I have come across anti-dialecticians claiming that Cantor's theorem is "rubbish" .

http://www.revleft.com/vb/showpost.p...&postcount=116

Given the tendency of the proponents of this pseudo-theory of creating strawmen and eventually proceeding to "negate" what they call Leninist or Maoist dialectics by mere word-play, I am interested in finding out exactly how they are attacking one of the most important theorems of mathematics.

Here is a relevant link:

http://www.ccas.ru/alexzen/papers/Ca...of_Cantor.html

Discuss.

Cor blimey, Guv'nor, I couldn't make head nor tails of that article on Cantor! What does it have to do with the dialectic of the class struggle?

Neither could I.

We should ask them to explain that.

http://anti-dialectics.co.uk/

http://anti-dialectics.co.uk/Godel_letter.htm

In simpler English, Cantor introduced the idea that, in some cases, two cases of infinity are the same size, but, in some other cases, one infinity can be larger than another infinity. For example, the set "all integers greater than 5" and the set "all lintegers greater than 100" are the same size, because 6 can be mapped to 101, 7 mapped to 102, etc. The one-to-one mapping makes them the two infinitely-large sets the same size. However, the infinite number of number of points in a plane is a higher order of infinity than the infinite number of points in a line, because a one-to-one correspondence cannot be expressed -- every single point in a line can be associated with a whole infinity of points in the second dimension.

The only connection I can think of is: the two topics are alike in that both topics may be mentioned by people who like to impress other people with the fact that they went to college.

What we are actually confused with is that how is Cantor's theorem a fallacy?

By the way, a one-to-one mapping from the line to the plane does exist and therefore they have equal cardinality.

That was a nice one.

I do not understand how there could exist a "higher order of infinity", compared to a "lower order of infinity". This seems like a paradox, given what is ordinarily meant by the word "infinity".

Can you give me an example of unequal cardinality?

(The simplest example you can think of, otherwise I won't understand it.)

Natural numbers and reals. Natural numbers form a sequence while reals don't. Moreover, the number of reals is equal to the number of subsets of the set of natural numbers. Hence, there are more reals than natural numbers.

Red Cat, we are quite right to question Cantor's mystical theory, based as it was on the defective diagonal 'proof'.

The 'proof' is based on a Platonist theory of classes/sets, and a novel use of the word "greater" -- so that if a set cannot be put in an equivalence relation with one of its own proper sub-sets, such as the set of counting numbers, it is uncountable and thus "greater" than that other infinite set.

I will, once my work on anti-dialectics is finished, publish a technical paper on this -- and one based partly on Hilbert's Hotel 'paradox'.

http://en.wikipedia.org/wiki/Hilbert...he_Grand_Hotel

I won't say any more here since that will give my thesis away.

This is all part of my aim to show that traditional ways of looking at such 'problems' (and not just dialectics) are defective from start to finish -- as Wittgenstein pointed out.

Finally, what 'strawman' have I ever constructed? I have only ever relied on what dialecticians themselves say -- whereas you just ignore it.

And this can't be a 'strawman' if the 20th century's greatest mathematician, Poincare, had this to say about it:

Mike:

And we could say the same about you scientists, couldn't we?

Actually I don't understand exactly why you claim Cantor's theorem to be false. I am looking forward to you explaining the counter-arguments.

Red Cat:

I have never claimed it to be false. His 'argument' depends on a verbal sleight-of-hand, a bit like Anselm's Ontological proof. So, it's confused, not false.

Why? In which step?

Red Cat:

In his enumerative definition of a set, in his use of "greater than", in his use of "infinite"..., and that's just for starters.

Just to be sure... are you talking about the proof of reals being uncountable, or the cardinality of a set being less than that of its power set in general, or both ?

Both (and more); the entire theory is based on what can only be called "mathematical mythology".

Let's start with the uncountability proof. Where do you think it goes wrong?

As I have said already, if and when I want to publish something on this, it will either appear at my site first, or in a journal that will accept it. I only intervened here to answer a few general points, not give my thesis away.

You can find a clue as to what it will be in some of the references I give at my site, and here:

http://www.uea.ac.uk/~j339/Godelselfref.htm

http://plato.stanford.edu/entries/wi...n-mathematics/

Too long and probably difficult to understand. If you don't explain things to us here then until you publish your thesis we will assume that both the proofs are correct.

Assume what you like...