riceissa(Issa Rice)

• riceissa 20 Dec 2019 8:35 UTC

  44 points

  on: We run the Cen­ter for Ap­plied Ra­tion­al­ity, AMA

  What are your thoughts on Dun­can Sa­bien’s Face­book post which pre­dicts sig­nifi­cant differ­ences in CFAR’s di­rec­tion now that he is no longer work­ing for CFAR?

• riceissa 20 Dec 2019 4:35 UTC

  33 points

  on: We run the Cen­ter for Ap­plied Ra­tion­al­ity, AMA

  Back in April, Oliver Habryka wrote:

  Anna Sala­mon has re­duced her in­volve­ment in the last few years and seems sig­nifi­cantly less in­volved with the broader strate­gic di­rec­tion of CFAR (though she is still in­volved in some of the day-to-day op­er­a­tions, cur­ricu­lum de­vel­op­ment, and more re­cent CFAR pro­gram­mer work­shops). [Note: After talk­ing to Anna about this, I am now less cer­tain of whether this ac­tu­ally ap­plies and am cur­rently con­fused on this point]

  Could some­one clar­ify the situ­a­tion? (Pos­si­ble sub-ques­tions: Why did Oliver get this im­pres­sion? Why was he con­fused even af­ter talk­ing talk­ing to Anna? To what ex­tent and in what ways has Anna re­duced her in­volve­ment in CFAR in the last few years? If Anna has re­duced her in­volve­ment in CFAR, what is she spend­ing her time on in­stead?)

• riceissa 28 Oct 2014 7:52 UTC

  31 points

  on: 2014 Less Wrong Cen­sus/​Survey

  I took the sur­vey.

• riceissa 20 Dec 2019 9:00 UTC

  24 points

  on: We run the Cen­ter for Ap­plied Ra­tion­al­ity, AMA

  I have seen/​heard from at least two sources some­thing to the effect that MIRI/​CFAR lead­er­ship (and Anna in par­tic­u­lar) has very short AI timelines and high prob­a­bil­ity of doom (and ap­par­ently hav­ing high con­fi­dence in these be­liefs). Here is the only pub­lic ex­am­ple that I can re­call see­ing. (Of the two ex­am­ples I can speci­fi­cally re­call, this is not the bet­ter one, but the other was not posted pub­li­cly.) Is there any truth to these claims?

• riceissa 28 Sep 2019 0:38 UTC

  LW: 23 AF: 7

  AF

  on: The strat­egy-steal­ing assumption

  Like Wei Dai, I am also find­ing this dis­cus­sion pretty con­fus­ing. To sum­ma­rize my state of con­fu­sion, I came up with the fol­low­ing list of ways in which prefer­ences can be short or long:

  1. time hori­zon and time dis­count­ing: how far in the fu­ture is the prefer­ence about? More gen­er­ally, how much weight do we place on the pre­sent vs the fu­ture?

  2. act-based (“short”) vs goal-based (“long”): us­ing the hu­man’s (or more gen­er­ally, the hu­man-plus-AI-as­sis­tants’; see (6) be­low) es­ti­mate of the value of the next ac­tion (act-based) or do­ing more open-ended op­ti­miza­tion of the fu­ture based on some goal, e.g. us­ing a util­ity func­tion (goal-based)

  3. amount of re­flec­tion the hu­man has un­der­gone: “short” would be the cur­rent hu­man (I think this is what you call “prefer­ences-as-elic­ited”), and this would get “longer” as we give the hu­man more time to think, with some­thing like CEV/​Long Reflec­tion/​Great De­liber­a­tion be­ing the “longest” in this sense (I think this is what you call “prefer­ence-on-ideal­ized-re­flec­tion”). This sense fur­ther breaks down into whether the hu­man it­self is ac­tu­ally do­ing the re­flec­tion, or if the AI is in­stead pre­dict­ing what the hu­man would think af­ter re­flec­tion.

  4. how far the search hap­pens: “short” would be a limited search (that lacks in­sight/​doesn’t see in­ter­est­ing con­se­quences) and “long” would be a search that has in­sight/​sees in­ter­est­ing con­se­quences. This is a dis­tinc­tion you made in a dis­cus­sion with Eliezer a while back. This dis­tinc­tion also isn’t strictly about prefer­ences, but rather about how one would achieve those prefer­ences.

  5. de dicto (“short”) vs de re (“long”): This is a dis­tinc­tion you made in this post. I think this is the same dis­tinc­tion as (2) or (3), but I’m not sure which. (But if my in­ter­pre­ta­tion of you be­low is cor­rect, I guess this must be the same as (2) or else a com­pletely differ­ent dis­tinc­tion.)

  6. un­der­stand­able (“short”) vs evaluable (“long”): A course of ac­tion is un­der­stand­able if the hu­man (with­out any AI as­sis­tants) can un­der­stand the ra­tio­nale be­hind it; a course of ac­tion is evaluable if there is some pro­ce­dure the hu­man can im­ple­ment to eval­u­ate the ra­tio­nale us­ing AI as­sis­tants. I guess there is also a “not even evaluable” op­tion here that is even “longer”. (Thanks to Wei Dai for bring­ing up this dis­tinc­tion, al­though I may have mi­s­un­der­stood the ac­tual dis­tinc­tion.)

  My in­ter­pre­ta­tion is that when you say “short-term prefer­ences-on-re­flec­tion”, you mean short in sense (1), ex­cept when the AI needs to gather re­sources, in which case ei­ther the hu­man or the AI will need to do more long-term plan­ning; short in sense (2); long in sense (3), with the AI pre­dict­ing what the hu­man would think af­ter re­flec­tion; long in sense (4); short in sense (5); long in sense (6). Does this sound right to you? If not, I think it would help me a lot if you could “fill in the list” with which of short or long you choose for each point.

  As­sum­ing my in­ter­pre­ta­tion is cor­rect, my con­fu­sion is that you say we shouldn’t ex­pect a situ­a­tion where “the user-on-re­flec­tion might be happy with the level of cor­rigi­bil­ity, but the user them­selves might be un­happy” (I take you to be talk­ing about sense (3) from above). It seems like the user-on-re­flec­tion and the cur­rent user would dis­agree about many things (that is the whole point of re­flec­tion), so if the AI acts in ac­cor­dance with the in­ten­tions of the user-on-re­flec­tion, the cur­rent user is likely to end up un­happy.

  What links here?

  • De­liber­a­tion as a method to find the “ac­tual prefer­ences” of humans by riceissa (22 Oct 2019 9:23 UTC; 24 points)

• riceissa 20 Dec 2019 23:44 UTC

  15 points

  on: We run the Cen­ter for Ap­plied Ra­tion­al­ity, AMA

  I had already seen all of those quotes/​links, all of the quotes/​links that Rob Bens­inger posts in the sibling com­ment, as well as this tweet from Eliezer. I asked my ques­tion be­cause those pub­lic quotes don’t sound like the pri­vate in­for­ma­tion I referred to in my ques­tion, and I wanted in­sight into the dis­crep­ancy.

• riceissa 28 Jul 2019 21:06 UTC

  15 points

  on: AALWA: Ask any LessWronger anything

  I was sur­prised to see, both on your web­site and the white pa­per, that you are part of Mer­ca­to­ria/​ICTP (al­though your level of in­volve­ment isn’t clear based on pub­lic in­for­ma­tion). My sur­prise is mainly be­cause you have a cou­ple of com­ments on LessWrong that dis­cuss why you have de­clined to join MIRI as a re­search as­so­ci­ate. You have also (to my knowl­edge) never joined any other ra­tio­nal­ity-com­mu­nity or effec­tive al­tru­ism-re­lated or­ga­ni­za­tion in any ca­pac­ity.

  My ques­tions are:

  1. What are the rea­sons you de­cided to join or sign on as a co-au­thor for Mer­ca­to­ria/​ICTP?

  2. More gen­er­ally, how do you de­cide which or­ga­ni­za­tions to as­so­ci­ate with? Have you con­sid­ered join­ing other or­ga­ni­za­tions, start­ing your own or­ga­ni­za­tion, or re­cruit­ing con­tract work­ers/​vol­un­teers to work on things you con­sider im­por­tant?

• riceissa 14 May 2019 20:04 UTC

  15 points

  on: Flash­cards for AI Safety?

  I’ve made around 250 Anki cards about AI safety. I haven’t pri­ori­tized shar­ing my cards be­cause I think find­ing a spe­cific card use­ful re­quires some­one to have read the source ma­te­rial gen­er­at­ing the card (e.g. if I made the card based on a blog post, one would need to read that ex­act blog post to get value out of re­view­ing the card; see learn be­fore you mem­o­rize). Since there are many AI safety blog posts and I don’t have the sense that lots of Anki users read any par­tic­u­lar blog post, it seems to me that the value gen­er­ated from shar­ing a set of cards about a blog post isn’t high enough to over­come the an­noy­ance cost of pol­ish­ing, pack­ag­ing, and up­load­ing the cards.

  More gen­er­ally, from a con­sumer per­spec­tive, I think peo­ple tend to be pretty bad at mak­ing good Anki cards (I’m of­ten em­bar­rassed at the cards I’ve cre­ated sev­eral months ago!), which makes it un­ex­cit­ing for me to spend a lot of effort try­ing to col­lab­o­rate with oth­ers on mak­ing cards (be­cause I ex­pect to re­ceive poorly-made cards in re­turn for the cards I provide). I think col­lab­o­ra­tive card-mak­ing can be done though, e.g. Michael Niel­sen and Andy Ma­tuschak’s quan­tum com­put­ing guide comes with pre-made cards that I think are pretty good.

  Differ­ent peo­ple also have differ­ent goals/​in­ter­ests so even given a sin­gle source ma­te­rial, the speci­fics one wants to Ankify can be differ­ent. For ex­am­ple, some­one who wants to un­der­stand the tech­ni­cal de­tails of log­i­cal in­duc­tion will want to Ankify the com­mon ob­jects used (mar­ket, pric­ing, trader, val­u­a­tion fea­ture, etc.), the the­o­rems and proof tech­niques, and so forth, whereas some­one who just wants a high-level overview and the “so what” of log­i­cal in­duc­tion can get away with Ankify­ing much less de­tail.

  Some­thing I’ve no­ticed is that many AI safety posts aren’t very good at ex­plain­ing things (not enough con­crete ex­am­ples, not enough em­pha­sis on com­mon mis­con­cep­tions and edge cases, not enough effort to an­swer what I think of as “ob­vi­ous” ques­tions); this fact is of­ten re­vealed by the com­ments peo­ple make in re­sponse to a post. This makes it hard to make Anki cards be­cause one doesn’t re­ally un­der­stand the con­tent of the post, at least not well enough to con­fi­dently gen­er­ate Anki cards (one of the benefits of be­ing an Anki user is hav­ing a greater sen­si­tivity to when one does not un­der­stand some­thing; see “illu­sion of ex­plana­tory depth” and re­lated terms). There are other prob­lems like con­flict­ing us­age of ter­minol­ogy (e.g. mul­ti­ple defi­ni­tions of “be­nign”, “al­igned”, “cor­rigible”) and the fact that some of the de­bates are on­go­ing/​some of the knowl­edge is still be­ing worked out.

  For “What would be a good strat­egy for gen­er­at­ing use­ful flash­cards?”: I try to read a post or a se­ries of posts and once I feel that I un­der­stand the ba­sic idea, I will usu­ally reread it to add cards about the ba­sic terms and ask my­self sim­ple ques­tions. Some ex­am­ple cards for iter­ated am­plifi­ca­tion:

  • what kind of train­ing does the Distill step use?

  • in the pseu­docode, what step gets re­peated/​iter­ated?

  • how do we get A[0]?

  • write A[1] in terms of H and A[0]

  • when Paul says IDA is go­ing to be com­pet­i­tive with tra­di­tional RL agents in terms of time and re­source costs, what ex­actly does he mean?

  • ad­van­tages of A[0] over H

  • sym­bolic ex­pres­sion for the overseer

  • why should the am­plified sys­tem (of hu­man + mul­ti­ple copies of the AI) be ex­pected to perform bet­ter than the hu­man alone?

• riceissa 25 Apr 2019 14:33 UTC

  15 points

  on: When is ra­tio­nal­ity use­ful?

  We can model suc­cess as a com­bi­na­tion of do­ing use­ful things and avoid­ing mak­ing mis­takes. As a par­tic­u­lar ex­am­ple, we can model in­tel­lec­tual suc­cess as a com­bi­na­tion of com­ing up with good ideas and avoid­ing bad ideas. I claim that ra­tio­nal­ity helps us avoid mis­takes and bad ideas, but doesn’t help much in gen­er­at­ing good ideas and use­ful work.

  Eliezer Yud­kowsky has made similar points in e.g. “Un­teach­able Ex­cel­lence” (“much of the most im­por­tant in­for­ma­tion we can learn from his­tory is about how to not lose, rather than how to win”, “It’s eas­ier to avoid du­pli­cat­ing spec­tac­u­lar failures than to du­pli­cate spec­tac­u­lar suc­cesses. And it’s of­ten eas­ier to gen­er­al­ize failure be­tween do­mains.”) and “Teach­ing the Un­teach­able”.

• riceissa 28 Jan 2020 6:06 UTC

  14 points

  on: The Episte­mol­ogy of AI risk

  As should be clear, this pro­cess can, af­ter a few iter­a­tions, pro­duce a situ­a­tion in which most of those who have en­gaged with the ar­gu­ments for a claim be­yond some depth be­lieve in it.

  This isn’t clear to me, given the model in the post. If a claim is false and there are suffi­ciently many ar­gu­ments for the claim, then it seems like ev­ery­one even­tu­ally ends up re­ject­ing the claim, in­clud­ing those who have en­gaged most deeply with the ar­gu­ments. The peo­ple who en­gage deeply “got lucky” by hear­ing the most per­sua­sive ar­gu­ments first, but even­tu­ally they also hear the weaker ar­gu­ments and coun­ter­ar­gu­ments to the claim, so they end up at a level of con­fi­dence where they don’t feel they should bother in­ves­ti­gat­ing fur­ther. Th­ese peo­ple can even have more ac­cu­rate be­liefs than the peo­ple who dropped out early in the pro­cess, de­pend­ing on the cut­off that is cho­sen.

• riceissa 3 Dec 2019 9:22 UTC

  14 points

  on: Open & Wel­come Thread—De­cem­ber 2019

  When think­ing about in­for­ma­tion asym­me­try in trans­ac­tions (e.g. in­surance, mar­ket for lemons), I can think of sev­eral axes for com­par­i­son:

  1. whether the thing hap­pens be­fore or af­ter the transaction

  2. whether the buyer or sel­ler has more information

  3. whether the asym­me­try is about a good or about an action

  In­surance-like trans­ac­tions pick “af­ter”, “buyer”, and “ac­tion”: the per­son buy­ing the in­surance can choose to act more care­lessly af­ter pur­chas­ing the in­surance.

  Mar­ket for lemons cases pick “be­fore”, “sel­ler”, and “good”: prior to the trans­ac­tion, the sel­ler of the good knows more about the qual­ity of the good.

  So in many typ­i­cal cases, the three axes “al­ign”, and the former is called a moral haz­ard and the lat­ter is called ad­verse se­lec­tion.

  But there are ex­am­ples like an all-you-can-eat buf­fet that sets a sin­gle price (which en­courages high-ap­petite peo­ple to eat there). This case picks “be­fore”, “buyer”, and “ac­tion”. So in this case, ²⁄₃ of the axes agree with the in­surance-like situ­a­tion, but this case is still clas­sified as ad­verse se­lec­tion be­cause the offi­cial dis­tinc­tion is about (1).

  Wikipe­dia states “Where ad­verse se­lec­tion de­scribes a situ­a­tion where the type of product is hid­den from one party in a trans­ac­tion, moral haz­ard de­scribes a situ­a­tion where there is a hid­den ac­tion that re­sults from the trans­ac­tion” (i.e. claims that (3) is the rele­vant axis) but then on the same page also states “For ex­am­ple, an all-you-can-eat buf­fet restau­rant that sets one price for all cus­tomers risks be­ing ad­versely se­lected against by high ap­petite” (i.e. clas­sifies this ex­am­ple as ad­verse se­lec­tion, even though clas­sify­ing ac­cord­ing to (3) would re­sult in call­ing this a moral haz­ard).

  Does any­one know why (1) is the most in­ter­est­ing axis (which I’m in­fer­ring based on how only this axis seems to have names for the two ends)?

• riceissa 19 Nov 2018 2:33 UTC

  LW: 14 AF: 4

  AF

  on: Topolog­i­cal Fixed Point Exercises

  My solu­tion for #3:

  Define $g:[0,1]\to R$ by $g\left(x\right)=f\left(x\right)-x$. We know that $g$ is con­tin­u­ous be­cause $f$ and the iden­tity map both are, and by the limit laws. Ap­ply­ing the in­ter­me­di­ate value the­o­rem (prob­lem #2) we see that there ex­ists $x\in [0,1]$ such that $g\left(x\right)=0$. But this means $f\left(x\right)=x$, so we are done.

  Coun­terex­am­ple for the open in­ter­val: con­sider $f:(0,1)\to (0,1)$ defined by $f\left(x\right)=x/2$. First, we can ver­ify that if $0<x<1$ then $0<x/2<1/2<1$, so $f$ in­deed maps to $(0,1)$. To see that there is no fixed point, note that the only solu­tion to $x/2=x$ in $R$ is $0$, which is not in $(0,1)$. (We can also view this graph­i­cally by plot­ting both $y=x$ and $y=x/2$ and check­ing that they do not in­ter­sect in $(0,1)$.)

• riceissa 12 Feb 2020 21:20 UTC

  13 points

  on: Did AI pi­o­neers not worry much about AI risks?

  Some other rele­vant links:

  His­tory of AI risk thought

  AI Risk & Op­por­tu­nity: A Timeline of Early Ideas and Arguments

  AI Risk and Op­por­tu­nity: Hu­man­ity’s Efforts So Far

  Quotes ex­perts about AI

• riceissa 17 Sep 2019 23:04 UTC

  13 points

  on: Why Subagents?

  When I ini­tially read this post, I got the im­pres­sion that “sub­agents = path-de­pen­dent/​in­com­plete DAG”. After work­ing through more ex­am­ples, it seems like all the work is be­ing done by “com­mit­tee re­quiring unan­i­mous agree­ment” rather than by the “sub­agents” part.

  Here are the ex­am­ples I thought about:

  1. Same as the mush­room/​pep­per­oni situ­a­tion, with the same two agents, but now each side can re­tal­i­ate/​hi­jack the rest of the mind if it doesn’t get what it wants. For ex­am­ple, if it starts at pep­per­oni, the mush­room-prefer­ring agent will hi­jack the rest of the mind to re­move the pep­per­oni, end­ing up at cheese. But if the agent starts at the “both” node, it will stay there (be­cause both agents are satis­fied). The prefer­ence re­la­tion can be rep­re­sented as $\text{pepperoni}\to \text{cheese}\leftarrow \text{mushroom}$ with an ex­tra ar­row from $\text{cheese}\to \text{both}$. This is still a DAG, and it’s still in­com­plete (in the sense that we can’t com­pare pep­per­oni vs mush­room) but it’s no longer path-de­pen­dent, be­cause no mat­ter where we start, we end up at cheese or “both” (I am as­sum­ing that top­pings-re­moval can always be done, whereas ac­quiring new top­pings can’t).

  2. Same as the pre­vi­ous ex­am­ple, ex­cept now only the mush­room-prefer­ring agent can re­tal­i­ate/​hi­jack (be­cause the pep­per­oni-prefer­ring agent is weak or nice). Now the prefer­ences are $\text{pepperoni}\to \text{cheese}\to \text{mushroom}\to \text{both}$. This is still a DAG, but now the prefer­ences are to­tal, so we can also view it as a (some­what weird) sin­gle agent. A re­al­is­tic ex­am­ple of this is given by An­drew Critch, where pep­per­oni=work, cheese=burnout (i.e. nei­ther work nor friend­ship), mush­room=friend­ship, and both=friend­ship-and-work.

  3. A mod­ified ver­sion of the Zyzzx Prime planet by Scott Alexan­der. Now when­ever we start out at pep­per­oni, the pep­per­oni-prefer­ring agent be­comes stupid/​weak, and loses dom­i­nance, so now there are edges from pep­per­oni to mush­room and “both”. (And similarly, mush­room points to both pep­per­oni and “both”.) Now we no longer have a DAG be­cause of the cy­cle be­tween pep­per­oni and mush­room.

  It seems like when peo­ple talk about the hu­man mind be­ing com­posed of sub­agents, the de­liber­a­tion pro­cess is not nec­es­sar­ily “com­mit­tee re­quiring unan­i­mous agree­ment”, so the re­sult­ing prefer­ence re­la­tions can­not nec­es­sar­ily be rep­re­sented us­ing path-de­pen­dent DAGs.

  It also seems like the gen­eral frame­work of view­ing sys­tems as sub­agents (i.e. not re­strict­ing to “com­mit­tee re­quiring unan­i­mous agree­ment”) is broad enough that it can ba­si­cally rep­re­sent any kind of di­rected graph. On one hand, this is sus­pi­cious (if ev­ery­thing can be viewed as a bunch of sub­agents, then maybe the sub­agents frame­work isn’t adding any­thing af­ter all). On the other hand, this sug­gests that claims of sub­agents are not re­ally about the re­sult­ing be­hav­ior/​prefer­ence or­der­ing of the sys­tem, but rather about the in­ter­nal dy­nam­ics of the sys­tem.

• riceissa 3 Jan 2019 7:13 UTC

  12 points

  on: What ex­er­cises go best with 3 blue 1 brown’s Lin­ear Alge­bra videos?

  Some other sources of ex­er­cises you might want to check out (that have solu­tions and that I have used at least partly):

  • Mul­ti­ple choice quizzes (the ones re­lated to lin­ear alge­bra are de­ter­mi­nants, el­e­men­tary ma­tri­ces, in­ner product spaces, lin­ear alge­bra, lin­ear sys­tems, lin­ear trans­for­ma­tions, ma­tri­ces, and vec­tor spaces)

  • Vipul Naik’s quizzes (dis­clo­sure: I am friends with Vipul and also do con­tract work for him)

  Re­gard­ing Axler’s book (since it has been men­tioned in this thread): there are sev­eral “lev­els” of lin­ear alge­bra, and Axler’s book is at a higher level (em­pha­sis on ab­stract vec­tor spaces and co­or­di­nate-free ways of do­ing things) than the 3Blue1Brown videos (more con­crete, work­ing in $R^n$). Axler’s book also as­sumes that the reader has had ex­po­sure to the lower level ma­te­rial (e.g. he does not talk about row re­duc­tion and el­e­men­tary ma­tri­ces). So I’m not sure I would recom­mend it to some­one start­ing out try­ing to learn the ba­sics of lin­ear alge­bra.

  Gra­tu­itous re­marks:

  • I think differ­ent re­sources cov­er­ing ma­te­rial in a differ­ent or­der and us­ing differ­ent ter­minol­ogy is in some sense a fea­ture, not a bug, be­cause it al­lows one to look at the sub­ject from differ­ent per­spec­tives. For in­stance, the “done right” in Axler’s book comes from one such change in per­spec­tive.

  • I find that learn­ing math­e­mat­ics well takes an un­in­tu­itively long time; it might be un­re­al­is­tic to ex­pect to learn the ma­te­rial well un­less one puts in a lot of effort.

  • I think there is a case to be made for the im­por­tance of strug­gling in learn­ing (dis­clo­sure: I am the au­thor of the page).

• riceissa 19 Nov 2018 1:54 UTC

  LW: 12 AF: 2

  AF

  on: Topolog­i­cal Fixed Point Exercises

  Here is my at­tempt, based on Hoagy’s proof.

  Let $n\ge 1$ be an in­te­ger. We are given that $f\left(0/n\right)=f\left(0\right)\le 0$ and $f\left(n/n\right)=f\left(1\right)\ge 0$. Now con­sider the points $0/n,1/n,\dots ,(n-1)/n,n/n$ in the in­ter­val $[0,1]$. By 1-D Sperner’s lemma, there are an odd num­ber of $j\in \{0,\dots ,n\}$ such that $f\left(j/n\right)\ge 0$ and $f\left(\right(j-1\left)/n\right)\le 0$ (i.e. an odd num­ber of “seg­ments” that be­gin be­low zero and end up above zero). In par­tic­u­lar, $0$ is an even num­ber, so there must be at least one such num­ber $j$. Choose the small­est and call this num­ber $j_n$.

  Now con­sider the se­quence $(j_n/n{)}_{n=1}^{\infty }$. Since this se­quence takes val­ues in $[0,1]$, it is bounded, and by the Bolzano–Weier­strass the­o­rem there must be some sub­se­quence $(k_n/n{)}_{n=1}^{\infty }$ that con­verges to some num­ber $x\in [0,1]$.

  Con­sider the se­quences $\left(f\right((k_n-1)/n){)}_{n=1}^{\infty }$ and $\left(f\right(k_n/n){)}_{n=1}^{\infty }$. We have $f\left(\right(k_n-1\left)/n\right)\le 0\le f\left(k_n/n\right)$ for each $n\ge 1$. By the limit laws, $(k_n-1)/n\to x$ as $n\to \infty$. Since $f$ is con­tin­u­ous, we have $f\left(\right(k_n-1\left)/n\right)\to f\left(x\right)$ and $f\left(k_n/n\right)\to f\left(x\right)$ as $n\to \infty$. Thus $f\left(x\right)\le 0$ and $f\left(x\right)\ge 0$, show­ing that $f\left(x\right)=0$, as de­sired.

• riceissa 16 Jan 2020 7:42 UTC

  11 points

  on: Open & Wel­come Thread—Jan­uary 2020

  I no­ticed that the par­li­a­men­tary model of moral un­cer­tainty can be framed as try­ing to im­port a “group ra­tio­nal­ity” mechanism into the “in­di­vi­d­ual ra­tio­nal­ity” set­ting, to deal with sub­agents/​sub­pro­cesses that ap­pear in the in­di­vi­d­ual set­ting. But usu­ally when the in­di­vi­d­ual ra­tio­nal­ity vs group ra­tio­nal­ity topic is brought up, it is to talk about how group ra­tio­nal­ity is much harder/​less un­der­stood than in­di­vi­d­ual ra­tio­nal­ity (here are two ex­am­ples of what I mean). I can’t quite ex­plain it, but I find it in­ter­est­ing/​counter-in­tu­itive/​para­dox­i­cal that given this gen­eral back­ground, there is a re­ver­sal here, where a solu­tion in the group ra­tio­nal­ity set­ting is be­ing im­ported to the in­di­vi­d­ual ra­tio­nal­ity set­ting. (I think this might be re­lated to why I’ve never found the par­li­a­men­tary model quite con­vinc­ing, but I’m not sure.)

  Has any­one thought about this, or more gen­er­ally about trans­fer­ring mechanisms be­tween the two set­tings?

• riceissa 14 Apr 2019 19:57 UTC

  11 points

  on: Di­ag­o­nal­iza­tion Fixed Point Exercises

  Thoughts on #10:

  I am con­fused about this ex­er­cise. The stan­dard/​mod­ern proof of Gödel’s sec­ond in­com­plete­ness the­o­rem uses the Hilbert–Ber­nays–Löb deriv­abil­ity con­di­tions, which are stated as (a), (b), (c) in ex­er­cise #11. If the ex­er­cises are meant to be solved in se­quence, this seems to im­ply that #10 is solv­able with­out us­ing the deriv­abil­ity con­di­tions. I tried do­ing this for a while with­out get­ting any­where.

  Maybe an­other way to state my con­fu­sion is that I’m pretty sure that up to ex­er­cise #10, noth­ing that dis­t­in­guishes Peano ar­ith­metic from Robin­son ar­ith­metic has been in­tro­duced (it is only with the in­tro­duc­tion of the deriv­abil­ity con­di­tions in #11 that this differ­ence be­comes ap­par­ent). It looks like there is a ver­sion of the sec­ond in­com­plete­ness the­o­rem for Robin­son ar­ith­metic, but the pa­per says “The proof is by the con­struc­tion of a non­stan­dard model in which this for­mula [i.e. for­mula ex­press­ing con­sis­tency] is false”, so I’m guess­ing this proof won’t work for Peano ar­ith­metic.

• riceissa 14 Apr 2019 19:41 UTC

  11 points

  on: Di­ag­o­nal­iza­tion Fixed Point Exercises

  My solu­tion for #12:

  Sup­pose for the sake of con­tra­dic­tion that such a for­mula $\varphi$ ex­ists. By the di­ag­o­nal lemma ap­plied to $\neg \varphi$, there is some sen­tence $\psi$ such that, prov­ably, $\neg \varphi (┌\psi ┐)\leftrightarrow \psi$. By the sound­ness of our the­ory, in fact $\neg \varphi (┌\psi ┐)\leftrightarrow \psi$. But by the prop­erty for $\varphi$ we also have $\varphi (┌\psi ┐)\leftrightarrow \psi$, which means $\neg \varphi (┌\psi ┐)\leftrightarrow \varphi (┌\psi ┐)$, a con­tra­dic­tion.

  This seems to be the “se­man­tic” ver­sion of the the­o­rem, where the prop­erty for $\varphi$ is stated out­side the sys­tem. There is also a “syn­tac­tic” ver­sion where the prop­erty for $\varphi$ is stated within the sys­tem.

• riceissa 19 Nov 2018 1:29 UTC

  LW: 11 AF: 2

  AF

  on: Topolog­i­cal Fixed Point Exercises

  I’m hav­ing trou­ble un­der­stand­ing why we can’t just fix $n=2$ in your proof. Then at each iter­a­tion we bi­sect the in­ter­val, so we wouldn’t be us­ing the “full power” of the 1-D Sperner’s lemma (we would just be us­ing some­thing close to the base case).

  Also if we are only given that $f$ is con­tin­u­ous, does it make sense to talk about the gra­di­ent?