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Psy-Kosh comments on Less Wrong Q&A with Eliezer Yudkowsky: Ask Your Questions - Less Wrong
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Could you (Well, "you" being Eliezer in this case, rather than the OP) elaborate a bit on your "infinite set atheism"? How do you feel about the set of natural numbers? What about its power set? What about that thing's power set, etc?
From the other direction, why aren't you an ultrafinitist?
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.
<Wince>. Uh-oh; if he takes this up, I may finally have to write that post I promised back in June!
Well obviously if a set is finite, no amount of taking its power set is going to change that fact.
I meant "the set of all natural numbers", IIRC, he's explicitly said he's not an ultrafinitist, so either he considers that as an acceptable infinite set, or he considers the natural numbers to exist, but not the set of them, or something.
I meant "if you accept countable infinities, where and how do you consider the whole cantor hierarchy to break down?"
What would it even mean for the natural numbers (the entire infinity of them) to "exist"?
What makes a set acceptable or not?
This question sounds weird to me.
I find it best not to speak about "existence", but speak instead of logical models that work. For example, we don't know if our concept of integers is consistent, but we have evolved a set of tools for reasoning about it that have been quite useful so far. Now we try to add new reasoning tools, new concepts, without breaking the system. For example, if we imagine "the set of all sets" and apply some common reasoning to it, we reach Russell's paradox; but we can't feed this paradox back into the integers to demonstrate their inconsistency, so we just throw the problematic concept away with no harm done.
It sounds weird to me too, which is why I asked it - because Psy-Kosh said EY said something about integers, or the set of integers, existing or not.
secondary sources? bah!
LMGTFY (or the full experience)