
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 > multiplication. Certainly not addition.
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 upthread? 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 3ary >>> 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 3ary predicate > P1, symbolized by '+' that would be used to construct the 2ary > predicate P2 symbolized by '<'.
??
> The point I was explaining to you > is that you've not constructed the 3ary 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

