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A quick sketch on how the Curry-Howard Isomorphism kinda appears to connect Algorithmic Information Theory with ordinal logics

I'd like to ask a not-too-closely related question, if you don't mind.

A Curry-Howard analogue of Gödel's Second Incompleteness Theorem is the statement "no total language can embed its own interpreter"; see the classic post by Conor McBride. But Conor also says (and it's quite obviously true), that total languages can still embed their coinductive interpreters, for example one that returns in the partiality monad.

So, my question is: what is the logical interpretation of a coinductive self-interpreter? I feel I'm not well-versed enough in mathematical logic for this. I imagine it would have a type like "forall (A : 'Type) -> 'Term A -> Partiality (Interp A)", where 'Type and 'Term denote types and terms in the object language and "Interp" returns a type in the meta-language. But what does "Partiality" mean logically, and is it anyhow related to the Löbstacle?

An Introduction to Löb's Theorem in MIRI Research

You're right, I mixed it up. Edited the comment.

An Introduction to Löb's Theorem in MIRI Research

Suppose "mathemathics would never prove a contradiction". We can write it out as ¬Provable(⊥). This is logically equivalent to Provable(⊥) → ⊥, and it also implies Provable(Provable(⊥) → ⊥) by the rules of provability. But Löb's theorem expresses ∀ A (Provable(Provable(A) → A) → Provable(A)), which we can instantiate to Provable(Provable(⊥) → ⊥)→ Provable(⊥), and now we can apply modus ponens and our assumption to get a Provable(⊥).

Preferences without Existence

To clarify my point, I meant that Solomonoff induction can justify caring less about some agents (and I'm largely aware of the scheme you described), but simultaneously rejecting Solomonoff and caring less about agents running on more complex physics is not justified.

L-zombies! (L-zombies?)

The unsimulated life is not worth living.

-- Socrates.

Preferences without Existence

You conflate two very different things here, as I see.

First, there are the preferences for simpler physical laws or simpler mathematical constructions. I don't doubt that they are real amongst humans; after all, there is an evolutionary advantage to using simpler models ceteris paribus since they are easier to memorize and easier to reason about. Such evolved preferences probably contribute to a matemathician's sense of elegance.

Second, there are preferences about the concrete evolutionarily relevant environment and the relevant agents in it. Naturally, this includes our fellow humans. Note here that we might also care about animals, uploads, AIs or aliens because of our evolved preferences and intuitions regarding humans. Of course, we don't care about aliens because of a direct evolutionary reason. Rather, we simply execute the adaptations that underlie our intuitions. For instance, we might disprefer animal suffering because it is similar enough to human suffering.

This second level has very little to do with the complexity of the underlying physics. Monkeys have no conception of cellular automata; you could run them on cellular automata of vastly differing complexity and they wouldn't care. They care about the kind of simplicity that is relevant to their day-to-day environment. Humans also care about this kind of simplicity; it's just that they can generalize this preference to more abstract domains.

(On a somewhat unrelated note, you mentioned bacteria. I think your point is a red herring; you can build agents with an assumption for the underlying physics, but that doesn't mean that the agent itself necessarily has any conception of the underlying physics, or even that the agent is consequentialist in any sense).

So, what I'm trying to get at: you might prefer simple physics and you might care about people, but it makes little sense to care less about people because they run on uglier physics. People are not physics; they are really high-level constructs, and a vast range of different universes could contain (more or less identical) instances of people whom I care about, or even simulations of those people.

If I assume Solomonoff induction, then it is in a way reasonable to care less about people running on convoluted physics, because then I would have to assign less "measure" to them. But you rejected this kind of reasoning in your post, and I can't exactly come to grips with the "physics racism" that seems to logically follow from that.


Donated $500.

March 2013 Media Thread

Very recently decent fansubs have surfaced for Mamoru Hosoda's new movie The Wolf Children Ame and Yuki. I wholeheartedly recommend it.

Participation in the LW Community Associated with Less Bias

I figured that ~36% discount over two months would be way too high.

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