Nice, I like this proof also. Maybe there's a clearer way to say thing, but your "unrolling one step" corresponds to my going from to . We somehow need to "look two possible worlds deep".
Here's a simple Kripke frame proof of Payor's lemma.
Let be a Kripke frame over our language, i.e. is a set of possible worlds, is an accessibility relation, and judges that a sentence holds in a world. Now, suppose for contradiction that but that , i.e. does not hold in some world .
A bit of De Morganing tells us that the hypothesis on is equivalent to , so . So, there is some world with such that . But again looking at our equivalent form for , we see that , so , a contradiction.
Both this proof and the proof in the post are very simple, but at least for me I feel like this proof tells me a bit more about what's going on, or at least tells me something about what's going on that the other doesn't. Though in a broader sense there's a lot I don't know about what's going on in modal fixed points.
Kripke frame-style semantics are helpful for thinking about lots of modal logic things. In particular, there are cool inductiony interpretations of the Gödel/Löb theorems. These are more complicated, but I'd be happy to talk about them sometime.
Theorem. Weak HCH (and similar proposals) contain EXP.
Proof sketch: I give a strategy that H can follow to determine whether some machine that runs in time accepts. Basically, we need to answer questions of the form "Does cell have value at time ." and "Was the head in position at time ?", where and are bounded by some function in . Using place-system representations of and , these questions have length in , so they can be asked. Further, each question is a simple function of a constant number of other such questions about earlier times as long as , and can be answered directly in the base case .
*I* think that there's a flaw in the argument.
I could elaborate, but maybe you want to think about this more, so for now I'll just address your remark about , where is refutable. If we assume that , then, since is false, must be false, so must be true. That is, you have proven that PA proves , that is, since is contradictory, PA proves its own inconsistency. You're right that this is compatible with PA being consistent--PA may be consistent but prove its own inconsistency--but this should still be worrying.
This reminds me of the Discourse on Method.
[T]here is seldom so much perfection in works composed of many separate parts, upon which different hands had been employed, as in those completed by a single master. Thus it is observable that the buildings which a single architect has planned and executed, are generally more elegant and commodious than those which several have attempted to improve, by making old walls serve for purposes for which they were not originally built. Thus also, those ancient cities which, from being at first only villages, have become, in course of time, large towns, are usually but ill laid out compared with the regularity constructed towns which a professional architect has freely planned on an open plain; so that although the several buildings of the former may often equal or surpass in beauty those of the latter, yet when one observes their indiscriminate juxtaposition, there a large one and here a small, and the consequent crookedness and irregularity of the streets, one is disposed to allege that chance rather than any human will guided by reason must have led to such an arrangement. And if we consider that nevertheless there have been at all times certain officers whose duty it was to see that private buildings contributed to public ornament, the difficulty of reaching high perfection with but the materials of others to operate on, will be readily acknowledged. In the same way I fancied that those nations which, starting from a semi-barbarous state and advancing to civilization by slow degrees, have had their laws successively determined, and, as it were, forced upon them simply by experience of the hurtfulness of particular crimes and disputes, would by this process come to be possessed of less perfect institutions than those which, from the commencement of their association as communities, have followed the appointments of some wise legislator. It is thus quite certain that the constitution of the true religion, the ordinances of which are derived from God, must be incomparably superior to that of every other. And, to speak of human affairs, I believe that the pre-eminence of Sparta was due not to the goodness of each of its laws in particular, for many of these were very strange, and even opposed to good morals, but to the circumstance that, originated by a single individual, they all tended to a single end. In the same way I thought that the sciences contained in books (such of them at least as are made up of probable reasonings, without demonstrations), composed as they are of the opinions of many different individuals massed together, are farther removed from truth than the simple inferences which a man of good sense using his natural and unprejudiced judgment draws respecting the matters of his experience. And because we have all to pass through a state of infancy to manhood, and have been of necessity, for a length of time, governed by our desires and preceptors (whose dictates were frequently conflicting, while neither perhaps always counseled us for the best), I farther concluded that it is almost impossible that our judgments can be so correct or solid as they would have been, had our reason been mature from the moment of our birth, and had we always been guided by it alone.
It is true, however, that it is not customary to pull down all the houses of a town with the single design of rebuilding them differently, and thereby rendering the streets more handsome; but it often happens that a private individual takes down his own with the view of erecting it anew, and that people are even sometimes constrained to this when their houses are in danger of falling from age, or when the foundations are insecure. With this before me by way of example, I was persuaded that it would indeed be preposterous for a private individual to think of reforming a state by fundamentally changing it throughout, and overturning it in order to set it up amended; and the same I thought was true of any similar project for reforming the body of the sciences, or the order of teaching them established in the schools: but as for the opinions which up to that time I had embraced, I thought that I could not do better than resolve at once to sweep them wholly away, that I might afterwards be in a position to admit either others more correct, or even perhaps the same when they had undergone the scrutiny of reason. I firmly believed that in this way I should much better succeed in the conduct of my life, than if I built only upon old foundations, and leaned upon principles which, in my youth, I had taken upon trust. For although I recognized various difficulties in this undertaking, these were not, however, without remedy, nor once to be compared with such as attend the slightest reformation in public affairs. Large bodies, if once overthrown, are with great difficulty set up again, or even kept erect when once seriously shaken, and the fall of such is always disastrous. Then if there are any imperfections in the constitutions of states (and that many such exist the diversity of constitutions is alone sufficient to assure us), custom has without doubt materially smoothed their inconveniences, and has even managed to steer altogether clear of, or insensibly corrected a number which sagacity could not have provided against with equal effect; and, in fine, the defects are almost always more tolerable than the change necessary for their removal; in the same manner that highways which wind among mountains, by being much frequented, become gradually so smooth and commodious, that it is much better to follow them than to seek a straighter path by climbing over the tops of rocks and descending to the bottoms of precipices.
...
And finally, as it is not enough, before commencing to rebuild the house in which we live, that it be pulled down, and materials and builders provided, or that we engage in the work ourselves, according to a plan which we have beforehand carefully drawn out, but as it is likewise necessary that we be furnished with some other house in which we may live commodiously during the operations, so that I might not remain irresolute in my actions, while my reason compelled me to suspend my judgement, and that I might not be prevented from living thenceforward in the greatest possible felicity, I formed a provisory code of morals, composed of three or four maxims, with which I am desirous to make you acquainted.
The first was to obey the laws and customs of my country, adhering firmly to the faith in which, by the grace of God, I had been educated from my childhood and regulating my conduct in every other matter according to the most moderate opinions, and the farthest removed from extremes, which should happen to be adopted in practice with general consent of the most judicious of those among whom I might be living. For as I had from that time begun to hold my own opinions for nought because I wished to subject them all to examination, I was convinced that I could not do better than follow in the meantime the opinions of the most judicious; and although there are some perhaps among the Persians and Chinese as judicious as among ourselves, expediency seemed to dictate that I should regulate my practice conformably to the opinions of those with whom I should have to live; and it appeared to me that, in order to ascertain the real opinions of such, I ought rather to take cognizance of what they practised than of what they said, not only because, in the corruption of our manners, there are few disposed to speak exactly as they believe, but also because very many are not aware of what it is that they really believe; for, as the act of mind by which a thing is believed is different from that by which we know that we believe it, the one act is often found without the other. Also, amid many opinions held in equal repute, I chose always the most moderate, as much for the reason that these are always the most convenient for practice, and probably the best (for all excess is generally vicious), as that, in the event of my falling into error, I might be at less distance from the truth than if, having chosen one of the extremes, it should turn out to be the other which I ought to have adopted. And I placed in the class of extremes especially all promises by which somewhat of our freedom is abridged; not that I disapproved of the laws which, to provide against the instability of men of feeble resolution, when what is sought to be accomplished is some good, permit engagements by vows and contracts binding the parties to persevere in it, or even, for the security of commerce, sanction similar engagements where the purpose sought to be realized is indifferent: but because I did not find anything on earth which was wholly superior to change, and because, for myself in particular, I hoped gradually to perfect my judgments, and not to suffer them to deteriorate, I would have deemed it a grave sin against good sense, if, for the reason that I approved of something at a particular time, I therefore bound myself to hold it for good at a subsequent time, when perhaps it had ceased to be so, or I had ceased to esteem it such.
My second maxim was to be as firm and resolute in my actions as I was able, and not to adhere less steadfastly to the most doubtful opinions, when once adopted, than if they had been highly certain; imitating in this the example of travelers who, when they have lost their way in a forest, ought not to wander from side to side, far less remain in one place, but proceed constantly towards the same side in as straight a line as possible, without changing their direction for slight reasons, although perhaps it might be chance alone which at first determined the selection; for in this way, if they do not exactly reach the point they desire, they will come at least in the end to some place that will probably be preferable to the middle of a forest. In the same way, since in action it frequently happens that no delay is permissible, it is very certain that, when it is not in our power to determine what is true, we ought to act according to what is most probable; and even although we should not remark a greater probability in one opinion than in another, we ought notwithstanding to choose one or the other, and afterwards consider it, in so far as it relates to practice, as no longer dubious, but manifestly true and certain, since the reason by which our choice has been determined is itself possessed of these qualities. This principle was sufficient thenceforward to rid me of all those repentings and pangs of remorse that usually disturb the consciences of such feeble and uncertain minds as, destitute of any clear and determinate principle of choice, allow themselves one day to adopt a course of action as the best, which they abandon the next, as the opposite.
(This is probably 5% of the text. There is more interesting stuff there, but it's less relevant to this post.)
Q5 is true if (as you assumed), the space of lotteries is the space of distributions over a finite set. (For a general convex set, you can get long-line phenomena.)
First, without proof, I'll state the following generalization.
Theorem 1. Let ⪯ be a relation on a convex space L satisfying axioms A1, A2, A3, and the following additional continuity axiom. For all A,B1,B2,C∈L, the set
{p∈[0,1]∣A≺pB1+(1−p)B2≺C}is open in [0,1]. Then, there exists a function u from L to the long line such that u(A)≤u(B) iff A⪯B.
The proof is not too different, but simpler, if we also assume A4. In particular, we no longer need the extra continuity axiom, and we get a stronger conclusion. Nate sketched part of the proof of this already, but I want to be clearer about what is stated and skip fewer steps. In particular, I'm not sure how Nate's hypotheses rule out examples that require long-line-valued functions—maybe he's assuming that the domain of the preference relation is a finite-dimensional simplex like I am, but none of his arguments use this explicitly.
Theorem 2. Let ⪯ be a relation on a finite-dimensional simplex L=ΔΩ satisfying axioms A1-A4. Then, there is a quasiconcave function u:L→R such that u(A)≤u(B) iff A⪯B.
First, I'll set up some definitions and a lemma. For any lotteries A, B, let [A,B] denote the line segment
{pA+(1−p)B∣p∈[0,1]}.We say that preferences are increasing along a line segment [A,B] if whenever p≤q, we have
(1−p)A+pB⪯(1−q)A+qB.We will also use open and half-open interval notation in the corresponding way.
Lemma. Let ⪯ be a preference relation on a finite-dimensional simplex L=ΔΩ satisfying axioms A1-A4. Then, there are ⪯-minimal and -maximal elements in L.
Proof. First, we show that there is a minimal element. Axiom A4 states that for any mixture C=pA+(1−p)B, either C⪰A or C⪰B. By induction, it follows more generally that any convex combination C of finitely many elements (Ai)i∈I satisfies C⪰Ai for some i∈I. But every element is a convex combination of the vertices of L, so some vertex of L is ⪯-minimal.
The proof that there is a maximal element is more complex. Consider the family of sets
F={{B∈L∣B⪰A}∣A∈L}.This is a prefilter, so since L is compact (L here carries the Euclidean metric), it has a cluster point B. Either B will be a maximal element, or we will find some other maximal element. In particular, take any A∈L. We are done if A is a maximal element; otherwise, pick A′≻A. By the construction of F, for every n∈N, we can pick some Cn⪰A′ within a distance of 1n from B. Now, if we show that B itself satisfies B⪰A, it will follow that B is maximal.
The idea is to pass from our sequence (Cn)n∈N, with limit B, to another sequence lying on a line segment with endpoint B. We can use axiom A4, which is a kind of convexity, to control the preference relation on convex combinations of our points Cn, so these are the points that we will construct along a line segment. Once we have this line segment, we can finish by using A3, which is a kind of continuity restricted to line segments, to control B itself.
Let S⊆L be the set of lotteries in the affine span of the set {Cn}n∈N. Then, if we take some index set I⊆N such that (Cn)n∈I is a maximal affinely independent tuple, it follows that {Cn}n∈I affinely generates S. Hence, the convex combination
D=∑n∈I1|I|Cn,i.e. the barycenter of the simplex with vertices at (Cn)n∈I, is in the interior of the convex hull of {Cn}n∈I relative to S, so we can pick some r>0 such that the r-ball around D relative to S is contained in this simplex.
Now, we will see that every lottery E in the set (B,D] satisfies E⪰A′. For any ε>0, pick k so that Ck is in the ε-ball around B. Since the tangent vector v=B−Ck has length less than ε, the lottery
F=D+rε(B−Ck)is in the r-ball around D, and it is in S, so it is in the simplex with vertices (Cn)n∈I. Then, F⪰A′ by A4, and Ck⪰A′ by hypothesis. So, applying A4 again,
A′⪯rr+εCk+εr+εF=rr+εB+εr+εD.Using A4 one more time, it follows that every lottery
E∈[rr+εB+εr+εD,D]satisfies E⪰A′, and hence every lottery E∈(B,D].
Now we can finish up. If B≺A then, using A3 and the fact that D⪰A′≻A, there would have to be some lottery in [B,D] that is ⪯-equivalent to A, but this would contradict what we just concluded. So, B⪰A, and so B is ⪯-maximal. □
Proof of Theorem 2. Let C be a ⪯-minimal and D a ⪯-maximal element of L. First, we will see that preferences are increasing on [C,D], and then we will use this fact to construct a function L→R and show that it has the desired properties. Suppose preferences we not increasing; then, there would be A,B∈[C,D] such that A is closer to C while B is closer to D, and A≻B. Then, B would be a convex combination of A and D, but B≺A⪯D by the maximality of D, contradicting A4.
Now we can construct our utility function u:L→R using A3; for each ∼-class [A], we have C⪯A⪯D, so there is some[1] p∈[0,1] such that
(1−p)C+pD∼A.Then, let u(A′)=p for all A′∈[A]. Since preferences are increasing on [C,D], it is immediate that if u(A)≤u(B), then A⪯B. Conversely, if A⪯B, we have two cases. If A≺B, then B⋠A, so u(B)≰u(A), and so u(A)≤u(B). Finally, if A∼B, then u(A)=u(B) by construction.
Finally, since for all A,B∈L we have u(A)≤u(B) iff A⪯B, it follows immediately that u is quasiconcave by A4. □
Nate mentions using choice in his answer, but here at least the use of choice is removable. Since ⪯ is monotone on [C,D], the intersection of the ∼-class [A] with [C,D] is a subinterval of [C,D], so we can pick p based on the midpoint of that interval