I am a proof theorist (or rather, I do research in programming languages, which has a very hazy boundary with proof theory). When you formalize induction in type theory, we really do formalize it the way you call omega-completeness. It’s very nearly a forced choice if you want to be faithful Martin-Loef’s judgmental method. (Roughly, you have to be able to give a logical connective introduction and elimination rules without reference to the other connectives, plus other stuff irrelevant to a blog comment.)

I can’t say a whole lot more, because I’m embarassingly ignorant of mathematical logic—I come at the subject from an odd angle.

Adding it makes the logic incomplete, end of story.

Well, no. Assuming standard first-order logic* and a basic description of the natural numbers (e.g. Robinson’s Q), it was already incomplete—you don’t need induction anywhere in Godel’s proof. In particular, you don’t need to a) add induction as a mathematical axiom to your system (hence Q not PA), nor b) apply induction metamathematically anywhere in the proof, as neither the Diagonalization Lemma nor the fragment of the Representation Lemma required to represent the relevant predicates necessitate induction.** Adding induction doesn’t “recomplete” it nor does it make it “incompleter”, of course—although I think the Turing degree of (the theory of) Q is strictly below the Turing degree of (the theory of) PA —but it doesn’t produce the incompleteness phenomenon you seem to be ascribing it.

• Things get more complicated in alternative logics simply because there are too many, well, alternatives, but the same gist applies to most of the major ones I know (2nd order, L_{kappa lambda}, etc.).

** This is a subtle but hugely important point: you only need Representation up to Sigma-n for some n large enough to diagonalize “not provable”. In fact, I think you can even get away with just diagonalizing “not provable”, or maybe its subformula closure.

This property holds for ordinary first-order logic (which is why it is semi-decidable),

I’m confused. Ordinary first-order logic is semi-decidable/c.e. because its notion of “proof” is explicitly decidable and hence “provable” is automatically semi-decidable. [Of which I’m sure you’re well aware, but it should be said nonetheless.] This holds for any decidable notion of “proof”—provided you don’t do something wacky like using K to encode formulas or whatever—irrespective of whether it possesses the subformula property or not.***

That said, I think I’ve found the problem: I think what you’re calling induction is more commonly called “omega-completeness” in the mathematical literature, the property that if T is a theory (extending Q or somesuch) which proves A(n) for each standard n then T proves (x) A(x). [I’m guessing here; you seem to be a proof-theorist whereas I’m a computability/set-theory guy.] This is an inherently infinitistic proof principle, absolutely, but it’s not induction, which applies equally (by definition) in standard and non-standard models of arithmetic. If that’s the principle that you’re invoking, fair enough—but it’s not required to get incompleteness, it’s almost never used in mathematical logic, and the problem isn’t “unbounded search” per se, it’s that it’s not even formalizable finistically except in a much stronger metatheory (e.g. ZFC or L_{omega_1 omega} or what have you) than one usually invokes in this context—because you need to diagonalize over predicates, as you noted above—and that formalization itself carries with it incompleteness and the standard/non-standard dilemma, just at a higher level.

*** I’d normally throw in the disclaimer that a “proof” ought to be finite too but I think, by Rice’s Theorem, that being decidable suffices.

The unbounded search I’m talking about doesn’t have anything to do with existentials. It arises from the failure of the subformula property, which is the property that any derivation of a provable proposition in a sequent calculus involves only subformulas of the goal proposition. This property holds for ordinary first-order logic (which is why it is semi-decidable), but it does not hold for calculi with induction.

This is because in order to use the principle of induction, you have to choose the induction hypothesis to apply it to (because it is schematic over predicates). Unfortunately, the hypothesis you need to make up is not necesarily a subformula of the goal; you may have to come up with a more general induction hypothesis. This creates the infinite search problem—your search space of hypotheses is all formulas, which is an infinite set.

This isn’t correct, fwiw. Robinson’s Q, which is basically Peano Arithmetic without induction, is already undecidable; and in fact Kripke showed, by a fairly simple extension of the proof of the Godel Incompleteness Theorem, that any theory encoding the quantifier-free truths of arithmetic is undecidable too. [In brief: the Godel statement G is Sigma-1, so if it were falsified you could actually pick an explicit witness to that effect.] You might be thinking of existential quantifers here—the unbounded search I think you’re talking about—in which case it depends on the base theory: unbounded search over, say, the axioms of a dense linear order is perfectly acceptable (DLO is complete, and hence decidable, up to the specification of endpoints), while an unbounded search over the usual axioms of arithmetic produces undecidability in the presence of negation.

Could you do something interesting there with counting symmetries (in the physicist’s sense of, roughly speaking, something that could demonstrably be more complex but isn’t)?

For example, the rules of chess are symmetrical between different chess games, between different moves in the same game (mostly, the ‘can castle only once’ rule is an exception), between different pieces of the same type, between different squares on the board, and between the two sides. None of those symmetries has any meaningful physical cause, none can be plausibly shown to be a consequnce of underlying phsyics symmetries.

That’s why it can be said that the rules of chess are a good explanation for a chess game.

Brian, you suggest we abstract human psychology away and see if we can find any authentic ontological features our macro-level causal concepts actually track. I’d say that perhaps all we could hope to find are the underlying microphysical phenomena as our idealized Martian physics (“MP”) would predict they would be experienced by beings like us. That is, if MP really does account completely for the way in which the microphysical processes and structure of reality play out dynamically in the generation of human-like consciousness (otherwise, I take it, it wouldn’t be a “complete” physics), then by hypothesis there would be a reliable translation between microphysical descriptions and predictable macrophysical percepts.

Obviously, this account of what’s been called in the discussion “gruesome predicates” isn’t purely psychology-free (for how could anything that required an assay of “understanding” in the explanans be purely psychology-free?), but the residue of psychology doesn’t appeal to the actual existence of percipient beings like us; it only requires that the underlying physics predicts what beings like us would perceive.

Now it may be that we find it impossible to imagine how such a physics could provide beings like our hypothetical Martians with an “understanding” of our own appeals to macro-level causal abstractions like “demand curves” or “society” or “Tuesday.” But this (as I suggested in an earlier comment) is entirely due to the construction of the hypothesis. There’s no reason to think their mode of understanding should be any more comprehensible to us than ours is supposed (by hypothesis) to be to them.

Of course, none of this is to say explanatory reductionism is true, but only that on the face of it MP seems to pose no challenge to it.

Again you are correct, however, I am not claiming that the real line is the canonical model for science, nor am I claiming that “jumps” are somehow inherintly computationally problematic. In my haste, I was simply being to imprecise about the sense of discontinuity I was using.
The universe may indeed be discrete, as the digital physics folks think. As Richard Feynman wrote in The Character of Physical Law:

It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do? So I have often made the hypotheses that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities.

My problem is another one: Why should the real line be a good model of science? If we are living in a discrete world, jumps wouldn’t be a problem for computability.

The detection of neutrinos requires considerable amounts of equipment, and for some energies, substantial amounts of computing power as well. No one really denies the neutrino’s existence, however.

The detection of (say), the mood of a friend requires a human being as a detector.

Why is a large particle detector and all the attendant foldorol considered to be philosophically different (we can agree that there are qualitative and quantitative differences; there always are), than a human “detector?”

If the Martians are that damn smart, they can learn the language and ask somebody.

Also, something is grue if it is green, but turns blue at some time in the future, i.e., something that cannot be observed currently, but can eventually be observed. The statement in the original post misses that distinction.

Wether they are homeomorphic depends on the topology chosen, and their is neither a natural predicate space nor a natural topology for it.

He present it this way:

Goodman’s point was that syntactic features invoked in accounts of relational support (e.g., “uniformity” of the input stream) are not preserved under translation, and hence cannot be objective, language-invariant features of the empirical problem itself. The solvability (and corresponding underdetermination) properties of the preceding problem persist no matter how one renames the inputs along the input streams (e.g., the feather [Baire space really] has the same branching structure whether the labels along the input streams are presented in the blue/green vocabulary or in the grue/bleen vocabulary). Both are immune, therefore, to Goodman-style arguments, as are all methodological recommendations based upon them. (from "The Logic of Success")

A dicontinuity of a real function with the usual topology, that is. I don’t see the relevance of the problem here.

Indeed, discontinuity in a real function presents is uncomputable. This becomes relevant to the current discussion if reduction is understood in the following Nagelian way: T reduces T′ just in case the laws of T′ are derivable from those of T. Derivability, then is taken as ‘computably decided from’. That is, given microphysical facts, there exists a computable function mapping these facts to the psychological, or other special sciences. What is derivable, is of course indexed to the capacities of the agent. In this case I suppose humans are Turing equivalent, though we can modify this assumption up or down (FSM or analog computer, say), and uncomputability/underivability will reassert itself. This, I think presents an intrinsic problem for reduction relative to the capacities of an agent, whereas gruesomeness does not. Gruesomness does, however, present a legitimate coding problem that can be treated information theoretically, but that is another very long story.
You can find some relevant philosophical papers on my abortive attempt at a formal epistemology blog. It’s ugly, but it has some classics.

If I understand Brad, he’s arguing that Martian axioms will consist of the rules of logic (which are the same for everyone) and basic physical law (which are also the same for everyone), and hence the Martians can deduce anything a human can.

This is, unfortunately, not true, even though I agree with him that Martians will use the same (up to some translation) logic as us, and will have the same laws of physics. The trouble is that the Martians will understand the principle of proof by induction, and when you add that principle to logic every proof search procedure become incomplete. This is because proof by induction can require inventing an induction hypothesis, and this is fundamentally a creative act. That is, there is no mechanical procedure for reliably cooking up the right induction hypothesis. (You can demonstrate this by showing that the subformula property fails to hold in a logic with induction over the natural numbers.)

So as a result, it’s entirely possible that we can come up with mathematical formalizations that the Martians would never have been able to, and vice-versa, even though they and we agree on what mathematics and physics are. We would be able to verify that a given Martian argument is correct, and likewise they would be able to verify that a given human argument is correct, but it’s entirely possible that the standard explanatory schemas we use are ones that Martians will never naturally come up with, and vice-versa.

Two robots play a game of chess inside a box. Everything inside the box is physically understood by the Martians – electrical signals, motors, chemical batteries, and so on.

Except they never happen to spot the rules of the game being played.