
Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Posted:
Jun 17, 2013 5:32 AM


Nam Nguyen wrote: > On 16/06/2013 10:34 PM, Nam Nguyen wrote: >> On 16/06/2013 5:55 PM, Nam Nguyen wrote: >>> On 16/06/2013 3:03 PM, Alan Smaill wrote: >>>> Nam Nguyen <namducnguyen@shaw.ca> writes: >>>> >>>>> On 15/06/2013 1:44 AM, Peter Percival wrote: >>>>>> Nam Nguyen wrote: >>>>>> >>>>>>> >>>>>>> No the inductive definition says that in your case of "{0, s(0), >>>>>>> s(s(0)), ... }" we'd have: >>>>>>> >>>>>>> (1) (0 e U) and (s(0) e U) and (s(s(0)) e U) >>>>>>> (2) (x e U) => (s(x) e U). >>>>>>> >>>>>>> In stipulation (2) it does _NOT_ say x must necessarily be finite. >>>>>> >>>>>> That is why you need a third clause that says (or has the effect >>>>>> that) >>>>>> the set being defined is the smallest such U. >>>>> >>>>> First, you should direct your technical "advice" here to Alan: that's >>>>> _his_ definition, _his_ defending of something, we're talking about. >>>> >>>> It's the standard definition, not mine. >>>> And this advice is of course correct. >>> >>> State it then in the technical language manner, as I did (1) and (2). >>> Let's postpone discussing anything else until you could do this. >>> >>> Remember you had "..." >> >> After that then of course you'd next show that your final U would >> contain _only_ individuals that are finitely encoded. >> >> I don't think that's possible but we'll wait to see how you might >> do that, would be my guess. > > Also, the following is just a note, a suggestion of mine which you're > free not to take it of course. > > Looking at Peter's "the set being defined is the smallest such U" > (above) reminds me of something, which is that for a 2 given infinite > sets S1 and S2, it might be the case that there are _two distinct_ > _senses_ of "smaller" between them:
Indeed so, "smaller" is ambiguous, which is why I have also offered a (standard, wellknown) precise characterisation: the set of natural numbers is the intersection of all sets U satisfying
(1) (0 e U) (2) (x e U) => (s(x) e U).
>  S1 is smaller that S2 in the sense that S1 is a _proper subset_ of S2, > but S1 and S2 are of the same cardinality, card(S1) = card(S2). > >  S1 is smaller that S2 in the sense that card(S1) < card(S2), but S1 > is _not a proper subset_ of S2. > > Given these 2 observations, it'd be interesting how you could come up > with the _construction_ of your final U, where all individuals be > finitely encoded.
There are other possibilities, it maybe that neither of S1 nor S2 is a subset of the other and their cardinalities are equal. So your
> Also, the following is just a note, a suggestion of mine which you're > free not to take it of course
is not worth "taking".
 I think I am an Elephant, Behind another Elephant Behind /another/ Elephant who isn't really there.... A.A. Milne

