purplecat

Well to be fair to the maths teacher I don't know exactly what happened. But the children obviously felt dismissed and wanted something to back up their argument. Omega, broadly speaking, is equivalent to aleph₀ and it depends on how you are defining numbers. Cardinal numbers define numbers in terms of the size of set (unlike defining them in terms of actual sets like the ordinal numbers do). Aleph₀ is the size of the set of natural numbers (and rational numbers), 2^aleph₀ is the size of the set of real numbers. Cantor showed that every cardinal number has a "next one up" so aleph₁ is the next cardinal number after aleph₀. It's an open question whether aleph₁ is the same thing as 2^aleph₀ - i.e. whether the real numbers form the next largest set after the natural numbers. This is called the continuum hypothesis. Last time I looked (which was about twenty years ago) it had been shown that most of mathematics was consistent whether the continuum hypothesis was true or whether it was false which suggests it may be undecidable in some sense.