Cantor Redux

I got criticised on my twitter stream for the Cantor article I posted yesterday. I was not called to order for being to harsh, @ianppreston criticised me, quite correctly, for not being harsh enough! As I don’t wish to create the impression that I’m becoming a wimp in my old age I thought I would give Ms Inglis-Arkell another brief kicking.

Strangely my major objection from yesterday has mysteriously disappeared from her post (did somebody tip her off that she was making a fool of herself?) but there remains enough ignorance and stupidity to amuse those with some knowledge of Cantorian set theory and transfinite arithmetic, knowledge, which Ms Inglis-Arkell apparently totally lacks.

Ms Inglis-Arkell’s dive into the depths of advance mathematics starts so:

Imagine a thin line, almost a thread, stretching to infinity in both directions. It runs to the end of the universe. It is, in essence, infinite. Now look at the space all around it. That also runs to the end of the universe. It’s also infinite. Both are infinite, yes, but are they the same? Isn’t one infinity bigger than the other?

The answer to the second question is actually no! Cantor demonstrated, counter-intuitively, that the number of points on a straight, the number of points in a square and the number of points in a cube are all infinite, all equal and all equal to ‘c’ the cardinality of the real numbers and the power set of aleph-nought.

After defining the infinite number of natural numbers as aleph-nought Ms Inglis-Arkell then writes the following:

But then what about real numbers? Real numbers include rational numbers, and irrational numbers (like the square root of five), and integers. This has to be a greater infinite number than all the other infinite numbers.

These three sentences contain three serious errors, one implied and two explicit. The rational numbers includes the integers so to state them separately when describing the real numbers is either wrong or at best tautologous. Secondly, and this is the implicit mistake, a set consisting of the rational numbers and those irrational numbers, which are also algebraic numbers i.e. describable with an algebraic equation for example X2 = 2, is also a countable infinite set that is equal to aleph-nought. Only when one includes the so called transcendental irrational numbers, those that cannot be described with an algebraic equation for example the circle constant π, that the infinite set become larger than aleph-nought. This result is again extremely counter-intuitive, as very few transcendental numbers have ever been identified. The final error is very serious because the cardinal number of the real numbers ‘c’ (for continuum) is by no means “a greater infinite number than all the other infinite numbers.

Cantor could demonstrate that the so-called power set of an infinite set, i.e. the set of all the subsets of the set, has a larger cardinality than the set itself. This newer set also has a larger power set and so on ad infinitum. As stated above c is equal to the power set of aleph-nought. There is in fact an infinite hierarchy of infinite sets each one larger than its predecessor. On of the great mysteries of Cantorian set theory is where exactly c fits into this hierarchy. Cantor asked the question whether c is equal to aleph-one where aleph-one is defined as the the cardinality of the set of all countable ordinal numbers (1)? He himself was not able to answer this question. It later turned out that it is in fact an undecidable question. In the axiomatic version of Cantorian set theory, the theory is consistent, i.e. free of contradictions, both when c is assumed to be equal to aleph-one and when they are assumed to be not equal. This produces two distinct set theories, the first with c equal to aleph-one is called Cantorian the other non-Cantorian.

Although my sketch of Cantorian set theory and transfinite arithmetic is only very basic I hope I have said enough to show that it is really not a subject about which one should write if, as appears to be the case with Ms Inglis-Arkell, one doesn’t have the necessary knowledge.

(1) Going beyond this and explaining exactly what this means goes futher than is healthy in normal life. For those who are curious I recommend Rudy Rucker’s Infinity and the Mind, Birkhäuser, 1982

Later additions: I have corrected the mistakes kindly pointed out by Sniffnoy in the comments. Note to self: Turn brain on before skating on the thin ice of transfinite arithmetic.

Filed under History of Logic, History of Mathematics

6 responses to “Cantor Redux”

1. Sniffnoy

I’m afraid there’s a few mistakes in your correction, too. Assuming we’re talking about a continuous line/square/cube of real numbers, the number of them would be 2^aleph_0, not aleph_0.

Secondly, aleph_1 is not defined to be 2^aleph_0, nor is aleph_2 defined to be 2^aleph_1, etc. If that were so the continuum hypothesis would be trivial, as certainly the continuum (2^aleph_0) is equal to 2^aleph_0. Rather, aleph_1 is the smallest cardinal larger than aleph_0 (and so forth). (Hence, the continuum hypothesis is the question of whether 2^aleph_0 immediately succeeds aleph_0, or if there is anything else inbetween.)

Also, to nitpick, of course both the assumption that the continuum hypothesis is true and the assumption that it is false are consistent only on the condition that set theory itself is consistent. 🙂

• Thony C.

Thank you kind sir for correcting the errors of an old man. Write in haste correct at leisure 😉

2. Peter

Very interesting continuation (especially the “correction” on io9). I have to nitpick the half-sentence

[…] as very few transcendental numbers have ever been identified.

It’s a little misleading since there are many known transcendental numbers.

Of course, those transcendental numbers we can actually describe (i.e., compute) again form only a countable set — which makes Cantor’s theorem so extremely counter-intuitive (and then Loewenheim-Skolem might take us even higher into the counter-intuitive…).

3. Thony C.

and then Loewenheim-Skolem might take us even higher into the counter-intuitive…

It might very well indeed