Sniffnoy comments on The Pascal's Wager Fallacy Fallacy - Less Wrong

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Comment author: [deleted] 27 April 2011 07:43:40PM 3 points [-]

My contrarian claim is that everyone could live with the nameable numbers

I don't understand. Those also seem to fall prey to

it turns out that you can't distinguish zero from nonzero, so the step function is actually uncomputable.

Also,

People who really strongly care about the uncountability of the reals have a hard time coming up with a concrete example of what they'd miss.

Lebesgue measure theory, Gal(C/R) = Z/2Z, and some pathological examples in the history of differential geometry without which the current definition of a manifold would have been much more difficult to ascertain.

Off the top of my head. There are certainly other things I would miss.

Comment author: Sniffnoy 28 April 2011 09:50:43PM 1 point [-]

Gal(C/R) = Z/2Z

I'm confused; this is true for any real closed field. What are you getting at with this?

Comment author: [deleted] 28 April 2011 10:05:25PM *  0 points [-]

A mistake. I was thinking of C as the so-called "generic complex numbers." You're right that if you replace C with the algebraic closure of whatever countable model's been dreamed up, then C = R[i] and that's it.

Admittedly I'm only conjecturing that Gal(C/K) will be different for some K countable, but I think there's good evidence in favor of it. After all, if K is the algebraic closure of Q, then Gal(C/K) is gigantic. It doesn't seem likely that one could "fix" the other "degrees of freedom" with only countably many irrationals.

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