Philosophy of mathematics/Georg Cantor

According to modern set theory, originally based on Cantor's seminal ideas, the union of two infinite disjoint classes, such as the odd and even natural numbers, has the same number of members as either constituent class. This violates the logical principle that the union of two disjoint classes is a distinct class, with a distinct cardinal number. The resulting theory of infinite classes is, therefore, constructed at the cost of abandoning this and related logical principles.

This casting aside of readily understood 'intuitive' principles in favour of a system of axioms and definitions constitutes the very foundation of modern mathematics, and largely precludes any challenge to the basis of Cantorian and subsequent theories of infinite classes or numbers. The danger of such a state of affairs is that the foundations of mathematics cannot be radically altered, thus precluding any advances which might result from a reappraisal of Cantor's theory of infinity.

An intuitive illustration

While this might seem "counter-intuitive" at first, a simple thought experiment can provide us with a feel for Cantor's theorem.

The "trick" is to simply count all of the odd (or even) natural numbers:

1 → 1

2 → 3

3 → 5

4 → 7

...

n → 2n-1

n+1 → 2(n+1) -1

...

What you'll realise over time (when you're tired of counting) is that you'll never run out of odd numbers to count, and that for any odd number n, it will appear at (n+1)/2 in the counting order.

A notable exception to this rule is that a set made up of infinitely many infinite sets will in fact be larger than any of the member sets, according to Hilbert's paradox of the Grand Hotel.