
Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Posted:
Jun 18, 2013 2:59 AM


Nam Nguyen wrote: > On 17/06/2013 4:37 AM, Alan Smaill wrote: >> Nam Nguyen <namducnguyen@shaw.ca> writes: >> >>> 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: >>> >>>  S1 is smaller that S2 in the sense that S1 is a _proper subset_ of >>> S2, >> >> That is the intended sense. > > Well then, what makes you so sure your constructed U couldn't be > uncountable
A set is countable if it can be put in onetoone correspondence with a subset of the natural numbers. So how could the set which defines the set of natural numbers be uncountable?
> and couldn't contain individuals that aren't finitely > encoded? After all, S1 being a proper subset of S2 says nothing about > their cardinality. >
 I think I am an Elephant, Behind another Elephant Behind /another/ Elephant who isn't really there.... A.A. Milne

