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Calculus, Infinity, And Dr. Dick

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Posted by Paul R. Martin on December 13, 2003 00:42:23 UTC

Hi Tim,

Sorry for being so slow to respond.

"here is another concern with respect to calculus.
with respect to giving infinity the ole heave ho how about the theorems of power series in calculus. they actually use infinity in the terms of the theorems. i should say i never fully understood power series but i need to ask the question anyway."


You are correct that the terms "infinity" and "infinite" appear in the theorems of "infinite" series, of which power series are a special case. But the use of these terms does not imply or necessitate the existence of any infinite sets. In the traditional foundations of Arithmetic, the Axiom of Choice has been accepted and included which implies that the set of natural numbers is infinite. But this assumption is not necessary for the definition of infinite series. Here's why.

An "Infinite" series is defined to be the limit of a sequence of partial sums. Note in particular that an infinite series is NOT defined to be a sum. Sums are defined only for pairs of summands. Using the axiom called the Associative Law, the sum of two numbers can be extended to produce the "sum" of any finite number of numbers. But there is no definition of a "sum" of an "infinite" set of numbers. Instead, the "sum" of a particular infinite series may be defined in some cases as the limit of a sequence of numbers. (Remember that a sequence is simply an ordered set of numbers.) The numbers making up the sequence are the partial sums of the series. That is, the nth number in the sequence is the sum of the first n numbers in the series. If, and only if, that sequence has a limit, then we define the "sum" of that series to be that limit. In that case, we also say that the sequence converges. Convergence is a necessary condition for the definition of the "sum" of an infinite series.

So this brings us back to the definition of a limit. The notion of a limit "as n goes to infinity" simply means that no matter how small a tolerance you specify around the limit, you can find a number, N, such that all partial sums where n>N will be within that tolerance. I claim that a meaningful and useful definition of such a limit can be made even for finite sets of numbers. I sketched this out in my previous posts.

So to net it out, modern math assumes the existence of infinite sets and the definition of a "limit"
is consistent with that assumption. I claim that if the existence of infinite sets were denied, a definition of "limit" could still be made which would yield the same theorems, albeit in a different domain of numbers.

"Again Paul..... May I Ask how your thoughts with respect to the concept of infinity square with the precepts put forth by Dr. Dick in Fundamentals of Physical Reality as quoted below:
"3. The number of subsets which make up the universe must be infinite. No matter how many subsets have been examined, ones model of the universe must allow for the existence of another subset not yet examined. This is the very definition of infinity.""


As you might imagine, Dick and I had quite a discussion on this point before we saw eye-to-eye. Even though I maintain that mathematics would be more useful if we denied the existence of infinity, I cannot deny that it is possible to conceive of the notion of infinity, to assume an axiom that allows for infinite sets, and to develop a theory of the peculiarities of infinite sets. This has been done now for about a hundred years.

In light of this possibility, Dick's theorem (as I see it, his formal work has demonstrated a theorem) needs to be consistent with the mathematical system. He has proved to my satisfaction that his theorem is true in conventional (i.e. infinite) mathematics. I am convinced that his theorem would also be true and provable in a sufficiently robust discrete (i.e. finite) mathematics.

In short, when in Rome, do as the Romans do. And, when in an infinite math, remain consistent with the axioms of that math.

"if you find time please address it an i promise (fingers crossed behind back) i wont ask any more of this nature."

I hope you didn't write that as a threat of punishment for being so tardy in answering your questions. If you can tolerate this slow response, I would love to have you ask more questions of this nature. Not many people want to talk about this stuff, you know.

Warm regards,

Paul

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