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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 18, 2013 2:59 AM

Nam Nguyen wrote:
> 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.

>>>>>>>
>>>>>>> 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

A set is countable if it can be put in one-to-one correspondence with a
subset of the natural numbers. So how could the set which defines the
set of natural numbers 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.
>

--
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