Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


fom
Posts:
1,968
Registered:
12/4/12


Re: Cantor's absurdity, once again, why not?
Posted:
Mar 14, 2013 6:17 AM


On 3/14/2013 4:47 AM, david petry wrote: > > In a previous aritcle a while back, "marcus_b" wrote: > > ** start quote ** > In math, the great pons asinorum now is Cantor's diagonal proof. There seem to be scads of people out there who just cannot quite get it and who yearn to achieve their rightful 15 minutes of fame by trying to shoot it down, thus thrusting themselves ahead of Cantor in the pantheon of mathematical geniuses. Can you shed some light on the motivation here? > ** end quote ** > > > The "cranks" who insist on pointing out the absurdity of Cantor's argument are puzzled that mathematicians "cannot quite get it". > > Cantor's argument relies on accepting the notion of an actual infinite, and that's problematic.
If you are going to paraphrase Cantor's argument, please do so correctly.
Cantor's "argument" is an argument scheme.
It is an argument for constructing a counterexample for some particular claim.
A particular claim must be made before Cantor's argument can even apply.
That particular claim is that there is a single infinity that is referred to in language as an object.
The referential notion is important because that underlies the fact that a completed infinity is being assumed in the claim.
Given that, the notion of infinity that will apply is the notion of the natural numbers since Euclid proved that there is no greatest prime number and every natural number has a prime factorization.
So, the person who will apply Cantor's argument can do nothing until the person referring to infinity accepts an enumeration of real numbers using some sort of infinitary naming scheme (such as any expansion based on long division).
Once an enumeration claiming to list the infinite collection of real numbers in an ordinal sequence is presented and acceptable to both parties as a starting point, Cantor's argument may be applied to show that the enumeration cannot be total.
One need not even believe in an actual infinite in order to apply Cantor's argument.
For example, Wittgenstein understood perfectly well how to apply Cantor's argument and he certainly is not thought of as believing in a completed infinity.
However, he also did not attack the mathematicians who conducted investigations along those lines. He recognized that he had a responsibility to pursue answers to accommodate mathematical practice. He also recognized that the ultimate difficulty and ground for criticism has to do with the nature of "all" and its interpretation.
Such considerations have led to fields of mathematics based on constructive ideas. Unfortunately, constructive mathematics has yet to be able to explain the successes of the differential calculus. In fact, it has yet to be able to prove the fundamental theorem of algebra.
It is all of the "other mathematics," including the difficulty with infinitesimals, that leads mathematicians to accept completed infinities as more reasonable than other alternatives.
In other words, there are responsible ways to address this problem and other ways which are less responsible.
You are making the latter choice.
But since you are, let me invite you to explain a resolution to Zeno's paradox without completed infinities. We can turn back the clock. We would just like you to run it back forward.
That is not so easy now, is it?



