On 19/06/2010 6:50 AM, WM wrote: > On 18 Jun., 09:37, Sylvia Else<syl...@not.here.invalid> wrote: >> On 18/06/2010 5:31 PM, |-|ercules wrote: >> >> >> >> >> >>> "Sylvia Else"<syl...@not.here.invalid> wrote ... >>>> On 18/06/2010 4:52 PM, |-|ercules wrote: >>>>> "Sylvia Else"<syl...@not.here.invalid> wrote ... >>>>>> On 18/06/2010 3:03 PM, |-|ercules wrote: >>>>>>> "Sylvia Else"<syl...@not.here.invalid> wrote >>>>>>>> On 18/06/2010 10:40 AM, Transfer Principle wrote: >>>>>>>>> On Jun 17, 6:56 am, Sylvia Else<syl...@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.
