The Continuum Hypothesis

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The continuum hypothesis (CH) states that there are no sets bigger than the integers and smaller than the real numbers. (A set X is "bigger" than a set Y if there is no one-to-one function from X to Y). CH was formulated by Georg Cantor around 1880.

This seems pretty straightforward, kind of like saying that there are no integers between 0 and 1. It should be easy to prove or disprove, right?

Wrong. No one made any progress on it, and in 1900 David Hilbert put it first on his famous list of 23 open problems.

Remarkably, it turns out that CH can be neither proved nor disproved from the current axioms of mathematics. The two parts of this assertion were proved by Kurt Godel (in 1940) and Paul Cohen (in 1963). For his part of the proof, Cohen invented a general-purpose technique called "forcing". When I retire, I vow to learn about forcing, starting by reading Forcing for Dummies.

In other words, unless a new axiom comes along - some basic, obvious fact that has somehow eluded mathematicians to date - we will never know if CH is true or false. This is analogous to the Heisenberg Uncertainty Principle, which proves (from the axioms of quantum mechanics) that we can't measure both the position and momentum of a particle. It imposes a hard upper bound on what we puny humans can know.

Independently of whether we can prove CH, does it really have a unique truth value? Is there an unique abstract universe of sets in which there either is or isn't something intermediate in size between the integers and reals? We now enter the realm of the Philosophy of mathematics. "Platonism" is the belief that there is such an abstract universe; there are many competing beliefs.

What's the word on the street about CH?

Paul Cohen though that CH was obviously false. His argument (paraphrased): there are incremental ways of making ever-bigger sets. The power-set axiom (which produces the reals from the integers) is not incremental - it's something different and much cruder. There must be an incremental way to make a set bigger than the integers (but smaller than the reals).

Hugh Woodin (a mathematician at UC Berkeley) has some results supporting the idea (but not proving) that CH is false. See here and here. (Warning: no one understands these papers).

Here's a video of a talk by Hugh Woodin.

Me personally? I'm an unabashed Platonist, and I lean towards thinking that CH is true. The incremental ways of making bigger sets (limits, products, etc.) can't go beyond the size of the integers. The power-set is the only way to make something bigger.

Copyright 2018 © David P. Anderson