The Continuum Hypothesis

The Continuum Hypothesis is very interesting. According to mathworld, It is a proposal (originally by George Cantor) that there is no infinite set with a cardinal number between that of the infinite set of integers and the larger infinite set of real numbers. It is easy to show that the second infinite set is larger than the first – 1st year students are shown this (think Cantor’s diagonal argument which would be familiar to most tertiary maths students who have studied topology or analysis).

Is the Continuum Hypothesis true? To quote from mathworld,

Gödel showed that no contradiction would arise if the continuum hypothesis were added to conventional Zermelo-Fraenkel set theory. However, using a technique called forcing, Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to set theory. Godel’s and Cohen’s results established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the Zermelo-Fraenkel axioms together with the axiom of choice).

Ok, I’ve just rewritten what’s in the link! You might enjoy clicking on the link and learning about the Axiom of Choice and set theory. One thing I’d like to do is to learn more about this.

In general, I like things to be as free and general as possible, so I like the idea (see the link) of adopting an axiom which (together with the Zermelo-Fraenkel axioms and the axiom of choice) would imply that the continuum hypothesis is false.

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One Response to The Continuum Hypothesis

  1. GMB says:

    Outside of maths its more productive and in fact you pretty much have to build your models up on the basis of near-axioms and generalisations that we couldn’t even call near-axiomatic. Since axioms are pretty thin on the ground in the natural world and the world of puny humans.

    The real trick would be able to find a way, to build such near-axiom-based models up in a systematic and reliable way.

    You see Sacha. I’m trying to push you toward those breakthroughs we were talking about. This troubled world needs your assistance.

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