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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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 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 16, 2013 5:22 PM

Nam Nguyen <namducnguyen@shaw.ca> writes:

> On 13/06/2013 4:17 AM, Alan Smaill wrote:
>> Nam Nguyen <namducnguyen@shaw.ca> writes:
>>

>>> On 12/06/2013 8:17 AM, Alan Smaill wrote:
>>>> But there is a language structure whose domain is, say,
>>>> the (least set with) terms {0, s(0), s(s(0)), ... }, given by an
>>>> inductive definition. Now show that there is only one way that addition
>>>> can be defined to satisfy the recursion equations for addition.

>>>
>>> Few technical problems (loopholes) here with your construction.
>>>
>>> First of all, let's call your domain of individuals (numbers) U:
>>>
>>> U = {0, s(0), s(s(0)), ... }.
>>>
>>> Then you intended U be finite and contain only "finite" elements
>>> ("terms" you said), but generalized inductive definition will _NOT_
>>> enable you to structure theoretically verify neither U is finite
>>> nor would contain only finite string elements.

>>
>> All I care abouty is that the case in which all elements are finite
>> is *possible*.

>
> Then show us the construction in "that case" of U, using the
> generalized inductive definition, where all individuals are
> finite (or finitely encoded). Specifically, show how the
> generalized inductive definition would _not_ admit any
> infinite individual from being a member of U!

You're missing my point;
you're the one claiming something is *impossible*;
so my question to you is:

is it *possible* that a language structure
given by a generalized individual domain can have as its domain
a set where all elements are finite?

>>> Secondly, a prime number can't be defined purely by the successor
>>> function and the addition function. Hence it's impossible
>>> to pin down (to verify) the ordering of the infinitely many primes,
>>> countably or uncountably.

>>
>> I didn't claim that; yes, you need addition and multiplication.

>
> No. _Constructing_ prime individuals need neither addition nor

I didn't say it was *necessary*;
I claim it is *possible* to characterise the prime numbers this way.

>> in the language structure while respecting the axioms of PA, there is
>> only one way to define multiplication.

>
> You haven't successfully _structure theoretically_ constructed what you
> called as the standard model (structure) for the language of
> arithmetic, let alone mentioning any formal system (e.g. PA).

Didn't I mention the language of PA up-thread?
If not, then I do that now (symbols for 0, successor, plus, time;
< defined via an abbreviation).

> Can you not mention a formal system in your own _structure theoretical_
> _discussion_ "there is a language structure whose domain is ..."?

There, the language is mentioned.

>>>> You claim that even the "<" relation is not pinned down by such a
>>>> structure, but since "x < y" is just "some z. x + z = y",

>>>
>>> Thirdly, this is a very common technical error that you and a few other
>>> posters frequently make: formula expression is _not_ a structure
>>> theoretical assertion. What you meant to say by "is just" is that
>>> you can define expression involving the symbol '<' by that involving
>>> the symbol '+'. Specifically, (x < y) df= (Ez[x + z = y]), but this
>>> _syntactical_ definition doesn't mean you have constructed a 3-ary
>>> predicate symbolized by '+' that one can verify that this predicate
>>> is indeed a function.

>>
>> What's the problem?
>> The predicate is indeed not a function ("+" is a function symbols,
>> "<" is not).

>
> The problem in this case is a function is a specialized 3-ary predicate
> P1, symbolized by '+' that would be used to construct the 2-ary
> predicate P2 symbolized by '<'.

??

> The point I was explaining to you
> is that you've not constructed the 3-ary predicate P1 yet, so you can't
> claim you've successfully constructed P2, for your symbol '<'.

If addition and multiplication is defined, over the specified domain,
then it is also defined, for given x,y, whether "some z. x + z = y" holds;
this uses the standard semantics for FOL that you find
in Shoenfield. (In this case, it's even a computable function.)

>> I'm sure Shoenfield explains about abbreviational definitions.
>
> I'm sure he explained that as a matter of syntactical issue.
> You confused that with our discussion about structure theoretical
> predicates (which are sets): this isn't a matter of formula
> syntax that Shoenfield was explaining.

But I am choosing to follow Shoenfield here.
And this is a *possible* route.

>> But you're claim *impossibility*, ie that thdere is *no* way
>> to deal with the notion of prime. So you need to deal
>> with all ways that anyone might use.

>
> I stated that you can't structure theoretically verify there are
> infinitely many primes.

You went further, claiming that it's impossible to verify trichotomy
for the usual order over the natural numbers. I'm claiming
that there is a language structure in which this can be done.

--
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