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I'd like an example of a field of order $\beth_2$ (that is, the cardinality of the power set of the continuum). I'd prefer more explicit constructions, if possible. This is just out of curiosity, as I was wondering what such a field would "look" like.

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up vote 8 down vote accepted

Let $X=\{x_\xi:\xi<\beth_2\}$ be a set of $\beth_2$ indeterminates. The ring $\Bbb Z[X]$ of finite polynomials over $X$ with integer coefficients is an integral domain of power $\beth_2$, and its field of quotients has power $\beth_2$.

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And this is about as explicit as it gets, since the transcendence degree of such a field over its prime field is $\beth_2$. – Andrés E. Caicedo Nov 2 '12 at 0:36

The set of all surreal numbers born before day $\beth_2$ is perhaps as "explicit" a construction as you'll get, since it's totally ordered and the surreals feel very "computational."

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Actually I'm not sure it's a field unless you adjoin them to the integers, in which case I guess we're in a situation similar to Brian's. – Daniel Briggs Nov 6 '12 at 2:47

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