"Sylvia Else" <sylvia@not.here.invalid> wrote > 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. > > 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
