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Topic: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Replies: 38   Last Post: Jun 21, 2013 6:16 AM

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 namducnguyen Posts: 2,777 Registered: 12/13/04
Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept

Posted: Jun 21, 2013 12:29 AM

On 20/06/2013 7:06 PM, Nam Nguyen wrote:
> On 20/06/2013 5:04 AM, Alan Smaill wrote:
>> 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,

>>>>>
>>>>> 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.

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

>>
>> 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.
>
> 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.

Another thing would help reminding you is that, a la Tarski, if only
on the basis of _assuming_ a meta statement P as true, then you can
not prove that neg(P) is _actually_ false. Naturally.

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

Date Subject Author
6/16/13 Alan Smaill
6/16/13 namducnguyen
6/17/13 namducnguyen
6/17/13 namducnguyen
6/17/13 Peter Percival
6/17/13 Peter Percival
6/16/13 Alan Smaill
6/17/13 Alan Smaill
6/17/13 namducnguyen
6/18/13 Peter Percival
6/17/13 Alan Smaill
6/17/13 namducnguyen
6/18/13 Peter Percival
6/19/13 Alan Smaill
6/19/13 namducnguyen
6/19/13 namducnguyen
6/19/13 Peter Percival
6/19/13 Alan Smaill
6/19/13 namducnguyen
6/19/13 namducnguyen
6/20/13 Peter Percival
6/20/13 Alan Smaill
6/20/13 namducnguyen
6/21/13 namducnguyen
6/21/13 Alan Smaill