Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Alan Smaill Posts: 1,103 Registered: 1/29/05
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,

>>>
>>> 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

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