Date: Jun 21, 2013 6:16 AM
Author: Alan Smaill
Subject: Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept

Nam Nguyen <namducnguyen@shaw.ca> writes:

> On 20/06/2013 5:04 AM, Alan Smaill wrote:
>> Nam Nguyen <namducnguyen@shaw.ca> writes:
...
>>> - 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.

>
> It is relevant: you just don't realize it.


You are the one claiming *impossibility*.
I don't have to prove anything.

> As long as you don't
> _cast away_ the informal symbol '...' in your constructed U as
> I've previously done on stipulations (1) and (2) [see the below quote]
> then your argument would go nowhere, and virtually every question
> would be relevant.
>
> <quote>
>
> (1) (0 e U) and (s(0) e U) and (s(s(0)) e U)
> (2) (x e U) => (s(x) e U).
>
> </quote>
>

>> We just *suppose* we are in situation (c).
>
> But "suppose" does _not_ necessarily grant you the right to
> prove a particular statement as true or false.


Of course not --
I'm not claiming that.

>> Is it *possible* that the only elements of U are those that can be
>> proved to be in U, using the inductive definition?

>
> You have changed the subject, the question: your question now no
> longer references about "finite elements", i.e. finitely encoded
> elements. So let's go back to where we were.


Yes, this is an additional question.
But why do you refuse to answer?

> Is it possible that Alan's constructed U (constructed with his '...')
> would contain only finitely encoded individuals, where '...'
> would refer to the Generalized Inductive Definition?
>
> The answer is Yes, it's possible.
>
> Can we prove that Alan's constructed U (constructed with his '...')
> would contain only finitely encoded individuals, where '...' would
> refer to Generalized Inductive Definition?
>
> The answer is No, we can not prove that.


Fine, let's go with that.

Now what about the question:

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