Eliezer_Yudkowsky comments on The Pascal's Wager Fallacy Fallacy - Less Wrong
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Only in first-order logic. In second-order logic, you can actually talk about the natural numbers as distinguished from any other collection, and the uncountable reals.
Amusingly, if you insist that we are only allowed to talk in first-order logic, it is impossible for you to talk about the property "finite", since there is no first-order formula which expresses this property. (Follows from the Compactness Theorem for first-order logic - any set of first-order formulae which are true of unboundedly large finite collections also have models of arbitrarily large infinite cardinality.) Without second-order logic there is no way to talk about this property of "finiteness", or for that matter "countability", which you seem to think is so important.
Yes, that's my understanding as well.
Proof theory for second-order logic seems to be problematic, and I have a formalist stance towards mathematics in general, which leads me to suspect that the standard definitions of second-order logic are somehow smuggling in uncountable infinities, rather than justifying them.
But I admit second-order logic is not something I've studied in depth.
Yeah, second-order logic is basically set theory in disguise. I'm not sure why Eliezer likes it. Example from the Wikipedia page:
You can capture the property "finite" with a first-order sentence over the "standard integers", I think. This leaves open the mystery of what exactly the "standard integers" are, which looks lightly less mysterious than the mystery of "sets" required for second-order logic.
An equivalent (and in my opinion less misleading) way of putting this is to say that there's no first-order formula which expresses the property of being infinite.