
Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Posted:
Jun 13, 2013 6:17 AM


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*. You admit that that one is possible, don't you? So let's use that. (Actually the usual use of inductive definition says that we get the least, just as we define the syntax of formulae by an inductive definition, and all our formulae 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. But just as there is only one way that addition can be defined in the language structure while respecting the axioms of PA, there is only one way to define multiplication.
>> 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).
> Hence you've _NOT_ constructed the 2ary predicate, symbolized by '<'. > >> we know how to tell for given denotations of x,y if x<y, >> and we do have trichotomy. > > Again you still don't have a predicate symbolized by '<' yet, let > alone a trichotomy.
I'm sure Shoenfield explains about abbreviational definitions.
>> Equally, don't define "prime" by a separate inductive definition, > > Again (as I showed to "fom" in a thread some months ago), prime > individuals can't be inductively defined but odd and even individuals > can.
I'm not disputing that. 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.
>> but by using properties of addition and multiplication. > > And in that thread, I showed that even, odd, prime individuals > can be defined purely by syntactical encoding of individuals, > independent of any notion of multiplication (or even addition).
Same comment as above.
>> So the >> only inductive aspect of the language structure is 0 and successor. > > That actually doesn't convey anything in this context.
 Alan Smaill

