Meta Math! by Gregory Chaitin

Author:Gregory Chaitin [Chaitin, Gregory]
Language: eng
Format: epub
ISBN: 978-0-307-48817-6
Publisher: Knopf Doubleday Publishing Group
Published: 2008-11-25T16:00:00+00:00

A century of work has not sufficed to solve this problem!

An important milestone was the proof by the combined efforts of Gödel and Paul Cohen that the usual axioms of axiomatic set theory (as opposed to the “naive” paradoxical original Cantorian set theory) do not suffice to decide one way or another. You can add a new axiom asserting there is a set with intermediate power, or that there is no such set, and the resulting system of axioms will not lead to a contradiction (unless there was already one there, without even having to use this new axiom, which everyone fervently hopes is not the case).

Since then there has been a great deal of work to see if there might be new axioms that set theorists can agree on that might enable them to settle Cantor's continuum problem. And indeed, something called the axiom of projective determinacy has become quite popular among set theorists, since it permits them to solve many open problems that interest them. However, it doesn't suffice to settle the continuum problem!

So you see, the continuum refuses to be tamed!

And now we'll see how the real numbers, annoyed at being “defined” by Cantor and Dedekind, got their revenge in the century after Cantor, the 20th century.

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