Why can no one in sci.math understand my simple point?
in [Logic]
|
From: Sylvia Else on 18 Jun 2010 22:25 On 19/06/2010 6:50 AM, WM wrote: > On 18 Jun., 09:37, Sylvia Else<syl...(a)not.here.invalid> wrote: >> On 18/06/2010 5:31 PM, |-|ercules wrote: >> >> >> >> >> >>> "Sylvia Else"<syl...(a)not.here.invalid> wrote ... >>>> On 18/06/2010 4:52 PM, |-|ercules wrote: >>>>> "Sylvia Else"<syl...(a)not.here.invalid> wrote ... >>>>>> On 18/06/2010 3:03 PM, |-|ercules wrote: >>>>>>> "Sylvia Else"<syl...(a)not.here.invalid> wrote >>>>>>>> On 18/06/2010 10:40 AM, Transfer Principle wrote: >>>>>>>>> On Jun 17, 6:56 am, Sylvia Else<syl...(a)not.here.invalid> wrote: >>>>>>>>>> On 15/06/2010 2:13 PM, |-|ercules wrote: >>>>>>>>>>> the list of computable reals contain every digit of ALL possible >>>>>>>>>>> infinite sequences (3) >>>>>>>>>> Obviously not - the diagonal argument shows that it doesn't. >> >>>>>>>>> But Herc doesn't accept the diagonal argument. Just because >>>>>>>>> Else accepts the diagonal argument, it doesn't mean that >>>>>>>>> Herc is required to accept it. >> >>>>>>>>> Sure, Cantor's Theorem is a theorem of ZFC. But Herc said >>>>>>>>> nothing about working in ZFC. To Herc, ZFC is a "religion" >>>>>>>>> in which he doesn't believe. >> >>>>>>>> Well, if he's not working in ZFC, then he cannot make statements >>>>>>>> about >>>>>>>> ZFC, and he should state the axioms of his system. >> >>>>>>> Can you prove from axioms that is what I should do? >> >>>>>>> If you want to lodge a complaint with The Eiffel Tower that the >>>>>>> lift is >>>>>>> broken >>>>>>> do you build your own skyscraper next to the Eiffel Tower to >>>>>>> demonstrate >>>>>>> that fact? >> >>>>>> That's hardly a valid analogy. >> >>>>>> If you're attempting to show that ZFC is inconsistent, then say that >>>>>> you are working within ZFC. >> >>>>>> If you're not working withint ZFC, then you're attempting to show that >>>>>> some other set of axioms is inconsistent, which they may be, but the >>>>>> result is uninteresting, and says nothing about ZFC. >> >>>>>> Sylvia. >> >>>>> That would be like finding a fault with the plans of The Leaning Tower >>>>> Of Piza. >> >>>>> I might look at ZFC at some point, but while you're presenting Cantor's >>>>> proof >>>>> in elementary logic I'll attack that logic. >> >>>>> Instead of 'constructing' a particular anti-diagonal, your proof should >>>>> work equally >>>>> well by giving the *form* of the anti-diagonal. >> >>>>> This is what a general diagonal argument looks like. >> >>>>> For any list of reals L. >> >>>>> CONSTRUCT a real such that >>>>> An AD(n) =/= L(n,n) >> >>>>> Now to demonstrate this real is not on L, it is obvious that >>>>> An AD(n) =/= L(n,n) >> >>>>> Therefore >>>>> [ An AD(n) =/= L(n,n) -> An AD(n) =/= L(n,n) ] proves superinfinity! >> >>>>> And THAT is Cantor's proof! >> >>>>> Want to see his other proof? That no box contains the box numbers (of >>>>> boxes) that >>>>> don't contain their own box number? >>>>> That ALSO proves superinfinity! >> >>>>> Great holy grail of mathematics you have there. >> >>>>> Herc >> >>>> What are you trying to prove? >> >>> There is only one type of infinity. >> >> Infinity is a mathematical construct. Before you can even being to >> discuss it, you have to have a set of axioms. > > What was the set that Cantor used? > Nevertheless he "proved". He certainly was using some. For example, the diagonal argument falls apart if the axioms don't permit the construction of a number by choosing digits different from those on the diagonal. It isn't even clear whether Herc is tying to invalidate Cantor's proof by finding a mistake in it, or to prove the inverse, which wouldn't invalidate Cantor's proof, but would only show that the axioms on which it is based are inconsistent. Herc cannot avoid the need to specify the set of axioms. Sylvia.
From: Tim Little on 18 Jun 2010 22:25 On 2010-06-19, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > I see that you have stopped responding to my arguments, and have > employed the rather lame tactic of suggesting I look at definitions > on the web. I don't care where you get your definitions from, so long as they are not from your own orifices. You are using mathematical terminology in a completely incorrect manner, and further discussion is unlikely to be fruitful until you learn what the words you are using mean. > In the mean time, how about answering two questions for me: > > 1. Do you believe it is possible to create a list of all computable > Reals? What do you mean by "create"? Such an list can be proven to exist, and I even provided a well-defined mapping from N to the set of computable reals earlier in this thread. If that doesn't answer your question, you'll have to clarify what you mean by it. > 2. Do you believe the computable Reals are countable? Obviously. - Tim
From: Tim Little on 18 Jun 2010 22:37 On 2010-06-19, Marshall <marshall.spight(a)gmail.com> wrote: > On Jun 18, 6:09 pm, Tim Little <t...(a)little-possums.net> wrote: >> On 2010-06-18, Peter Webb <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: >> >> > The number that is produced is clearly "computable", because we have >> > computed it. >> >> I see you still haven't consulted a definition of "computable number". >> No worries, let me know when you have. > > I suggest it would be more persuasive if you made whatever > point you have in mind about the definition of computable number > directly. Simply repeating this one-liner makes it seem like > you might not have a point. True. I was simply losing patience. I had in fact provided the relevant point three times already, but the point was ignored each time. One suitable definition: a computable real x is one for which there exists a Turing machine that given a natural number n, will output the n'th symbol in the decimal representation of x. (There are other equivalent definitions, but this one seems most relevant to the current discussion) The relevant point: the *only* input to the Turing machine in the definition is n. The rest of the tape must is blank. Peter apparently believes that a number is still computable even if the Turing machine must be supplied with an infinite amount of input (the list of reals). - Tim
From: |-|ercules on 18 Jun 2010 22:45 "Sylvia Else" <sylvia(a)not.here.invalid> wrote > On 19/06/2010 6:50 AM, WM wrote: >> On 18 Jun., 09:37, Sylvia Else<syl...(a)not.here.invalid> wrote: >>> On 18/06/2010 5:31 PM, |-|ercules wrote: >>> >>> >>> >>> >>> >>>> "Sylvia Else"<syl...(a)not.here.invalid> wrote ... >>>>> On 18/06/2010 4:52 PM, |-|ercules wrote: >>>>>> "Sylvia Else"<syl...(a)not.here.invalid> wrote ... >>>>>>> On 18/06/2010 3:03 PM, |-|ercules wrote: >>>>>>>> "Sylvia Else"<syl...(a)not.here.invalid> wrote >>>>>>>>> On 18/06/2010 10:40 AM, Transfer Principle wrote: >>>>>>>>>> On Jun 17, 6:56 am, Sylvia Else<syl...(a)not.here.invalid> wrote: >>>>>>>>>>> On 15/06/2010 2:13 PM, |-|ercules wrote: >>>>>>>>>>>> the list of computable reals contain every digit of ALL possible >>>>>>>>>>>> infinite sequences (3) >>>>>>>>>>> Obviously not - the diagonal argument shows that it doesn't. >>> >>>>>>>>>> But Herc doesn't accept the diagonal argument. Just because >>>>>>>>>> Else accepts the diagonal argument, it doesn't mean that >>>>>>>>>> Herc is required to accept it. >>> >>>>>>>>>> Sure, Cantor's Theorem is a theorem of ZFC. But Herc said >>>>>>>>>> nothing about working in ZFC. To Herc, ZFC is a "religion" >>>>>>>>>> in which he doesn't believe. >>> >>>>>>>>> Well, if he's not working in ZFC, then he cannot make statements >>>>>>>>> about >>>>>>>>> ZFC, and he should state the axioms of his system. >>> >>>>>>>> Can you prove from axioms that is what I should do? >>> >>>>>>>> If you want to lodge a complaint with The Eiffel Tower that the >>>>>>>> lift is >>>>>>>> broken >>>>>>>> do you build your own skyscraper next to the Eiffel Tower to >>>>>>>> demonstrate >>>>>>>> that fact? >>> >>>>>>> That's hardly a valid analogy. >>> >>>>>>> If you're attempting to show that ZFC is inconsistent, then say that >>>>>>> you are working within ZFC. >>> >>>>>>> If you're not working withint ZFC, then you're attempting to show that >>>>>>> some other set of axioms is inconsistent, which they may be, but the >>>>>>> result is uninteresting, and says nothing about ZFC. >>> >>>>>>> Sylvia. >>> >>>>>> That would be like finding a fault with the plans of The Leaning Tower >>>>>> Of Piza. >>> >>>>>> I might look at ZFC at some point, but while you're presenting Cantor's >>>>>> proof >>>>>> in elementary logic I'll attack that logic. >>> >>>>>> Instead of 'constructing' a particular anti-diagonal, your proof should >>>>>> work equally >>>>>> well by giving the *form* of the anti-diagonal. >>> >>>>>> This is what a general diagonal argument looks like. >>> >>>>>> For any list of reals L. >>> >>>>>> CONSTRUCT a real such that >>>>>> An AD(n) =/= L(n,n) >>> >>>>>> Now to demonstrate this real is not on L, it is obvious that >>>>>> An AD(n) =/= L(n,n) >>> >>>>>> Therefore >>>>>> [ An AD(n) =/= L(n,n) -> An AD(n) =/= L(n,n) ] proves superinfinity! >>> >>>>>> And THAT is Cantor's proof! >>> >>>>>> Want to see his other proof? That no box contains the box numbers (of >>>>>> boxes) that >>>>>> don't contain their own box number? >>>>>> That ALSO proves superinfinity! >>> >>>>>> Great holy grail of mathematics you have there. >>> >>>>>> Herc >>> >>>>> What are you trying to prove? >>> >>>> There is only one type of infinity. >>> >>> Infinity is a mathematical construct. Before you can even being to >>> discuss it, you have to have a set of axioms. >> >> What was the set that Cantor used? >> Nevertheless he "proved". > > He certainly was using some. For example, the diagonal argument falls > apart if the axioms don't permit the construction of a number by > choosing digits different from those on the diagonal. > > It isn't even clear whether Herc is tying to invalidate Cantor's proof > by finding a mistake in it, or to prove the inverse, which wouldn't > invalidate Cantor's proof, but would only show that the axioms on which > it is based are inconsistent. > > Herc cannot avoid the need to specify the set of axioms. > > Sylvia. How would one dispute axiomatic deductions if that were the case? Are you saying all mathematical facts are either axioms or the result of (X & X->Y) -> Y ? Herc
From: Peter Webb on 18 Jun 2010 22:53
"Tim Little" <tim(a)little-possums.net> wrote in message news:slrni1o9rt.jrj.tim(a)soprano.little-possums.net... > On 2010-06-19, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: >> Bijectable with N and "listable" are not the same. To be "listable" >> the set must be countable and recursively enumerable. > > There is no such requirement for recursive enumerability in Cantor's > work. Correct. > The concept was not even introduced into mathematics until some > time after his death. Correct. > Your insistence that Cantor required lists to > be recursively enumerable is bizarre. > Which of these two statements do you agree with: 1. You cannot form a list of computable Reals. 2. The computable Reals are countable. > > - Tim You have failed to explain why Cantor's diagonal proof "proves" the Reals are uncountable, but the same proof applied to computable Reals does *not* prove the Computable Reals are uncountable. I believe that this is because the set of computable numbers is not recursively enumerable, but if you have a better explanation I would love to hear it. |