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Topic: Re: Fundamental Theorem of Calculus: derivative is inverse to integral #7 textbook 5th ed. : TRUE CALCULUS; without the phony limit concept
Replies: 34   Last Post: Jun 21, 2013 6:39 PM

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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 12, 2013 9:18 AM

On 12/06/2013 4:16 AM, Alan Smaill wrote:
> Peter Percival <peterxpercival@hotmail.com> writes:
>

>> Nam Nguyen wrote:
>>

>>> Yes. Something unclear or vague you did have: what _are_ the
>>> "natural numbers" ?

>>
>> The natural numbers are the elements of the intersection of all sets M
>> such that:
>> (1) 0 in M
>> (2) if x in M then x+1 in M.
>> It's strange that you yourself have come close to this definition
>> (though you've never got it quite right). So what is your objection
>> to the[*] right version?
>>
>> [* Or 'a right version', there is more than one way to do it.]

>
> we can of course ask what "0" and "+" refer to ...
> this is (very roughly) one of Nam's points, I think.

That's right. My main theme here is the natural numbers aren't just
individuals of the universe symbolized as U but are also part of a
_language structure_ symbolized as N(U).

But as with any language structure, all relevant predicates have
to be constructed, enough to the point that if one is given a sentence
(formula) F that must be either true or false in N(U), then F must
must necessarily be verifiable (in principle at least), as true or as
false in N(U).

That's because the whole definition of a language structure is dedicated
to this single purpose: _to ASSERT in a VERIFIABLE manner a sentence as_
_true or as false_ .

> He was sort of persuaded that the denotation of "0" is recognisable
> in any given particular "language structure" for PA.
>
> The notion of standard model makes sense, even if "0" does not
> correspond to the empty set.

Exactly. The natural numbers collectively is just a language structure
constructed to assert, in a verifiable manner, (sentence) formulas as
true or false, by definition of a structure.

Therefore if there's a sentence that it's impossible to verifiable as
true or false, then - by definition - we don't have a structure, however
intended.

The key "strategy" of my argument in this respect is that, to the extend
that we have to construct an infinite (language) structure, as N(U) is
supposed to be, generalized-inductive-definition constructions of
predicate-sets are _NOT_ sufficient to assert the truth value of certain
formulas.

We don't have the concept _one_ modulo arithmetic: for each modulo
arithmetic, there's a fixed non-zero number m such that the sentence
formula m+m=0 is true, but is also false in _another_ modulo arithmetic,
all modulo arithmetic structure being finite of course.

Then, _the concept of the natural numbers_ is the concept of _AN_
_infinite modulo arithmetic_ where the the modulo-formula isn't
m+m=0 but cGC, ~cGC, or any of infinitely many such "impossible"
formulas.

Iow, there's indeed a class of infinitely many concepts each of
which can be named (known) as "the natural numbers".

From this class, mathematical relativity can be established.

--
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There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
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