# Second-Order Logic: The Controversy

**Followup to**: [Godel's Completeness and Incompleteness Theorems](https://www.lesswrong.com/lw/g1y/godels_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](https://www.lesswrong.com/lw/i4/belief_in_belief/), 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](http://wiki.lesswrong.com/wiki/Highly_Advanced_Epistemology_101_for_Beginners)_

Previous post: "[Godel's Completeness and Incompleteness Theorems](https://www.lesswrong.com/lw/g1y/godels_completeness_and_incompleteness_theorems/)"