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Warrigal comments on Less Wrong Q&A with Eliezer Yudkowsky: Ask Your Questions - Less Wrong
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Earlier today, I pondered whether this infinite set atheism thing is something Eliezer merely claims to believe as some sort of test of basic rationality. It's a belief that, as far as I can tell, makes no prediction.
But here's what I predict that I would say if I had Eliezer's opinions and my mathematical knowledge: I'm a fan of thinking of ZFC as being its countably infinite model, in which the class of all sets is enumerable, and every set has a finite representation. Of course, things like the axiom of infinity and Cantor's diagonal argument still apply; it's just that "uncountably infinite set" simply means "set whose bijection with the natural numbers is not contained in the model".
(Yes, ZFC has a countable model, assuming it's consistent. I would call this weird, but I hate admitting that any aspect of math is counterintuitive.)
ZFC's countable model isn't that weird.
Imagine a computer programmer, watching a mathematician working at a blackboard. Imagine asking the computer programmer how many bytes it would take to represent the entities that the mathematician is manipulating, in a form that can support those manipulations.
The computer programmer will do a back of the envelope calculation, something like: "The set of all natural numbers" is 30 characters, and essentially all of the special symbols are already in Unicode and/or TeX, so probably hundreds, maybe thousands of bytes per blackboard, depending. That is, the computer programmer will answer "syntactically".
Of course, the mathematician might claim that the "entities" that they're manipulating are more than just the syntax, and are actually much bigger. That is, they might answer "semantically". Mathematicians are trained to see past the syntax to various mental images. They are trained to answer questions like "how big is it?" in terms of those mental images. A math professor asking "How big is it?" might accept answers like "it's a subset of the integers" or "It's a superset of the power set of reals". The programmer's answer of "maybe 30 bytes" seems, from the mathematical perspective, about as ridiculous as "It's about three feet long right now, but I can write it longer if you want".
The weirdly small models are only weirdly small if what you thought were manipulating was something other than finite (and therefore Godel-numberable) syntax.
Models are semantics. The whole point of models is to give semantic meaning to syntactic strings.
I haven't studied the proof of the Löwenheim–Skolem theorem, but I would be surprised if it were as trivial as the observation that there are only countably many sentences in ZFC. It's not at all clear to me that you can convert the language in which ZFC is expressed into a model for ZFC in a way that would establish the Löwenheim–Skolem theorem.
I have studied the proof of the (downward) Lowenheim-Skolem theorem - as an undergraduate, so you should take my conclusions with some salt - but my understanding of the (downward) Lowenheim-Skolem theorem was exactly that the proof builds a model out of the syntax of the first-order theory in question.
I'm not saying that the proof is trivial - what I'm saying is that holding Godel-numberability and the possibility of a strict formalist interpretation of mathematics in your mind provides a helpful intuition for the result.
I've said this before in many places, but I simply don't do that sort of thing. If I want to say something flawed just to see how my readers react to it, I put it into the mouth of a character in a fictional story; I don't say it in my own voice.
I swear I meant to say that, knowing you, you probably wouldn't do such a thing.