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dankane comments on Logical Pinpointing - Less Wrong
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Your idea of pinning down the natural numbers using second order logic is interesting, but I don't think that it really solves the problem. In particular, it shouldn't be enough to convince a formalist that the two of you are talking about the same natural numbers.
Even in second order PA, there will still be statements that are independent of the axioms, like "there doesn't exist a number corresponding to a Godel encoding of a proof that 0=S0 under the axioms of second order PA". Thus unless you are assuming full semantics (i.e. that for any collection of numbers there is a corresponding property), there should be distinct models of second order PA for which the veracity of the above statement differs.
Thus it seems to me that all you have done with your appeal to second order logic is to change my questions about "what is a number?" into questions about "what is a property?" In any case, I'm still not totally convinced that it is possible to pin down The Natural Numbers exactly.
I'm assuming full semantics for second-order logic (for any collection of numbers there is a corresponding property being quantified over) so the axioms have a semantic model provably unique up to isomorphism, there are no nonstandard models, the Completeness Theorem does not hold and some truths (like Godel's G) are semantically entailed without being syntactically entailed, etc.
OK then. As soon as you can explain to me exactly what you mean when you say "for any collection of numbers there is a corresponding property being quantified over", I will be satisfied. In particular, what do you mean when you say "any collection"?
Are you claiming that this term is ambiguous? In what specially favored set theory, in what specially favored collection of allowed models, is it ambiguous? Maybe the model of set theory I use has only one set of allowable 'collections of numbers' in which case the term isn't ambiguous. Now you could claim that other possible models exist, I'd just like to know in what mathematical language you're claiming these other models exist. How do you assert the ambiguity of second-order logic without using second-order logic to frame the surrounding set theory in which it is ambiguous?
I'm not entirely sure what you're getting at here. If we start restricting properties to only cut out sets of numbers rather than arbitrary collections, then we've already given up on full semantics.
If we take this leap, then it is a theorem of set theory that all set-theoretic models of the of the natural numbers are isomorphic. On the other hand, since not all statements about the integers can be either proven or disproven with the axioms of set theory, there must be different models of set theory which have different models of the integers within them (in fact, I can build these two models within a larger set theory).
On the other hand, if we continue to use full semantics, I'm not sure how you clarify to be what you mean when you say "a property exists for every collection of numbers". Telling me that I should already know what a collection is doesn't seem much more reasonable than telling me that I should already know what a natural number is.
Doesn't the proof of the Completeness Theorem / Compactness Theorem incidentally invoke second-order logic itself? (In the very quiet way that e.g. any assumption that the standard integers even exist invokes second-order logic.) I'm not sure but I would expect it to, since otherwise the notion of a "consistent" theory is entirely dependent on which models your set theory says exist and which proofs your integer theory says exist. Perhaps my favorite model of set theory has only one model of set theory, so I think that only one model exists. Can you prove to me that there are other models without invoking second-order logic implicitly or explicitly in any called-on lemma? Keep in mind that all mathematicians speak second-order logic as English, so checking that all proofs are first-order doesn't seem easy.
I am admittedly in a little out of my depth here, so the following could reasonably be wrong, but I believe that the Compactness Theorem can be proved within first order set theory. Given a consistent theory, I can use the axiom of choice to extend it to a maximal consistent set of statements (i.e. so that for every P either P or (not P) is in my set). Then for every statement that I have of the form "there exists x such that P(x)", I introduce an element x to my model and add P(x) to my list of true statements. I then re-extend to a maximal set of statements, and add new variables as necessary, until I cannot do this any longer. What I am left with is a model for my theory. I don't think I invoked second order logic anywhere here. In particular, what I did amounts to a construction within set theory. I suppose it is the case that some set theories will have no models of set theory (because they prove that set theory is inconsistent), while others will contain infinitely many.
My intuition on the matter is that if you can state what you are trying to say without second order logic, you should be able to prove it without second order logic. You need second order logic to even make sense of the idea of the standard natural numbers. The Compactness Theorem can be stated in first order set theory, so I expect the proof to be formalizable within first order set theory.
If you're already fine with the alternating quantifiers of first-order logic, I don't see why allowing branching quantifiers would cause a problem. I could describe second order logic in terms of branching quantifiers.
Huh. That's interesting. Are you saying that you can actually pin down The Natural Numbers exactly using some "first order logic with branching quantifiers"? If so, I would be interested in seeing it.
Sure:
It is not the case that: there exists a z such that for every x and x’, there exists a y depending only on x and a y’ depending only on x’ such that Q(x,x’,y,y’,z) is true
where Q(x,x’,y,y’,z) is ((x=x' ) → (y=y' )) ∧ ((Sx=x' ) → (y=y' )) ∧ ((x=0) → (y=0)) ∧ ((x=z) → (y=1))
Cool. I agree that this is potentially less problematic than the second order logic approach. But it does still manage to encode the idea of a function in it implicitly when it talks about "y depending only on x", it essentially requires that y is a function of x, and if it's unclear exactly which functions are allowed, you will have problems. I guess first order logic has this problem to some degree, but with alternating quantifiers, the functions that you might need to define seem closer to the type that should necessarily exist.
I think this is his way of connecting numbers to the previous posts. If "a property" is defined as a causal relation, which all properties are, then I think this makes sense. It doesn't provide some sort of ultimate metaphysical justification for numbers or properties or anything, but it clarifies connections between the two and such a justification isn't really possible anyways.
I don't think that I understand what you mean here.
How can these properties represent causal relations? They are things that are satisfied by some numbers and not by others. Since numbers are aphysical, how do we relate this to causal relations.
On the other hand, even with a satisfactory answer to the above question, how do we know that "being in the first chain" is actually a property, since otherwise we still can't show that there is only one chain.
You just begged the question. Eliezer answered you in the OP:
I can't think of an example, but I'm thinking that if a property existed then it would be a causal relation. A property wouldn't represent a causal relation, it would be one. I wasn't thinking mathematically but instead in terms of a more commonplace understanding of properties as things like red and yellow and blue.
The argument made by the simple idea of truth might be a way to get us from physical states (which are causal relations) to numbers. If you believe that counting sheep is a valid operation, then quantifying color also seems fine. The reason I spoke in terms of causal relations is because I believe understanding qualities as causal relations between things allows us to deduce properties about things through a combination of Salmonoff Induction and the method described in this post.
Are you questioning the idea that numbers or properties are a quality about objects? If so, what are they?
I'm feeling confused though. If the definition of property used here doesn't connect to or means something completely different than facts about objects, then I'm way off base. I might also be off base for other reasons. Not sure.
I am questioning the idea that numbers (at least the things that this post refers to as numbers) are a quality about objects. Numbers, as they are described here, are an abstract logical construction.