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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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 Peter Percival Posts: 2,623 Registered: 10/25/10
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 5:32 AM

Nam Nguyen wrote:
> 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:

Indeed so, "smaller" is ambiguous, which is why I have also offered a
(standard, well-known) precise characterisation: the set of natural
numbers is the intersection of all sets U satisfying

(1) (0 e U)
(2) (x e U) => (s(x) e U).

> - 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 are other possibilities, it maybe that neither of S1 nor S2 is a
subset of the other and their cardinalities are equal. So your

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

is not worth "taking".

--
I think I am an Elephant,
Behind another Elephant
Behind /another/ Elephant who isn't really there....
A.A. Milne

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