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Problems with infinities

Amos Ng's picture

Amos Ng
2020-04-12

Thanks for making this book public! I’m glad that you’ve written an interesting summary of what this section is to be, even though it’s not complete yet.

Could you say what the “ problems with infinities” you mention here are, so that I can look them up myself? I assume you aren’t talking about Gödel’s incompleteness theorem?

Any additional pointers to resources on the historical ramifications of such revolutions in mathematics would also be appreciated.

Thanks! No worries if it’s too much trouble :)

Russell's paradox, for example

Hi, Amos,

This Wikipedia article section (on the “foundational crisis”) may be a helpful brief overview.

In the late 1800s and early 1900s, several problems showed up in the theory of infinite sets pioneered by Georg Cantor. One was Russell’s Paradox. The Wikipedia article I linked there gives a good explanation of why this caused so much trouble. Another was the continuum hypothesis.

These didn’t lead directly to Gödel’s and Turing’s no-go theorems, but those were part of the same general milieu (the Hilbert Program).

Enjoy reading about this! Apparently many great mathematicians found it awful at the time, but in retrospect it seems just really interesting.

For an overview of the crisis

frank's picture

frank
2020-05-05

For an overview of the crisis in mathematics that non-math folk can understand, I’d recommend linking the extremely good graphic novel Logicomix, which follows Bertrand Russell as he tries to uncover foundational . As someone who’s pretty mediocre at maths, this work was a really good introduction to the topic, the art is really good.

https://en.wikipedia.org/wiki/Logicomix

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