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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 17, 2013 2:27 AM

On 16/06/2013 10:34 PM, Nam Nguyen wrote:
> On 16/06/2013 5:55 PM, Nam Nguyen wrote:
>> On 16/06/2013 3:03 PM, Alan Smaill wrote:
>>> Nam Nguyen <namducnguyen@shaw.ca> writes:
>>>

>>>> On 15/06/2013 1:44 AM, Peter Percival wrote:
>>>>> Nam Nguyen wrote:
>>>>>

>>>>>>
>>>>>> No the inductive definition says that in your case of "{0, s(0),
>>>>>> s(s(0)), ... }" we'd have:
>>>>>>
>>>>>> (1) (0 e U) and (s(0) e U) and (s(s(0)) e U)
>>>>>> (2) (x e U) => (s(x) e U).
>>>>>>
>>>>>> In stipulation (2) it does _NOT_ say x must necessarily be finite.

>>>>>
>>>>> That is why you need a third clause that says (or has the effect that)
>>>>> the set being defined is the smallest such U.

>>>>
>>>> First, you should direct your technical "advice" here to Alan: that's
>>>> _his_ definition, _his_ defending of something, we're talking about.

>>>
>>> It's the standard definition, not mine.
>>> And this advice is of course correct.

>>
>> State it then in the technical language manner, as I did (1) and (2).
>> Let's postpone discussing anything else until you could do this.
>>

>
> After that then of course you'd next show that your final U would
> contain _only_ individuals that are finitely encoded.
>
> I don't think that's possible but we'll wait to see how you might
> do that, would be my guess.

Also, the following is just a note, a suggestion of mine which you're
free not to take it of course.

Looking at Peter's "the set being defined is the smallest such U"
(above) reminds me of something, which is that for a 2 given infinite
sets S1 and S2, it might be the case that there are _two distinct_
_senses_ of "smaller" between them:

- S1 is smaller that S2 in the sense that S1 is a _proper subset_ of S2,
but S1 and S2 are of the same cardinality, card(S1) = card(S2).

- S1 is smaller that S2 in the sense that card(S1) < card(S2), but S1
is _not a proper subset_ of S2.

Given these 2 observations, it'd be interesting how you could come up
with the _construction_ of your final U, where all individuals be
finitely encoded.

--
----------------------------------------------------
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