Re: Ultimate debunking of Cantor's Theory

Calvin wrote:

On Jul 15, 1:31 pm, Rotwang <sg...@xxxxxxxxxxxxx> wrote:


Of course there's no reason why you should believe me and not
them when I say this, if you don't think my posts stand up on their
own (which you evidently don't).



I just looked over all of your posts, and to the
extent that I can understand them, I don't have
any disagreements, except two:

You seemed to be saying that an analogy could be
made between saying 'set of all sets' and saying
'infinite set of natural numbers'.

And you said you doubted that one could prove
that the natural numbers form an infinite set
without the axiom of infinity.

As to the latter, I was wrong to question your
motivation, and I'm sorry.

But this thing about the axiom of infinity is
illustrative of what I was complaining about in
the post that you just responded to.
There are things that the dullest among us can
know and understand and prove conclusively without
knowing anything at all about set theory, and
the unlimited nature of the total collection of
natural numbers is, I believe, one of them.

If one becomes interested in the set-theoretical
basis of the foundations of mathemetics, then by
all means, study all necessary and sufficient
axioms.

But don't try to deny Joe Blow the simple pleasure
of doing what he can do, correctly and conclusively,
without going into set theory at all.


There are two things to consider here.

1) The original post of this thread was a joke, really. Chris Heckman, aka Proginoskes, is a well-respected mathematician around here. The post was certainly meant to be an ironic response to a long line of cranks. Among others, many of these deny Cantor's theorem and the uncountability of the reals; at least one denies the existence of infinity at all, and there have been some who insist that the sets {0, 1, 2, ...} and {1, 2, 3,...} have different "sizes." Many regular readers tire of such cranks. I, for one, usually filter out their posts.

2) The fact that there are infinitely many integers was clear to Augustine at least. The point is that if one wants to talk about such things rigorously, one has to set down axioms, in the same way Euclid laid the foundations of the geometry associated with his name. Modern mathematics lays it foundations with the axioms of set theory. If something we believe and need to be true cannot be proved from other stated axioms, then we must state another axiom.

Mathematicians have mostly settled on the axiom system called Zermelo-Frankel with Choice (ZFC). If you remove the Axiom of Infinity from this system, one can still show that there are infinitely many natural numbers (i.e., nonnegative integers). One can also do this with an earlier axiomatization of arithmetic called Peano's Axioms (PA). (You might still need concepts of sets added to PA; I really don't know much about the subject.) However, without the Axiom of Infinity, one cannot talk of a "set" of natural numbeers, and this is contrary to our desires and needs. For example, without this axiom, it seems that one cannot even prove the existence of negative numbers in our mathematical universe. (Or, if one can, the development would be unnecessarily more complicated than the way we do now.)

An undergraduate professor (Walter H. Gottschalk) of mine once defined mathematics as the study of mathematical objects, and he defined a mathematical object as a set. "Set" is an undefined term; sets are the inhabitants of the universe described by ZFC, just as points, lines, planes, and space are the inhabitants of the universe of Euclidean geometry. (Some may take issue with Gottschalk's definition. What about logical formulae, categories, and classes? Is an algebraic expression a mathematical object? An equation? A problem?)

So it is very useful to be able to speak of the *set* of natural numbers and more generally of infinite sets. Thus, we include the Axiom of Infinity in our foundation.

--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan

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