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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: 2,699
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 20, 2013 9:06 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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,
>>>>> 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.


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.

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

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


Date Subject Author
6/16/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Alan Smaill
6/16/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/17/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/17/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/17/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Peter Percival
6/17/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Peter Percival
6/16/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Alan Smaill
6/17/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Alan Smaill
6/17/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/18/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Peter Percival
6/17/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Alan Smaill
6/17/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/18/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Peter Percival
6/19/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Alan Smaill
6/19/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/19/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/19/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Peter Percival
6/19/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Alan Smaill
6/19/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/19/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/20/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Peter Percival
6/20/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Alan Smaill
6/20/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/21/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral
#7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
namducnguyen
6/21/13
Read Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Alan Smaill

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