## Very Basic Model Theory

In this post I'll discuss some basic results of model theory. It may be helpful to read through my previous post if you haven't yet. Model Theory is an implicit context for the Heavily Advanced Epistemology sequence and for a few of the recent MIRI papers, so casual readers may find this brief introduction useful. And who knows, maybe it will pique your interest:

## A tale of two logics

**propositional logic** is the "easy logic", built from basic symbols and the connectives "and" and "not". Remember that all other connectives can be built from these two: With Enough NAND Gates You Can Rule The World and all that. Propositional logic is sometimes called the "sentential logic", because it's not like any other logics are "of or relating to sentences" (/sarcasm).

**first order logic** is the "nice logic". It has quantifiers ("there exists", "for all") and an internal notion of equality. Its sentences contain constants, functions, and relations. This lets you say lots of cool stuff that you can't say in propositional logic. First order logic turns out to be quite friendly (as we'll see below). However, it's not strong enough to talk about certain crazy/contrived ideas that humans cook up (such as "the numbers").

There are many other logics available (second order logic AKA "the heavy guns", ω-logic AKA "please just please can I talk about numbers", and many more). In this post we'll focus on propositional and first-order logics.

## Mental Context for Model Theory

I'm reviewing the books on the MIRI course list. After my first four book reviews I took a week off, followed up on some dangling questions, and upkept other side projects. Then I dove into *Model Theory*, by Chang and Keisler.

It has been three weeks. I have gained a decent foundation in model theory (by my own assessment), but I have not come close to completing the textbook. There are a number of other topics I want to touch upon before December, so I'm putting *Model Theory* aside for now. I'll be revisiting it in either January or March to finish the job.

In the meantime, I do not have a complete book review for you. Instead, this is the first of three posts on my experience with model theory thus far.

This post will give you some framing and context for model theory. I had to hop a number of conceptual hurdles before model theory started making sense — this post will contain some pointers that I wish I'd had three weeks ago. These tips and realizations are somewhat general to learning any logic or math; hopefully some of you will find them useful.

Shortly, I'll post a summary of what I've learned so far. For the casual reader, this may help demystify some heavily advanced parts of the Heavily Advanced Epistemology sequence (if you find it mysterious), and it may shed some light on some of the recent MIRI papers. On a personal note, there's a lot I want to write down & solidify before moving on.

In follow-up post, I'll discuss my experience struggling to learn something difficult on my own — model theory has required significantly more cognitive effort than did the previous textbooks.

## A Voting Puzzle, Some Political Science, and a Nerd Failure Mode

In grade school, I read a series of books titled *Sideways Stories from Wayside School *by Louis Sachar, who you may know as the author of the novel *Holes* which was made into a movie in 2003. The series included two books of math problems, *Sideways Arithmetic from Wayside School *and *More Sideways Arithmetic from Wayside School, *the latter of which included the following problem (paraphrased):

The students have Mrs. Jewl's class have been given the privilege of voting on the height of the school's new flagpole. She has each of them write down what they think would be the best hight for the flagpole. The votes are distributed as follows:

- 1 student votes for 6 feet.
- 1 student votes for 10 feet.
- 7 students vote for 25 feet.
- 1 student votes for 30 feet.
- 2 students vote for 50 feet.
- 2 students vote for 60 feet.
- 1 student votes for 65 feet.
- 3 students vote for 75 feet.
- 1 student votes for 80 feet, 6 inches.
- 4 students vote for 85 feet.
- 1 student votes for 91 feet.
- 5 students vote for 100 feet.

At first, Mrs. Jewls declares 25 feet the winning answer, but one of the students who voted for 100 feet convinces her there should be a runoff between 25 feet and 100 feet. In the runoff, each student votes for the height closest to their original answer. But after that round of voting, one of the students who voted for 85 feet wants *their *turn, so 85 feet goes up against the winner of the *previous *round of voting, and the students vote the same way, with each student voting for the height closest to their original answer. Then the same thing happens again with the 50 foot option. And so on, with each number, again and again, "very much like a game of tether ball."

Question: if this process continues until it settles on an answer that can't be beaten by any other answer, how tall will the new flagpole be?

Answer (rot13'd): fvkgl-svir srrg, orpnhfr gung'f gur zrqvna inyhr bs gur bevtvany frg bs ibgrf. Naq abj lbh xabj gur fgbel bs zl svefg rapbhagre jvgu gur zrqvna ibgre gurberz.

Why am I telling you this? There's a minor reason and a major reason. The minor reason is that this shows it is possible to explain little-known academic concepts, at least certain ones, in a way that grade schoolers will understand. It's a data point that fits nicely with what Eliezer has written about how to explain things. The major reason, though, is that a month ago I finished my systematic read-through of the sequences and while I generally agree that they're awesome (perhaps moreso than most people; I didn't see the problem with the metaethics sequence), I thought the mini-discussion of political parties and voting was on reflection weak and indicative of a broader nerd failure mode.

TLDR (courtesy of lavalamp):

- Politicians probably conform to the median voter's views.
- Most voters are not the median, so most people usually dislike the winning politicians.
- But people dislike the politicians for different reasons.
- Nerds should avoid giving advice that boils down to "behave optimally". Instead, analyze the reasons for the current failure to behave optimally and give more targeted advice.

## Robust Cooperation in the Prisoner's Dilemma

I'm proud to announce the preprint of Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic, a joint paper with Mihaly Barasz, Paul Christiano, Benja Fallenstein, Marcello Herreshoff, Patrick LaVictoire (me), and Eliezer Yudkowsky.

This paper was one of three projects to come out of the 2nd MIRI Workshop on Probability and Reflection in April 2013, and had its genesis in ideas about formalizations of decision theory that have appeared on LessWrong. (At the end of this post, I'll include links for further reading.)

Below, I'll briefly outline the problem we considered, the results we proved, and the (many) open questions that remain. Thanks in advance for your thoughts and suggestions!

## Background: Writing programs to play the PD with source code swap

(If you're not familiar with the Prisoner's Dilemma, see here.)

The paper concerns the following setup, which has come up in academic research on game theory: say that you have the chance to write a computer program **X**, which takes in one input and returns either *Cooperate* or *Defect*. This program will face off against some other computer program **Y**, but with a twist: **X** will receive the source code of **Y** as input, and **Y** will receive the source code of **X** as input. And you will be given your program's winnings, so you should think carefully about what sort of program you'd write!

Of course, you could simply write a program that defects regardless of its input; we call this program **DefectBot**, and call the program that cooperates on all inputs **CooperateBot**. But with the wealth of information afforded by the setup, you might wonder if there's some program that might be able to achieve mutual cooperation in situations where **DefectBot** achieves mutual defection, without thereby risking a sucker's payoff. (Douglas Hofstadter would call this a perfect opportunity for superrationality...)

## Previously known: CliqueBot and FairBot

And indeed, there's a way to do this that's been known since at least the 1980s. You can write a computer program that knows its own source code, compares it to the input, and returns *C* if and only if the two are identical (and *D* otherwise). Thus it achieves mutual cooperation in one important case where it intuitively ought to: when playing against itself! We call this program **CliqueBot**, since it cooperates only with the "clique" of agents identical to itself.

There's one particularly irksome issue with **CliqueBot**, and that's the fragility of its cooperation. If two people write functionally analogous but syntactically different versions of it, those programs will defect against one another! This problem can be patched somewhat, but not fully fixed. Moreover, mutual cooperation might be the best strategy against some agents that are not even functionally identical, and extending this approach requires you to explicitly delineate the list of programs that you're willing to cooperate with. Is there a more flexible and robust kind of program you could write instead?

As it turns out, there is: in a 2010 post on LessWrong, cousin_it introduced an algorithm that we now call **FairBot**. Given the source code of **Y**, **FairBot** searches for a proof (of less than some large fixed length) that **Y** returns *C* when given the source code of **FairBot**, and then returns *C* if and only if it discovers such a proof (otherwise it returns *D*). Clearly, if our proof system is consistent, **FairBot** only cooperates when that cooperation will be mutual. But the really fascinating thing is what happens when you play two versions of **FairBot** against each other. Intuitively, it seems that *either* mutual cooperation or mutual defection would be stable outcomes, but it turns out that if their limits on proof lengths are sufficiently high, they will achieve mutual cooperation!

The proof that they mutually cooperate follows from a bounded version of Löb's Theorem from mathematical logic. (If you're not familiar with this result, you might enjoy Eliezer's Cartoon Guide to Löb's Theorem, which is a correct formal proof written in much more intuitive notation.) Essentially, the asymmetry comes from the fact that both programs are searching for the same outcome, so that a short proof that one of them cooperates leads to a short proof that the other cooperates, and vice versa. (The opposite is not true, because the formal system can't know it won't find a contradiction. This is a subtle but essential feature of mathematical logic!)

## Generalization: Modal Agents

Unfortunately, **FairBot** isn't what I'd consider an ideal program to write: it happily cooperates with **CooperateBot**, when it could do better by defecting. This is problematic because in real life, the world isn't separated into agents and non-agents, and any natural phenomenon that doesn't predict your actions can be thought of as a **CooperateBot** (or a **DefectBot**). You don't want your agent to be making concessions to rocks that happened not to fall on them. (There's an important caveat: some things have utility functions that you care about, but don't have sufficient ability to predicate their actions on yours. In that case, though, it wouldn't be a true Prisoner's Dilemma if your values actually prefer the outcome (*C*,*C*) to (*D*,*C*).)

However, **FairBot** belongs to a promising class of algorithms: those that decide on their action by looking for short proofs of logical statements that concern their opponent's actions. In fact, there's a really convenient mathematical structure that's analogous to the class of such algorithms: the modal logic of provability (known as GL, for Gödel-Löb).

So that's the subject of this preprint: **what can we achieve in decision theory by considering agents defined by formulas of provability logic?**

## Tiling Agents for Self-Modifying AI (OPFAI #2)

An early draft of publication #2 in the Open Problems in Friendly AI series is now available: Tiling Agents for Self-Modifying AI, and the Lobian Obstacle. ~20,000 words, aimed at mathematicians or the highly mathematically literate. The research reported on was conducted by Yudkowsky and Herreshoff, substantially refined at the November 2012 MIRI Workshop with Mihaly Barasz and Paul Christiano, and refined further at the April 2013 MIRI Workshop.

**Abstract:**

We model self-modication in AI by introducing 'tiling' agents whose decision systems will approve the construction of highly similar agents, creating a repeating pattern (including similarity of the offspring's goals). Constructing a formalism in the most straightforward way produces a Godelian difficulty, the Lobian obstacle. By technical methods we demonstrate the possibility of avoiding this obstacle, but the underlying puzzles of rational coherence are thus only partially addressed. We extend the formalism to partially unknown deterministic environments, and show a very crude extension to probabilistic environments and expected utility; but the problem of finding a fundamental decision criterion for self-modifying probabilistic agents remains open.

Commenting here is the preferred venue for discussion of the paper. This is an early draft and has not been reviewed, so it may contain mathematical errors, and reporting of these will be much appreciated.

The overall agenda of the paper is introduce the conceptual notion of a self-reproducing decision pattern which includes reproduction of the goal or utility function, by exposing a particular possible problem with a tiling logical decision pattern and coming up with some partial technical solutions. This then makes it conceptually much clearer to point out the even deeper problems with "We can't yet describe a probabilistic way to do this because of non-monotonicity" and "We don't have a good bounded way to do this because maximization is impossible, satisficing is too weak and Schmidhuber's swapping criterion is underspecified." The paper uses first-order logic (FOL) because FOL has a lot of useful standard machinery for reflection which we can then invoke; in real life, FOL is of course a poor representational fit to most real-world environments outside a human-constructed computer chip with thermodynamically expensive crisp variable states.

As further background, the idea that something-like-proof might be relevant to Friendly AI is not about achieving some chimera of absolute safety-feeling, but rather about the idea that the total probability of catastrophic failure should not have a significant conditionally independent component on each self-modification, and that self-modification will (at least in initial stages) take place within the highly deterministic environment of a computer chip. This means that statistical testing methods (e.g. an evolutionary algorithm's evaluation of average fitness on a set of test problems) are not suitable for self-modifications which can potentially induce catastrophic failure (e.g. of parts of code that can affect the representation or interpretation of the goals). Mathematical proofs have the property that they are as strong as their axioms and have no significant conditionally independent per-step failure probability if their axioms are semantically true, which suggests that something like mathematical reasoning may be appropriate for certain particular types of self-modification during some developmental stages.

Thus the content of the paper is very far off from how a realistic AI would work, but conversely, if you can't even answer the kinds of simple problems posed within the paper (both those we partially solve and those we only pose) then you must be very far off from being able to build a stable self-modifying AI. Being able to say how to build a theoretical device that would play perfect chess given infinite computing power, is very far off from the ability to build Deep Blue. However, if you can't even say how to play perfect chess given infinite computing power, you are confused about the rules of the chess or the structure of chess-playing computation in a way that would make it entirely hopeless for you to figure out how to build a bounded chess-player. Thus "In real life we're always bounded" is no excuse for not being able to solve the much simpler unbounded form of the problem, and being able to describe the infinite chess-player would be substantial and useful conceptual progress compared to *not *being able to do that. We can't be absolutely certain that an analogous situation holds between solving the challenges posed in the paper, and realistic self-modifying AIs with stable goal systems, but every line of investigation has to start somewhere.

Parts of the paper will be easier to understand if you've read Highly Advanced Epistemology 101 For Beginners including the parts on correspondence theories of truth (relevant to section 6) and model-theoretic semantics of logic (relevant to 3, 4, and 6), and there are footnotes intended to make the paper somewhat more accessible than usual, but the paper is still essentially aimed at mathematically sophisticated readers.

## Reflection in Probabilistic Logic

Paul Christiano has devised **a new fundamental approach** to the "Löb Problem" wherein Löb's Theorem seems to pose an obstacle to AIs building successor AIs, or adopting successor versions of their own code, that trust the same amount of mathematics as the original. (I am currently writing up a more thorough description of the *question *this preliminary technical report is working on answering. For now the main online description is in a quick Summit talk I gave. See also Benja Fallenstein's description of the problem in the course of presenting a different angle of attack. Roughly the problem is that mathematical systems can only prove the soundness of, aka 'trust', weaker mathematical systems. If you try to write out an exact description of how AIs would build their successors or successor versions of their code in the most obvious way, it looks like the mathematical strength of the proof system would tend to be stepped down each time, which is undesirable.)

Paul Christiano's approach is inspired by the idea that whereof one cannot prove or disprove, thereof one must assign probabilities: and that although no mathematical system can contain its own *truth* predicate, a mathematical system might be able to contain a reflectively consistent *probability* predicate. In particular, it looks like we can have:

∀a, b: (a < P(φ) < b) ⇒ P(a < P('φ') < b) = 1

∀a, b: P(a ≤ P('φ') ≤ b) > 0 ⇒ a ≤ P(φ) ≤ b

Suppose I present you with the human and probabilistic version of a Gödel sentence, the Whitely sentence "You assign this statement a probability less than 30%." If you disbelieve this statement, it is true. If you believe it, it is false. If you assign 30% probability to it, it is false. If you assign 29% probability to it, it is true.

Paul's approach resolves this problem by restricting your belief about your own probability assignment to within epsilon of 30% for any epsilon. So Paul's approach replies, "Well, I assign *almost* exactly 30% probability to that statement - maybe a little more, maybe a little less - in fact I think there's about a 30% chance that I'm a tiny bit under 0.3 probability and a 70% chance that I'm a tiny bit over 0.3 probability." A standard fixed-point theorem then implies that a consistent assignment like this should exist. If asked if the probability is over 0.2999 or under 0.30001 you will reply with a definite yes.

## Second-Order Logic: The Controversy

**Followup to**: Godel's Completeness and Incompleteness Theorems

"So the question you asked me last time was, 'Why does anyone bother with first-order logic at all, if second-order logic is so much more powerful?'"

Right. If first-order logic can't talk about finiteness, or distinguish the size of the integers from the size of the reals, why even bother?

"The first thing to realize is that first-order theories can still have a *lot *of power. First-order arithmetic does narrow down the possible models by a lot, even if it doesn't narrow them down to a *single *model. You can prove things like the existence of an infinite number of primes, because *every *model of the first-order axioms has an infinite number of primes. First-order arithmetic is never going to prove anything that's *wrong *about the standard numbers. Anything that's true in *all *models of first-order arithmetic will also be true in the *particular *model we call the standard numbers."

Even so, if first-order theory is strictly weaker, why bother? Unless second-order logic is just as incomplete relative to third-order logic, which is weaker than fourth-order logic, which is weaker than omega-order logic -

"No, surprisingly enough - there's tricks for making second-order logic encode any proposition in third-order logic and so on. If there's a collection of third-order axioms that characterizes a model, there's a collection of second-order axioms that characterizes the same model. Once you make the jump to second-order logic, you're *done *- so far as anyone knows (so far as I know) there's *nothing *more powerful than second-order logic in terms of *which models it can characterize*."

Then if there's one spoon which can eat anything, why not just use the spoon?

"Well... this gets into complex issues. There are mathematicians who don't believe there *is *a spoon when it comes to second-order logic."

Like there are mathematicians who don't believe in infinity?

"Kind of. Look, suppose you *couldn't* use second-order logic - you belonged to a species that doesn't have second-order logic, or anything like it. Your species doesn't have any native mental intuition you could use to construct the notion of 'all properties'. And then suppose that, after somebody used first-order set theory to prove that first-order arithmetic had many possible models, you stood around shouting that you believed in only *one *model, what you called the *standard *model, but you couldn't explain what made this model different from any other model -"

Well... a lot of times, even in math, we make statements that genuinely mean something, but take a while to figure out how to define. I think somebody who talked about 'the numbers' would mean something even before second-order logic was invented.

"But here the hypothesis is that you belong to a species that *can't* invent second-order logic, or think in second-order logic, or anything like it."

Then I suppose you want me to draw the conclusion that this hypothetical alien is just standing there shouting about standardness, but its words don't mean anything because they have no way to pin down one model as opposed to another one. And I expect this species is also magically forbidden from talking about all possible subsets of a set?

"Yeah. They can't talk about the largest powerset, just like they can't talk about the smallest model of Peano arithmetic."

Then you could arguably deny that shouting about the 'standard' numbers would mean anything, to the members of this particular species. You might as well shout about the 'fleem' numbers, I guess.

"Right. Even if all the members of this species *did *have a built-in sense that there was a special model of first-order arithmetic that was fleemer than any other model, if that fleem-ness wasn't bound to anything else, it would be meaningless. Just a floating word. Or if all you could do was define fleemness as floobness and floobness as fleemness, you'd have a loop of floating words; and that might give you the impression that each particular word had a meaning, but the loop as a whole wouldn't be connected to reality. That's why it doesn't help to say that the standard model of numbers is the smallest among all possible models of Peano arithmetic, if you can't define 'smallest possible' any more than you can define 'connected chain' or 'finite number of predecessors'."

But second-order logic *does *seem to have consequences first-order logic doesn't. Like, what about all that Godelian stuff? Doesn't second-order arithmetic semantically imply... its own Godel statement? Because the unique model of second-order arithmetic doesn't contain any number encoding a proof of a contradiction from second-order arithmetic? Wait, now I'm confused.

"No, that's correct. It's not paradoxical, because there's no effective way of finding all the *semantic *implications of a collection of second-order axioms. There's no analogue of Godel's Completeness Theorem for second-order logic - no *syntactic* system for deriving *all *the semantic consequences. Second-order logic is *sound*, in the sense that anything syntactically provable from a set of premises, is true in any model obeying those premises. But second-order logic isn't *complete*; there are semantic consequences you can't derive. If you take second-order logic at face value, there's no effectively computable way of deriving all the consequences of what you say you 'believe'... which is a major reason some mathematicians are suspicious of second-order logic. What does it mean to believe something whose consequences you can't derive?"

But second-order logic clearly has *some *experiential consequences first-order logic doesn't. Suppose I build a Turing machine that looks for proofs of a contradiction from first-order Peano arithmetic. If PA's consistency isn't provable in PA, then by the Completeness Theorem there must exist nonstandard models of PA where this machine halts after finding a proof of a contradiction. So first-order logic doesn't tell me that this machine runs forever - maybe it has nonstandard halting times, i.e., it runs at all standard N, but halts at -2* somewhere along a disconnected chain. Only second-order logic tells me there's no proof of PA's inconsistency *and therefore* this machine runs forever. Only second-order logic tells me I should *expect to see this machine keep runnin*g, and *not expect* - note falsifiability - that the machine ever halts.

"Sure, you just used a second-order theory to derive a consequence you didn't get from a first-order theory. But that's not the same as saying that you can *only *get that consequence using second-order logic. Maybe another first-order theory would give you the same prediction."

Like what?

"Well, for one thing, first-order *set theory* can prove that first-order *arithmetic *has a model. Zermelo-Fraenkel set theory can prove the existence of a set such that all the first-order Peano axioms are true about that set. It follows within ZF that sound reasoning on first-order arithmetic will never prove a contradiction. And since ZF can prove that the syntax of classical logic is sound -"

What does it mean for *set theory* to prove that *logic *is sound!?

"ZF can quote formulas as structured, and encode models as sets, and then represent a finite ZF-formula which says whether or not a set of quoted formulas is true about a model. ZF can then prove that no step of classical logic goes from premises that are true inside a set-model, to premises that are false inside a set-model. In other words, ZF can represent the idea 'formula X is semantically true in model Y' and 'these syntactic rules never produce semantically false statements from semantically true statements'."

Wait, then why can't ZF prove *itself *consistent? If ZF believes in all the axioms of ZF, and it believes that logic never produces false statements from true statements -

"Ah, but ZF can't prove that there exists any set which is a model of ZF, so it can't prove the ZF-axioms are consistent. ZF *shouldn't* be able to prove that some set is a model of ZF, because that's not true in all models. Many models of ZF don't contain any *individual *set well-populated enough for that one set to be a model of ZF all by itself."

I'm kind of surprised in a Godelian sense that ZF *can *contain sets as large as the universe of ZF. Doesn't any given set have to be smaller than the whole universe?

"Inside *any particular model* of ZF, every set *within *that model is smaller than that model. But not all models of ZF are the same size; in fact, models of ZF of *every *size exist, by the Lowenheim-Skolem theorem. So you can have *some *models of ZF - some universes in which all the elements collectively obey the ZF-relations - containing individual sets which are larger than *other *entire models of ZF. A set that large is called a *Grothendieck universe* and assuming it exists is equivalent to assuming the existence of 'strongly inaccessible cardinals', sizes so large that you provably can't prove inside set theory that anything that large exists."

Whoa.

(Pause.)

But...

"But?"

I agree you've shown that *one *experiential consequence of second-order arithmetic - namely that a machine looking for proofs of inconsistency from PA, won't be seen to halt - can be derived from first-order set theory. Can you get *all *the consequences of second-order arithmetic in some particular first-order theory?

"You can't get all the *semantic *consequences of second-order logic, taken at face value, in *any *theory or *any *computable reasoning. What about the halting problem? Taken at face value, it's a semantic consequence of second-order logic that any given Turing machine either halts or doesn't halt -"

Personally I find it rather intuitive to imagine that a Turing machine either halts or doesn't halt. I mean, if I'm walking up to a machine and watching it run, telling me that its halting or not-halting 'isn't entailed by my axioms' strikes me as not describing any actual experience I can have with the machine. Either I'll see it halt eventually, or I'll see it keep running forever.

"My point is that the statements we *can *derive from the syntax of current second-order logic is limited by that syntax. And by the halting problem, we shouldn't ever expect a computable syntax that gives us *all *the semantic consequences. There's no possible theory you can *actually use* to get a correct advance prediction about whether an arbitrary Turing machine halts."

Okay. I agree that no computable reasoning, on second-order logic or anything else, should be able to solve the halting problem. Unless time travel is possible, but even then, you shouldn't be able to solve the expanded halting problem for machines that use time travel.

"Right, so the *syntax *of second-order logic can't prove everything. And in fact, it turns out that, in terms of what you can *prove syntactically* using the standard syntax, second-order logic is identical to a many-sorted first-order logic."

Huh?

"Suppose you have a first-order logic - one that doesn't claim to be able to quantify over all possible predicates - which does allow the universe to contain two different sorts of things. Say, the logic uses lower-case letters for all type-1 objects and upper-case letters for all type-2 objects. Like, '∀x: x = x' is a statement over all type-1 objects, and '∀Y: Y = Y' is a statement over all type-2 objects. But aside from that, you use the same syntax and proof rules as before."

Okay...

"Now add an element-relation x∈Y, saying that x is an element of Y, and add some first-order axioms for making the type-2 objects behave like collections of type-1 objects, including axioms for making sure that most describable type-2 collections exist - i.e., the collection X containing just x is guaranteed to exist, and so on. What you can *prove syntactically* in this theory will be identical to what you can prove using the standard syntax of second-order logic - even though the theory doesn't claim that *all possible* collections of type-1s are type-2s, and the theory will have models where many 'possible' collections are missing from the type-2s."

Wait, now you're saying that second-order logic is no more powerful than first-order logic?

"I'm saying that the supposed power of second-order logic derives from *interpreting* it a particular way, and taking on faith that when you quantify over *all properties*, you're actually talking about all properties."

But then second-order arithmetic is no more powerful than first-order arithmetic in terms of what it can actually *prove*?

"2nd-order arithmetic is *way *more powerful than first-order arithmetic. But that's because first-order set theory is more powerful than arithmetic, and adding the syntax of second-order logic corresponds to adding axioms with set-theoretic properties. In terms of which consequences can be *syntactically *proven, second-order arithmetic is more powerful than first-order arithmetic, but *weaker* than first-order set theory. First-order set theory can prove the existence of a model of second-order arithmetic - ZF can prove there's a collection of numbers and sets of numbers which models a many-sorted logic with syntax corresponding to second-order logic - and so ZF can prove second-order arithmetic consistent."

But first-order logic, including first-order set theory, can't even *talk about* the standard numbers!

"Right, but first-order set theory can *syntactically prove *more statements about 'numbers' than second-order arithmetic can prove. And when you combine that with the *semantic *implications of second-order arithmetic not being computable, and with any second-order logic being syntactically identical to a many-sorted first-order logic, and first-order logic having neat properties like the Completeness Theorem... well, you can see why some mathematicians would want to give up entirely on this whole second-order business."

But if you deny second-order logic you *can't even say the word 'finite'*. You would have to believe the word 'finite' was a *meaningless noise*.

"You'd define finiteness relative to whatever first-order model you were working in. Like, a set might be 'finite' only on account of the model not containing any one-to-one mapping from that set onto a smaller subset of itself -"

But that set wouldn't *actually *be finite. There wouldn't actually be, like, only a billion objects in there. It's just that all the mappings which could *prove *the set was infinite would be mysteriously missing.

"According to some *other *model, maybe. But since there is no one true model -"

How is this not crazy talk along the lines of 'there is no one true reality'? Are you saying there's no *really *smallest set of numbers closed under succession, without all the extra infinite chains? Doesn't talking about how these theories have multiple possible models, imply that those possible models are *logical thingies* and one of them actually *does *contain the largest powerset and the smallest integers?

"The mathematicians who deny second-order logic would see that reasoning as invoking an implicit background universe of set theory. Everything you're saying makes sense *relative *to some *particular model* of set theory, which would contain possible models of Peano arithmetic as sets, and could look over those sets and pick out the smallest *in that model*. Similarly, that set theory could look over a proposed model for a many-sorted logic, and say whether there were any subsets *within *the larger universe which were missing from the many-sorted model. Basically, your brain is insisting that it lives inside some *particular *model of set theory. And then, from that standpoint, you could look over some *other *set theory and see how it was missing subsets that *your* theory had."

Argh! No, damn it, I live in the set theory that *really does* have all the subsets, with no mysteriously missing subsets or mysterious extra numbers, or denumerable collections of all possible reals that could like totally map onto the integers if the mapping that did it hadn't gone missing in the Australian outback -

"But *everybody *says that."

Okay...

"Yeah?"

Screw set theory. I live in the *physical universe* where when you run a Turing machine, and keep watching forever *in the physical universe*, you never *experience a time* where that Turing machine outputs a proof of the inconsistency of Peano Arithmetic. Furthermore, I live in a universe where space is *actually *composed of a real field and space is *actually *infinitely divisible and contains *all *the points between A and B, rather than space only containing a denumerable number of points whose existence can be proven from the first-order axiomatization of the real numbers. So to talk about *physics *- forget about mathematics - I've got to use second-order logic.

"Ah. You know, that particular response is not one I have seen in the previous literature."

Yes, well, I'm not a pure mathematician. When I ask whether I want an Artificial Intelligence to think in second-order logic or first-order logic, I wonder how that affects what the AI does in *the actual physical universe*. Here in the *actual physical universe* where times are followed by successor times, I *strongly suspect* that we should only expect to experience *standard *times, and not experience any nonstandard times. I think time is *connected*, and global connectedness is a property I can only talk about using second-order logic. I think that every *particular* time is finite, and yet time has no upper bound - that there are all finite times, but only finite times - and that's a property I can only characterize using second-order logic.

"But if you can't ever tell the difference between standard and nonstandard times? I mean, *local *properties of time can be described using first-order logic, and you can't directly *see *global properties like 'connectedness' -"

But I *can *tell the difference. There are only nonstandard times where a proof-checking machine, running forever, outputs a proof of inconsistency from the Peano axioms. So I don't expect to experience seeing a machine do that, since I expect to experience only standard times.

"Set theory can also prove PA consistent. If you use set theory to define time, you similarly won't expect to see a time where PA is proven inconsistent - those nonstandard integers don't exist in any model of ZF."

Why should I anticipate that my physical universe is restricted to having only the nonstandard times allowed by a *more *powerful set theory, instead of nonstandard times allowed by first-order arithmetic? If I then talk about a nonstandard time where a proof-enumerating machine proves ZF inconsistent, will you say that only nonstandard times allowed by some still more powerful theory can exist? I think it's clear that the way you're deciding which experimental outcomes you'll have to excuse, is by secretly assuming that *only standard times* exist regardless of which theory is required to narrow it down.

"Ah... hm. Doesn't physics say this universe is going to run out of negentropy before you can do an infinite amount of computation? Maybe there's only a bounded amount of time, like it stops before googolplex or something. That can be characterized by first-order theories."

We don't know that for certain, and I wouldn't want to build an AI that just *assumed* lifespans had to be finite, in case it was wrong. Besides, should I use a different *logic *than if I'd found myself in Conway's Game of Life, or something else really infinite? Wouldn't the same sort of creatures evolve in that universe, having the same sort of math debates?

"Perhaps no universe like that *can *exist; perhaps only finite universes can exist, because only finite universes can be uniquely characterized by first-order logic."

You just used the word 'finite'! Furthermore, taken at face value, our own universe *doesn't* look like it has a finite collection of entities related by first-order logical rules. Space and time both look like infinite collections of points - continuous collections, which is a second-order concept - and then to characterize the *size *of that infinity we'd need second-order logic. I mean, by the Lowenheim-Skolem theorem, there aren't just *denumerable *models of first-order axiomatizations of the reals, there's also *unimaginably large cardinal infinities* which obey the same premises, and that's a possibility straight out of H. P. Lovecraft. Especially if there are any *things *hiding in the *invisible cracks of space*."

"How could *you *tell if there were inaccessible cardinal quantities of points hidden inside a straight line? And anything that *locally *looks continuous each time you try to split it at a describable point, can be axiomatized by a first-order schema for continuity."

That brings up another point: Who'd *really *believe that the reason Peano arithmetic works on physical times, is because that whole infinite axiom *schema *of induction, containing for every Φ a *separate *rule saying...

(Φ(0) ∧ (∀x: Φ(x) → Φ(Sx))) → (∀n: Φ(n))

...was used to specify our universe? How improbable would it be for an* infinitely long* list of rules to be true, if there wasn't a unified reason for all of them? It seems much more likely that the *real *reason first-order PA works to describe time, is that all *properties *which are true at a starting time and true of the successor of any time where they're true, are true of all later times; and this generalization over *properties* makes induction hold for *first-order formulas* as a special case. If my native thought process is first-order logic, I wouldn't see the connection between each individual formula in the axiom schema - it would take separate evidence to convince me of each one - they would feel like independent mathematical facts. But after doing *scientific *induction over the fact that many *properties *true at zero, with succession preserving truth, seem to be true everywhere - after generalizing the simple, compact *second-order* theory of numbers and times - *then *you could invent an infinite first-order theory to approximate it.

"Maybe that just says you need to adjust whatever theory of scientific induction you're using, so that it can more easily induct infinite axiom schemas."

But why the heck would you *need *to induct infinite axiom schemas in the first place, if Reality *natively *ran on first-order logic? Isn't it far more likely that the way we ended up with these infinite lists of axioms was that Reality was manufactured - forgive the anthropomorphism - that Reality was manufactured using an underlying schema in which time is a *connected *series of events, and space is a *continuous* field, and these are properties which happen to require second-order logic for humans to describe? I mean, if you picked out first-order theories at random, what's the chance we'd end up inside an infinitely long axiom schema that just *happened *to look like the projection of a compact second-order theory? Aren't we ignoring a sort of *hint*?

"A hint to what?"

Well, I'm not that sure myself, at this depth of philosophy. But I would currently say that finding ourselves in a physical universe where times have successor times, and space looks continuous, seems like a possible *hint *that the Tegmark Level IV multiverse - or the way Reality was manufactured, or whatever - might look more like *causal universes characterizable by compact second-order theories* than *causal universes characterizable by first-order theories*.

"But since any second-order theory can just as easily be *interpreted *as a many-sorted first-order theory with quantifiers that can range over either elements or sets of elements, how could using second-order syntax actually *improve *an Artificial Intelligence's ability to handle a reality like that?"

Good question. One obvious answer is that the AI would be able to induct what *you* would call an infinite axiom schema, as a single axiom - a simple, finite hypothesis.

"There's all *sorts *of obvious hacks to scientific induction of first-order axioms which would let you assign nonzero probability to computable infinite sequences of axioms -"

Sure. So beyond that... I would currently guess that the basic assumption behind 'behaving as if' second-order logic is true, says that the AI should act as if only the 'actually smallest' numbers will ever appear in physics, relative to some 'true' mathematical universe that it thinks it lives in, but knows it can't fully characterize. Which is roughly what I'd say human mathematicians do when they take second-order logic at face value; they assume that there's some *true *mathematical universe in the background, and that second-order logic lets them talk about it.

"And what behaviorally, experimentally distinguishes the hypothesis, 'I live in the true ultimate math with fully populated powersets' from the hypothesis, 'There's some random model of first-order set-theory axioms I happen to be living in'?"

Well... one behavioral consequence is suspecting that your time obeys an infinitely long list of first-order axioms with induction schemas for *every formula*. And then moreover believing that you'll never experience a time when a proof-checking machine outputs a proof that Zermelo-Fraenkel set theory is inconsistent - even though there's *provably *some models with times like that, which fit the axiom schema you just inducted. That sounds like secretly believing that there's a background 'true' set of numbers that you think characterizes physical time, and that it's the *smallest *such set. Another suspicious behavior is that as soon as you suspect Zermelo-Fraenkel set theory is consistent, you suddenly expect not to experience any *physical *time which ZF proves isn't a standard number. You don't think you're in the nonstandard time of some weaker theory like Peano arithmetic. You think you're in the minimal time expressible by *any and all theories*, so as soon as ZF can prove some number doesn't exist in the minimal set, you think that 'real time' lacks such a number. All of these sound like behaviors you'd carry out if you thought there was a single 'true' mathematical universe that provided the best model for describing all *physical *phenomena, like time and space, which you encounter - and believing that this 'true' backdrop used the *largest *powersets and *smallest *numbers.

"How *exactly *do you formalize all that reasoning, there? I mean, how would you *actually *make an AI reason that way?"

Er... I'm still working on that part.

"That makes your theory a bit hard to criticize, don't you think? Personally, I wouldn't be surprised if any such *formalized* reasoning looked just like believing that you live inside a first-order set theory."

I suppose I wouldn't be *too *surprised either - it's hard to argue with the results on many-sorted logics. But if comprehending the physical universe is best done by assuming that real phenomena are characterized by a 'true' mathematics containing *the *powersets and *the *natural numbers - and thus expecting that no mathematical model we can formulate will ever contain any larger powersets or smaller numbers than those of the 'true' backdrop to physics - then I'd call that a moral victory for second-order logic. In first-order logic we aren't even supposed to be able to talk about such things.

"Really? To me that sounds like believing you live inside a model of first-order set theory, and believing that all models of any theories *you *can invent must *also *be sets in the larger model. You can prove the Completeness Theorem inside ZF plus the Axiom of Choice, so ZFC already proves that all consistent theories have models which are sets, although of course it can't prove that ZFC itself is such a theory. So - anthropomorphically speaking - no model of ZFC *expects *to encounter a theory that has fewer numbers or larger powersets than itself. No model of ZFC expects to encounter any quoted-model, any set that a quoted theory entails, which contains larger powersets than the ones in its own Powerset Axiom. A first-order set theory can even make the leap from the finite statement, 'P is true of all my subsets of X', to believing, 'P is true of all my subsets of X that can be described by this denumerable collection of formulas' - it can encompass the jump from a single axiom over 'all my subsets', to a quoted axiom *schema *over formulas. I'd sum all that up by saying, 'second-order logic is how first-order set theory feels from the inside'."

Maybe. Even in the event that neither human nor superintelligent cognition will ever be able to 'properly talk about' unbounded finite times, global connectedness, particular infinite cardinalities, or true spatial continuity, it doesn't follow that Reality is similarly limited in which physics it can privilege.

Part of the sequence *Highly Advanced Epistemology 101 for Beginners*

Previous post: "Godel's Completeness and Incompleteness Theorems"

## Godel's Completeness and Incompleteness Theorems

**Followup to**: Standard and Nonstandard Numbers

So... last time you claimed that using first-order axioms to rule out the existence of nonstandard numbers - other chains of numbers besides the 'standard' numbers starting at 0 - was *forever and truly impossible*, even unto a superintelligence, no matter *how *clever the first-order logic used, even if you came up with an entirely different way of axiomatizing the numbers.

"Right."

How could you, in your finiteness, possibly know that?

"Have you heard of Godel's Incompleteness Theorem?"

Of course! Godel's Theorem says that for every consistent mathematical system, there are statements which are *true *within that system, which can't be *proven* within the system itself. Godel came up with a way to encode theorems and proofs as numbers, and wrote a purely numerical formula to detect whether a proof obeyed proper logical syntax. The basic trick was to use prime factorization to encode lists; for example, the ordered list <3, 7, 1, 4> could be uniquely encoded as:

2^{3} * 3^{7} * 5^{1} * 7^{4}

And since prime factorizations are unique, and prime powers don't mix, you could inspect this single number, 210,039,480, and get the unique ordered list <3, 7, 1, 4> back out. From there, going to an encoding for logical formulas was easy; for example, you could use the 2 prefix for NOT and the 3 prefix for AND and get, for any formulas Φ and Ψ encoded by the numbers #Φ and #Ψ:

¬Φ = 2^{2} * 3^{#Φ}

Φ ∧ Ψ = 2^{3} * 3^{#Φ} * 5^{#Ψ}

It was then possible, by dint of crazy amounts of work, for Godel to come up with a gigantic formula of Peano Arithmetic [](p, c) meaning, 'P encodes a valid logical proof using first-order Peano axioms of C', from which directly followed the formula []c, meaning, 'There exists a number P such that P encodes a proof of C' or just 'C is provable in Peano arithmetic.'

Godel then put in some *further *clever work to invent statements which referred to *themselves*, by having them contain sub-recipes that would reproduce the entire statement when manipulated by another formula.

And then Godel's Statement encodes the statement, 'There does not exist any number P such that P encodes a proof of (this statement) in Peano arithmetic' or in simpler terms 'I am not provable in Peano arithmetic'. If we assume first-order arithmetic is consistent and sound, then no *proof *of this statement *within *first-order arithmetic exists, which means the statement is *true *but can't be proven within the system. That's Godel's Theorem.

"Er... no."

No?

"No. I've heard rumors that Godel's Incompleteness Theorem is horribly misunderstood in your Everett branch. Have you heard of Godel's *Completeness *Theorem?"

Is that a thing?

"Yes! Godel's Completeness Theorem says that, for any collection of first-order statements, *every semantic implication of those statements is syntactically provable within first-order logic*. If something is a genuine implication of a collection of first-order statements - if it actually *does *follow, in the models pinned down by those statements - then you can *prove *it, *within *first-order logic, using *only* the syntactical rules of proof, from those axioms."

## Standard and Nonstandard Numbers

**Followup to**: Logical Pinpointing

"Oh! Hello. Back again?"

Yes, I've got another question. Earlier you said that you *had *to use second-order logic to define the numbers. But I'm pretty sure I've heard about something called 'first-order Peano arithmetic' which is also supposed to define the natural numbers. Going by the name, I doubt it has any 'second-order' axioms. Honestly, I'm not sure I understand this second-order business at all.

"Well, let's start by examining the following model:"

"This model has three properties that we would expect to be true of the standard numbers - 'Every number has a successor', 'If two numbers have the same successor they are the same number', and '0 is the only number which is not the successor of any number'. All three of these statements are true in this model, so in that sense it's quite numberlike -"

And yet this model clearly is *not* the numbers we are looking for, because it's got all these mysterious extra numbers like C and -2*. That C thing even loops around, which I certainly wouldn't expect any number to do. And then there's that infinite-in-both-directions chain which isn't corrected to anything else.

"Right, so, the difference between first-order logic and second-order logic is this: In first-order logic, we can get rid of the ABC - make a statement which *rules out* any model that has a loop of numbers like that. But we can't get rid of the infinite chain underneath it. In second-order logic we can get rid of the extra chain."

## Logical Pinpointing

**Followup to**: Causal Reference, Proofs, Implications and Models

The fact that one apple added to one apple invariably gives two apples helps in the teaching of arithmetic, but has no bearing on the truth of the proposition that 1 + 1 = 2.

-- James R. Newman, The World of Mathematics

*Previous meditation 1:* If we can only meaningfully talk about parts of the universe that can be pinned down by chains of cause and effect, where do we find the fact that 2 + 2 = 4? Or did I just make a meaningless noise, there? Or if you claim that "2 + 2 = 4"*isn't* meaningful or true, then what alternate property does the sentence "2 + 2 = 4" have which makes it so much more useful than the sentence "2 + 2 = 3"?

*Previous **meditation** 2:* It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just *assuming* that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equalled two apples before there was anyone to count them, and that we couldn't make it 5 just by assuming differently.

Speaking conventional English, we'd say the sentence 2 + 2 = 4 is "true", and anyone who put down "false" instead on a math-test would be marked wrong by the schoolteacher (and not without justice).

But what can *make* such a belief true, what is the belief *about,* what is the truth-condition of the belief which can make it true or alternatively false? The sentence '2 + 2 = 4' is true if and only if... what?

In the previous post I asserted that the study of logic is the study of which conclusions follow from which premises; and that although this sort of inevitable implication is sometimes called "true", it could more specifically be called "valid", since checking for inevitability seems quite different from comparing a belief to our own universe. And you could claim, accordingly, that "2 + 2 = 4" is 'valid' because it is an inevitable implication of the axioms of Peano Arithmetic.

And yet thinking about 2 + 2 = 4 doesn't really *feel* that way. Figuring out facts about the natural numbers doesn't feel like the operation of making up assumptions and then deducing conclusions from them. It feels like the numbers are just *out* there, and the only point of making up the axioms of Peano Arithmetic was to *allow* mathematicians to talk about them. The Peano axioms might have been convenient for *deducing* a set of theorems like 2 + 2 = 4, but really all of those theorems were true *about *numbers to begin with. Just like "The sky is blue" is true about the sky, regardless of whether it follows from any particular assumptions.

So comparison-to-a-standard does seem to be at work, just as with *physical* truth... and yet this notion of 2 + 2 = 4 seems different from "stuff that makes stuff happen". Numbers don't occupy space or time, they don't arrive in any order of cause and effect, there are no *events* in numberland.

**Meditation: **What are we talking *about* when we talk about numbers? We can't navigate to them by following causal connections - so how do we get there from here?

View more: Next