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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 11:20 PM

On 17/06/2013 4:37 AM, Alan Smaill wrote:
> Nam Nguyen <namducnguyen@shaw.ca> writes:
>

>> 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,

>
> That is the intended sense.

Well then, what makes you so sure your constructed U couldn't be
uncountable and couldn't contain individuals that aren't finitely
encoded? After all, S1 being a proper subset of S2 says nothing about
their cardinality.

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