= 27d567bc2d48d9f40e9ccc20ea370b15)
Eliezer_Yudkowsky comments on The Pascal's Wager Fallacy Fallacy - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (121)
You reference a popular idea, something like "The integers are countable, but the real number line is uncountable." I apologize for nitpicking, but I want to argue against philosophers (that's you, Eliezer) blindly repeating this claim, as if it was obvious or uncontroversial.
Yes, it is strictly correct according to current definitions. However, there was a time when people were striving to find the "correct" definition of the real number line. What people ended up with was not the only possibility, and Dedekind cuts (or various other things) are a pretty ugly, arbitrary construction.
The set containing EVERY number that you might, even in principle, name or pick out with a definition is countable (because the set of names, or definitions, is a subset of the set of strings, which is countable).
The Lowenheim-Skolem theorem says (loosely interpreted) that even if you CLAIM to be talking about uncountably infinite things, there's a perfectly self-consistent interpretation of your talk that refers only to finite things (e.g. your definitions and proofs themselves).
You don't get magical powers of infinity just from claiming to have them. Standard mathematical talk is REALLY WEIRD from a computer science perspective.
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.