So apparently Richard Feynman once said:

If, in some cataclysm, all scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or atomic fact, or whatever you wish to call it) that all things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence you will see an enormous amount of information about the world, if just a little imagination and thinking are applied.

I could be missing something, but this strikes me as a terrible answer.

When was the atomic hypothesis confirmed?  If I recall correctly, it was only when chemists started noticing that the outputs of chemical reactions tended to factorize a certain way, which is to say that it took millennia after Democritus to get the point where the atomic hypothesis started making clearly relevant experimental predictions.

How about, "Stop trying to sound wise and come up with theories that make precise predictions about things you can measure in numbers."

I noticed this on Marginal Revolution, so I shall also state my candidate for the one most important sentence about macroeconomics:  "You can't eat gold, so figure out how the heck money is relevant to making countries actually produce more or less food."  This is a pretty large advance on how kings used to think before economics.  I mean, Scott Sumner is usually pretty savvy (so is Richard Feynman btw) but his instruction to try to understand money is likely to fall on deaf ears, if it's just that one sentence.  Think about money?  Everyone wants more money!  Yay, money!  Let's build more gold mines!  And "In the short run, governments are not households"?  Really, Prof. Cowen, that's what you'd pass on to the next generation as they climb up from the radioactive soil?

*Cough.*  Okay, I'm done.  Does anyone want to take their own shot at doing better than Feynman did for their own discipline?

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 running, 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"

http://wiki.lesswrong.com/wiki/Deletion_policy

This is my attempt to codify the informal rules I've been working by.

I'll leave this post up for a bit, but strongly suspect that it will have to be deleted not too long thereafter.  I haven't been particularly encouraged to try responding to comments, either.  Nonetheless, if there's something I missed, let me know.

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:

2³ * 3⁷ * 5¹ * 7⁴

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 = 2² * 3^#Φ

Φ ∧ Ψ = 2³ * 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."

New proposed censorship policy:

Any post or comment which advocates or 'asks about' violence against sufficiently identifiable real people or groups (as opposed to aliens or hypothetical people on trolley tracks) may be deleted, along with replies that also contain the info necessary to visualize violence against real people.

Reason: Talking about such violence makes that violence more probable, and makes LW look bad; and numerous message boards across the Earth censor discussion of various subtypes of proposed criminal activity without anything bad happening to them.

More generally: Posts or comments advocating or 'asking about' violation of laws that are actually enforced against middle-class people (e.g., kidnapping, not anti-marijuana laws) may at the admins' option be censored on the grounds that it makes LW look bad and that anyone talking about a proposed crime on the Internet fails forever as a criminal (i.e., even if a proposed conspiratorial crime were in fact good, there would still be net negative expected utility from talking about it on the Internet; if it's a bad idea, promoting it conceptually by discussing it is also a bad idea; therefore and in full generality this is a low-value form of discussion).  

This is not a poll, but I am asking in advance if anyone has non-obvious consequences they want to point out or policy considerations they would like to raise. In other words, the form of this discussion is not 'Do you like this?' - you probably have a different cost function from people who are held responsible for how LW looks as a whole - but rather, 'Are there any predictable consequences we didn't think of that you would like to point out, and possibly bet on with us if there's a good way to settle the bet?'

Yes, a post of this type was just recently made.  I will not link to it, since this censorship policy implies that it will shortly be deleted, and reproducing the info necessary to say who was hypothetically targeted and why would be against the policy.

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."

Followup to: Mixed Reference: The Great Reductionist Project

Humans need fantasy to be human.

"Tooth fairies? Hogfathers? Little—"

Yes. As practice. You have to start out learning to believe the little lies.

"So we can believe the big ones?"

Yes. Justice. Mercy. Duty. That sort of thing.

"They're not the same at all!"

You think so? Then take the universe and grind it down to the finest powder and sieve it through the finest sieve and then show me one atom of justice, one molecule of mercy.

- Susan and Death, in Hogfather by Terry Pratchett

Suppose three people find a pie - that is, three people exactly simultaneously spot a pie which has been exogenously generated in unclaimed territory. Zaire wants the entire pie; Yancy thinks that 1/3 each is fair; and Xannon thinks that fair would be taking into equal account everyone's ideas about what is "fair".

Followup to: Logical Pinpointing, Causal Reference

Take the universe and grind it down to the finest powder and sieve it through the finest sieve and then show me one atom of justice, one molecule of mercy.

- Death, in Hogfather by Terry Pratchett

Meditation: So far we've talked about two kinds of meaningfulness and two ways that sentences can refer; a way of comparing to physical things found by following pinned-down causal links, and logical validity by comparison to models pinned-down by axioms. Is there anything else that can be meaningfully talked about? Where would you find justice, or mercy?

We can always use more case studies of insanity that aren't religion, right?

Well, Miracle Mineral Supplement is my new go-to example for Bad Things happening to people with low epistemic standards. "MMS" is a supposed cure for everything ranging from the common cold to HIV to cancer. I just saw it recommended in another Facebook thread to someone who was worried about malaria symptoms.

It's industrial-strength bleach. Literally just bleach. Usually drunk, sometimes injected, and yes, it often kills you. It is every bit as bad as it sounds if not worse.

This is beyond Poe's Law. Medieval blood draining via leeches was far more of an excusable error than this, they had far less evidence it was a bad idea. I think if I was trying to guess what was the dumbest alternative medicine on the planet, I still would not have guessed this low. My brain is still not pessimistic enough about human stupidity.

http://en.wikipedia.org/wiki/Miracle_Mineral_Supplement