
Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Posted:
Jun 20, 2013 7:04 AM


Nam Nguyen <namducnguyen@shaw.ca> writes:
> On 19/06/2013 8:20 AM, Alan Smaill wrote: >> Nam Nguyen <namducnguyen@shaw.ca> writes: >> >>> On 19/06/2013 3:02 AM, Alan Smaill wrote: >>>> Nam Nguyen <namducnguyen@shaw.ca> writes: >>>> >>>>> On 17/06/2013 4:36 AM, Alan Smaill wrote: >>>> >>>> [on the possibility that some members of the set {0,s(0),s(s(0)),...} >>>> are not finite] >>>> >>>>>> My debate with you does not depend on this being possible, however. >>>>> >>>>> I don't know what you're trying to say here. >>>>> >>>>> My statement here is that your constructed set: >>>>> >>>>> U = {0, s(0), s(s(0)), ... } >>>>> >>>>> could be uncountable and could contain elements that aren't finitely >>>>> encoded. >>>>> >>>>> Do you accept or refute my statement here. If you refute, please note >>>>> that I had a request (above): >>>> >>>> For purposes of argument, I accept it. >>>> >>>> My question to you is: is it possible that the set in question >>>> contains only finite elements. >>>> >>>> Do you accept or reject my statement here. If you reject, >>>> please explain why. >>> >>> As is, with your '...' being syntactically unformalized, then Yes, >>> the followings are possible: >>> >>> (a) U is finite: containing only finite elements. >>> (b) U is finite: containing also infinite elements. >>> (c) U is infinite: containing only finite elements. >>> (d) U is infinite: containing also infinite elements. >>> >>> _All_ those are the possibilities. _Which of those 4 possibilities_ >>> can you _specifically construct that one can verify_ ? >> >> I am specifically *not* claiming that I can persuade you that >> some specific structure has property (c). It is enough >> that you admit that (c) is possible. > > Sure. As I've stated it's 1 out of 4 possibilities: so (c) is > a possibility.
OK
>>> You might have (c) in mind, but then from the unformalized and >>> _unverifiable_ notion of (c), how could you _verify_ the existences >>> of certain predicate and function sets, hence _verify_ as true or >>> false the truth values of certain formulas? >> >> Since you admit (c) is possible, let's consider that case. > > Sure. > >  In this of (c) you can _verify_ that 0, s(0), s(s(0)) are > finite individuals, in your constructed set named "U". > >  In this of (c) you can _NOT verify_ x is a finite individual > given x is in your constructed set named "U". > > Agree? If not, please refute my above by clearly _constructing a set_ > named "U", per the possibility (c), _without_ your '...' symbol.
The question is irrelevant to my argument. We just *suppose* we are in situation (c).
Is it *possible* that the only elements of U are those that can be proved to be in U, using the inductive definition?
 Alan Smaill

