then shows that at least one Real is missing. It assumes that a list exists, then shows a Real is missing. If you didn't supply the list in advance, he wouldn't be able to construct a Real missing from the list. In fact, if you don't say what is on the list, it you can't prove soimething is missing from it. There is no Real which is missing from every list, but every list misses at least one Real. You do believe Cantor's original proof that the Reals cannot be listed, don't you? > - Tim |