
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 11:20 PM


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 and couldn't contain individuals that aren't finitely encoded? After all, S1 being a proper subset of S2 says nothing about their cardinality.
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 

