"By all means say it better then."

"0 + 0 = 0.  \k, k = ~0 -> k + k ~= k."

"Progress.  But '~0' isn't a thing in this language, 'not' takes in propositions, which have the values of truth or falsehood, and spits out falsehood or truth.  'Not four' isn't a number - or if you wanted it to talk about the collection of all numbers except four, we'd have to start introducing collections and that's a big ol' subject.  ~= isn't already a symbol in our language either, and in fact you don't particularly need to define a new symbol for it.  Next rewrite?" 

\k, (k = 0 /\ k + k =k) \/ ~(k + k = k) 

She writes this rather than saying it, because it seems like it'd be quite unpleasant to say and harder to tweak while speaking. She writes it with Prestidigitation because she has better control and precision than a student and they ought to remember it.

"Good try, but your statement doesn't quite narrow down the possible worlds to where you wanted - it includes worlds where ~(k + k = k) is true of every number, including the one you called zero.  Can anyone see how to fix it?"

"Can't you just add ~(k=0) to the second part?"

"Works unless I've missed something myself, but do you want to write out exactly what you mean there, to make sure it's not just my own imagination supplying the answer I think is correct?"

\k, (k = 0 /\ k + k =k) \/ ~(k=0) /\ ~(k + k = k) 

He supposes it's good that people are finding so many detailed ways to be wrong, exhibiting them early and getting them out of the way.

"We haven't said anything about adding - outside the system, up at our level - rules for adding in parentheses that weren't in the written formula.  So that could mean either of -"

\k. ((k=0 /\ k + k = k) \/ ~(k=0)) /\ ~(k + k = k)
\k. (k=0 /\ k + k = k) \/ (~(k=0) /\ ~(k + k = k))

"Second one," the girl says.

"Then it looks good to me.  Keeping in mind that these don't exactly match standard forms I learned, since we're making them up as we go, and my own intelligence is not at the level where I will reliably spot errors on the first pass.  I'm not sure quite why I feel the need to say this - it seems like the sort of thing that should be obvious? - but if I'm the one who makes an error, or it just looks like that, speak up.  If you're right, you get to be impressive, and if you're wrong, you need to know which mistake you made."

Keltham keeps prodding the group for a while, dropping hints as needed, until he's pretty sure they've written enough random rules to yield in their combination all the constraints of first-order arithmetic except for the induction axiom schema.  If anybody from Cheliax brilliantly pulls the induction axiom schema out of their ass, he's going to be sure they're getting it from somewhere; maybe dath ilani geniuses can pull that kind of shit at their age (he doubts it), but the geniuses of this world are only as smart as him unless they're wearing intelligence headbands.

They do not brilliantly pull the induction axiom schema out of their asses or out of his mind, which they are not reading. They do mostly manage to follow along through all the rules of first-order arithmetic, and they seem to be having fun about it.

Once they've got a nearly full set of rules, Keltham remarks that the last puzzle piece for identifying the numbers, as well as they can ever be identified in a certain sense he's not going into right now, is one he really doesn't expect them to get unhinted.

And then he drops on them the infinite axiom schema for induction, trying as best he can to explain why you'd need it to pinpoint the numbers.

After clearing up any misapprehensions about that as best he can, Keltham is ready to move on to his next point.

"We're running through things a lot faster than I went through them as a kid, and I'm probably accidentally leaving out important ideas along the way - all of this would take more like a month, if you were eight years old and doing the exercises, even if you were doing nothing else.  But you may recall that some time earlier, I posed some puzzles about asking for examples of necessary truths, and why they were ever good for anything, and what it means to say that one equals one is a necessary truth - what you ought to expect to see happen as a result - especially given that a necessary truth should still end up being true no matter what happens to you, if it could happen inside any illusion depicted in full detail."

"We now have a language in which I can give some of my own answers there.  But before then, does anybody want to take a renewed shot at saying what it is we're talking about, and what we should expect to see happen, if we say that one equals one is a necessary truth?"

"...that if it's not true we just can't do any reasoning in a formal system at all?"

"Does reality need to care what you can't reason about?  Perhaps you can depict an illusion in full detail in which one does not equal one, and we will need to construct a new logic which does not take as primitive the assertion that every object equals itself, in order to describe that illusion."

" - you can't depict an illusion in full detail in which one does not equal one."

"What about clouds drifting across the sky and sometimes separating?  One cloud equals two clouds, it doesn't equal one cloud!  Divide both sides of the equation by cloud and there you have it."

They stare concernedly at him.

"More to the point, how would you depict one, the successor of zero, inside an illusion?  You can depict one cherry in a bowl of fruit, in an illusion.  How would you depict one, the successor of zero, as it appears in our collections of statements?"

"- you'd just have to put the symbols."

"Then we have merely depicted the symbols talking about one, in our illusion, not depicted one itself.  That's like making an illusion of a piece of paper with 'cherry' written on it, and saying you made an illusion of a cherry."

Some particularly daring girls, in the middle of the discussion of the induction axiom, sent around a crumbled piece of paper, as girls do; it flew between desks, as wizarding girls do; its text was in Infernal, as Chelish wizarding girls do. It read 'is Keltham a sadist? y/n'

The vote leaned yes.

Keltham's audience squirms anxiously at this question.

Keltham does not have the sharpness of unaided vision, let alone interpretation capacity, that would be required to perceive these nigh-imperceptible squirms.  He lives in a mental universe very far away from this reality, a mental universe where uncomfortable or unhappy students will of course speak up and tell you this fact as soon as they realize it themselves.  Though he has noticed his researchers' apparent lack of any visible emotions besides competitive enthusiasm, and is starting to wonder if they used a magical spell that's the equivalent of a mind-affecting drug that made them fixedly enthusiastic.

Well, at least this time he's about to say some words where enthusiasm seems appropriate.  Keltham tries to channel the demeanor of an appropriately specialized Watcher as best he can; the sort of Watcher who tries to make sure that kids get the full impact of things, and aren't cheated out of awesome stuff by older kids mistakenly trying to act like it's no big deal.

"In my language, we'd say that the subject matter of our discussion, when we talk about math, is which conclusions follow from which premises.  When we discuss numbers and say 2 + 3 = 5, it implies that if we observe cherries and come to believe that our number-axioms describe in reality the operations of combining bowls of cherries, we will expect to see in reality that pouring a bowl of two cherries into a bowl of three cherries yields a bowl of five cherries.  If we get six cherries instead, we might think we made a mistake in the math.  Or we might suspect that watching the bowl closely would let us see an extra cherry popping into existence.  In the latter case our beliefs about which conclusions validly follow from the addition-premises would have been right, but our guess that the addition-premises applied to combining bowls of cherries would have been wrong.  Being the fragile creatures that we are, and sometimes making mistakes in our reasoning, when we do math about a bridge and then the bridge falls down, we might be observing that the bridge disobeyed the premises about which we did math; or we might be observing that we made an error about what was a necessary connection, and our conclusion didn't follow even though all the premises were true."

"All of this is to say that we can observe the consequences, the shadows, of necessary truths, when we watch the empirical world; even to the point of our observations leading us to suspect errors in our own reasoning about what was necessary.  But observing the number 1 itself?  At best, maybe, somebody could make an illusion of an object representing zero, not the successor of any other object, connected by a successor relationship to the object that would therefore be one," and Keltham draws some green dots connected by red arrows to other dots.  "This would give us an illusion that would map very directly, in our external interpretation, onto a partial model that fits the number axioms.  But it doesn't make sense to say that the illusion is depicting the number one; there isn't a single thingy like that out there floating in the void, just a set of premises that actually existent things might obey, in which case we'd expect them to behave like the number one."

"The facts about which conclusions follow inevitably from which premises can't be said to be older than the temporal universe, because they're outside of time entirely; it's not that they existed before the universe began, or that they'll last after the universe perishes, but that they are somewhere above or below that.  Temporal and physical processes draw on necessities, mirror them, but cannot change them between one time and another.  There's a certain sense in which, in controlling our own decisions, we are controlling links between premises and conclusions - we are controlling, given the premise of a person like ourselves, the conclusion of which decision we come to - but this doesn't mean we are changing mathematical facts between one time and another.  An alternate plane of existence - or so I would suspect - can obey different premises in its physical behavior; it cannot alter which conclusions follow necessarily from which premises.  Those facts are, not eternal, but outside of time entirely.  This is one of the ideas invoked by a word in my language, which translates into your word 'Lawful': the concept of drawing on and mirroring the level of existence where certain facts may be viewed as absolute and unalterable."

"You have seen now how we can start with two apparently different concepts of validity - one that preserves truths about properties of objects connected by 'and' or 'implies', one that produces true numerical equations from true equations - and, at least for the case of whole numbers, reduced the equational subject matter to the predicate-logic subject matter.  Just like we were able to reduce 'and' to 'or', or 'or' to 'implies'.  I will tell you now a point that you will not be ready to prove yourselves for a while: the system of predicate logic I've introduced to you is one of several systems that are complete in the sense that all mathematics can be translated into them.  The topology you learned as wizards, unless it deals with some phenomenon absolutely foreign to dath ilan whose mere existence refutes this entire philosophy - which I am mostly not expecting, to be clear - is just another kind of math that could be translated into this system, or translated into several other systems I haven't shown you, all of which could also be translated into this system, and into which this system could also translate, moving between them as freely as we rewrite an 'implies' connector as an 'or' connector."

"This is one reason I could pop into another dimension and expect to have a reasonable conversation with Lrilatha, who belongs to a species that doesn't exist in my world.  If different formulations of validity can so freely translate between each other, it would seem more reasonable to hope that I, to the extent I had those concepts right, would use a version mappable to Lrilatha's, who is a Lawful being; so that her arguments would make sense to me, and my arguments make sense to her.  Either a conclusion follows with necessity from its premises or it does not.  Only mistakes about that subject matter could differ between people, between factions, between planes.  The right answer is the same everywhere.  And this is also part of the concept in my language, which the translation spell translates into your word 'Lawful'."

It is probably optimistic to, before she has even properly learned this herself, conclude that Cheliax could be teaching it so much better, but - Cheliax could be teaching this so much better that it suddenly hurts to realize how badly she learned it before. 

There is a right way to be. Devils are it; mortals aren't. Mortals were, for a while, at least controlled by gods, who are, but the control broke, and now mortals just wander around, missing the concepts that make up the right way to be, for the most part not smart enough to learn them. Cheliax emphasizes - that this is disappointing to Asmodeus, that it makes mortals less valuable to Asmodeus, and of course that's the angle from which Asmodeus cares about it, but the angle from which the humans ought to care about it is that they are worse. It is in their interests to be perfected, not because some god will get them when they die and the other gods waste even more of them, but because there is a commonality among all perfect beings, a shared language that the perfect can speak across planes, across time -

How is it that Chelish children learn they'll be perfected and are scared, instead of learning they'll be perfected and are joyful, impatient, full of the longing that filled Carissa when Contessa Lrilatha spoke - why aren't we teaching it like this - but the answer of course must be that Asmodeus couldn't divine the result of thousands of years of experiments across an entire other world, at least not more cheaply than He could grab someone from there - 

Keltham pauses, at least partially to catch his own mental breath.  He should probably call a break sometime soon, but he was working around to a point, some time earlier before the whole digression into Validity and first-order arithmetic, and he feels like it would be only polite to actually finish the digression and work around to what he meant to say, before then.  There's still some distance left to go before he can pop the stack - he wasn't actually expecting, on some wordless level, that it would take this long to teach grown teenagers first-order logic and how to axiomatize arithmetic.

"In saying all this, I'm jumping ahead in a dath ilani education and skipping over a number of exercises required to actually understand everything and a dozen dozen precautions we were given against common mistakes, some of which now seem pretty silly and obvious to me, but which might turn out to be a lot more necessary to otherwise unprepared minds, I don't know."

"For example, if somebody throws a ball and you need to catch it, trying to translate your thoughts into predicate logic about the ball having the property of flying and this implying an eventual fall given gravity, is going to utterly fail to help you.  You would be, first of all, better off thinking in your mind's native wordless language that tries to visualize the ball's fall and run to there, because your native language is faster and the ball will fall before you can think of anything logical.  You would, second of all, be engaging in a particularly naive kind of jumping ahead of your real capabilities, by trying to translate your thoughts about the ball directly into a logic with a falling predicate.  What you would actually need to predict the ball's fall using serious math is calculus, and not simple calculus either; it would include terms like how the resistance of the air to the ball's flight changed with the ball's speed.  If you try to summarize all that by saying that the 'falling' predicate on the ball was true, when that 'falling' statement would be equally true if gravity was pulling on the ball a different amount or if the air was more resistant, you're throwing away details that actually matter in order to try to squeeze down the issue into predicate logic - an amount of predicate logic that you find easy to handle, which is too little to actually solve the problem.  You also wouldn't be absolutely certain about the ball's position or trajectory, and managing uncertainty inside of mathematics is a whole separate topic I haven't broached to you."

"Actually solving the ball's flight using a full written-on-paper account of how conclusions about what's probably true, follow necessarily from some unnecessary premises you already believe about what's probably true, would require setting up a complicated problem in calculus and probability that would describe how to infer the ball's trajectory from what your eyes had seen of the ball.  And if you wanted that said in pure logic with all premises spelled out, you'd have to axiomatize that calculus problem into the more universal language of logic.  It is a whole lot easier to just run and catch the ball by thinking about it in your native brain's native style."

"I mean, maybe if you were a god you could solve the whole problem using explicitly valid reasoning - or not, I'm not sure how mentally powerful your gods are, exactly.  And if you were a god and you could do that, then you could get tossed into another plane of existence where gravity works differently, and rapidly recalculate how to catch flying balls from scratch and do that successfully on your first try, instead of having to relearn the rules your brain uses through a lot of trying and failing.  But I am not a god, yet, and you are not gods, also possibly yet, and I doubt that any intelligence headbands we can afford in the foreseeable future will let us do it either."

"So don't get ahead of yourselves just because logic is absolute, timeless, and universal.  If you can't think fast enough to solve a problem using explicit logic, or if it would require humanly unmanageable huge problem statements even in principle, or if you simply don't know how, then all that absolute and timeless stuff won't help you.  The fact there exists a better method that gods or super-gods could use to solve a problem doesn't mean that you can solve it that way, or that you can get closer to the super-god's solution by using bad, clownlike, and tiny imitations of the outer forms of the ideal methods they'd use."  Gods seem like a surprisingly useful metaphor for ideal cognitive processes, now that he's in a world that has gods; Keltham is a little surprised it wasn't a more common explanatory metaphor used in dath ilan.

"I don't know if any of you actually needed me to say that, to be clear.  But it's the sort of thing they tell you when you're 8 years old, and you get into your head the idea that if logic is so great, you should be able to use it to crush your opponents at sportsball by making only sharply logical muscle movements.  I mean, actually they just let me make the mistake and lose the sportsball game, but then afterwards, that was what they told me when I asked what I'd been doing wrong and how to do it correctly."