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Viliam - LessWrong 2.0 viewerxml-emitteren-usComment by Viliam on John_Maxwell's Shortform
https://www.greaterwrong.com/posts/fagKRreeqwsb7xG4p/john_maxwell-s-shortform#comment-WZgQ6RWf4pdHCfSvE
<p>Sequences: <a href="/tag/beisutsukai">Beisutsukai</a></p><p>One year later: <a href="/posts/LgavAYtzFQZKg95WC/extreme-rationality-it-s-not-that-great">Extreme Rationality: It’s Not That Great</a></p>ViliamWZgQ6RWf4pdHCfSvEThu, 04 Mar 2021 21:04:23 +0000Comment by Viliam on Open & Welcome Thread – March 2021
https://www.greaterwrong.com/posts/CtAPvY7zz8FFheXSQ/open-and-welcome-thread-march-2021#comment-HPuCbiyHgg3GDttAP
<blockquote><p>I am interested in how learning rationalist techniques could help me navigate making complex life choices.</p></blockquote><p>Have you already found some useful answers?</p><p>In my opinion, the greatest value usually comes from a few rather simple techniques, assuming that you <i>actually do them</i>. In other words, the typical failures are “not even trying” and “thinking about what needs to be done (and reading about it), but never actually doing it”.</p><p>A few simple techniques:</p><ul><li><p>actually spend 5 minutes by clock trying to answer the question</p></li><li><p>ask a smart and trusted person</p></li><li><p>use Google (to find answers to factual questions)</p></li><li><p>write a letter to an imaginary smart advisor, imagine their response (e.g. asking for more details, or making an obvious conclusion “well if you say X is better, why aren’t you already doing X?”), write another letter, etc.</p></li><li><p>imagine it is actually your friend having the problem, what would you advise them?</p></li><li><p>imagine you chose X and later you regretted the outcome. why? (say the most likely reason)</p></li><li><p>decide by flipping a coin (actually flip the coin), now contemplate how you feel about the outcome</p></li></ul>ViliamHPuCbiyHgg3GDttAPThu, 04 Mar 2021 20:29:45 +0000Comment by Viliam on Naïve Set Theory - Part 1: Construction of Sets
https://www.greaterwrong.com/posts/qGAYpZRHkbvKRDkqc/naive-set-theory-part-1-construction-of-sets#comment-cjPhxuscG4Qk8BttM
<p>Yes, by “urelements” I meant “elements that are not sets”. However, this was just a different way to express the question that if we treat <i>a, b, c…</i> as <strong>black boxes</strong>, i.e. ignoring the question of what is inside them; whether from the facts that <i>a</i> is a set, <i>b</i> is a set, <i>c</i> is a set… inevitably follows that <i>{a, b, c...}</i> is also a set in given model of set theory.</p><p>For example, using the axiom of Pair, we can prove that <i>{a, b}</i> or <i>{g, x}</i> are also sets. Using also the axiom of Union, we can prove that any <i>finite</i> collections, such as <i>{a, b, g, h, x}</i> are sets. But using Pair and Union alone, we cannot prove that about any <i>infinite</i> collection.</p><p>The axiom of Infinity only proves the existence of <i>one</i> specific infinite set: ω.</p><p>My guess is that using all axioms together, we can only prove existence of sets that involve a finite number of <i>a, b, c...</i></p><p>This needs to be put a bit more precisely, though, because by the very fact of them being an infinitely descending chain, including one of them means involving an infinite amount of them; for example <i>{{}, c}</i> is simultaneously <i>{{}, {d}}</i>, which is also <i>{{}, {{e}}}</i>, etc. But suppose that we stop expanding the expression whenever we hit one of these chain-sets. Thus, the set <i>{{}, c}</i> would involve <i>c</i> <strong>explicitly</strong>, but <i>d, e…</i> only <strong>implicitly</strong>.</p><p>Now we can rephrase my guess, that using all axioms together, we can only prove existence of sets that involve <i>explicitly</i> a <i>finite</i> number of <i>a, b, c…</i> That means, <i>{a, b, c...}</i> is not among them.</p><p>Axioms of Existence, Infinity, Pair, and Powerset do not increase the number of <i>a, b, c…</i> used explicitly. Axioms of Union, Comprehension can only “unpack” these sets one level deeper. -- Union applied to <i>c</i> would only prove the existence of <i>d</i>. Comprehension applied to <i>c</i> would result in either empty set or <i>d</i>. Each of them applied to a structure that explicitly contains a finite number of <i>a, b, c…</i> would result in a structure that also explicitly contains a finite, albeit maybe twice larger, number of <i>a, b, c...</i></p><p>Axiom of Foundation should be okay with such structures, because even if you have a set containing <i>finitely</i> many of <i>a, b, c...</i>, just choose the one furthest down the chain. For example, for set <i>{{{}}, a, c, e}</i>, choose <i>e = {f}</i>, and the intersection is empty.</p><p>How to construct the universe: Take a ZF universe, and for each set in that universe also create all possible sets that replace some of the empty sets in the structure by sets <i>a, b, c…</i> under condition that during each replacement you only use a finite number of <i>a, b, c…</i> (but with each of the selected finite few, you can replace an arbitrary, finite or infinite, number of the empty sets in the original structure). My hypothesis is that this, together with the rules <i>a = {b}, b = {c}…</i> is also a ZF universe, and it is one containing an infinite descending chain of sets (<i>a, b, c...</i>).</p>ViliamcjPhxuscG4Qk8BttMThu, 04 Mar 2021 18:48:17 +0000Comment by Viliam on Naïve Set Theory - Part 1: Construction of Sets
https://www.greaterwrong.com/posts/qGAYpZRHkbvKRDkqc/naive-set-theory-part-1-construction-of-sets#comment-3czZmsrompD7jpxBM
<p>Yes, you understood my question correctly… and I need to spend some time thinking about your answer. (Mostly because it’s past midnight here, so I am leaving my computer for now.)</p><p>Thank you! It was a pleasure to be understood—unlike when I e.g. post a question on Stack Exchange. :D</p>Viliam3czZmsrompD7jpxBMThu, 04 Mar 2021 00:28:19 +0000Comment by Viliam on Enabling Children
https://www.greaterwrong.com/posts/g7EroWAw8T7MJrmbQ/enabling-children#comment-AvjSZpqshXDox4omW
<p>The main problem with point 2 in my opinion is with not being <i>honest</i> about one’s priorities. Otherwise, why would they marry and divorce, instead of staying single, or staying legally married but living separately?</p><p>There are rich men who essentially have a deal with their lovers: “I will knock you up, you will take all care of the baby, and I will abandon you at some moment in future, but I will give you enough money to raise the child without needing to get a job”. And both sides seem to be happy with the deal: the man keeps his career, and is happy about having reproduced biologically; the woman gets an early retirement.</p><p>Problem is with men who want the same deal, without being able/willing to pay enough money to make it a great deal also for the woman. (Assuming, stereotypically, that it’s the man who follows his career, and woman who stays with the kids.)</p>ViliamAvjSZpqshXDox4omWThu, 04 Mar 2021 00:03:56 +0000Comment by Viliam on Enabling Children
https://www.greaterwrong.com/posts/g7EroWAw8T7MJrmbQ/enabling-children#comment-Gi2cxQbGvtLLzKc6y
<p>Generally, strategically living with other people (having your friends live next door, rather than strangers next door and friends in other parts of town) seems like something that could tremendously increase quality of life, but is difficult to organize. Kids just add another level of complexity to this problem.</p><p>The added complexity of living with kids is that you also need good schools nearby (unless your plan is to homeschool), and playgrounds. But more importantly, the place with kids needs to be <i>larger</i> than the place without kids. So for example, if you would already have a nice rationalist block of flats, with flats large enough for a pair of adults, they would not be large enough for a pair of adults with kids. So you either need to move after the kids are born, or you need to live in an unnecessarily large place before the kids are born. (Buying an unnecessarily large place, and then <i>renting</i> the extra rooms while you don’t have the kids, is a possible way to do it.)</p><p>If you are a millionaire who dreams about having a rationalist community house, you could probably pay someone to <i>build</i> a house according to your specification—you wouldn’t need to wait until one spontaneously appears on the market. There are already companies that build houses, they would probably be happy to have a guaranteed buyer; you might even get a discount. Though it might be difficult/expensive to find a place to build in a good location.</p><p>The problem of people needing different amount of place in different stages of life, could be solved by having places of different size in the house, so that you move to a different place when your kids are born, and move again when they leave home. Alternatively, some larger places could be shared by more couples without kids.</p><p>It would probably be good to have a silent “work/study room” where people could temporarily go with their computer to avoid all noise. And on the opposite side of house, a shared “playground”. Maybe also a “gym”; and some common room for talking / dancing / drinking, i.e. noisy activities for adolescents and grownups.</p><p>In my opinion, unless there are conflicts with neighbors, living with people you know is always better than living alone among strangers. To prevent conflicts, as they say: good fences make good neighbors. Every family should have their private place, where they make the ultimate decisions. Plus, there can be communal spaces. Living close to each other, but with the ability to close the door and leave everyone else outside, when you need it.</p><p>Kids aged 0 to 2 need to be with their parents almost constantly. But even then, friends living near can help with all <i>other</i> activities, such as shopping, cooking… or just being there and talking to you, to prevent you from going crazy.</p><p>After 3, kids can play together. An adult should supervise them, but the ratio of adults to kids improves dramatically. (In many aspects, having two kids is <i>easier</i> than having one. Two kids aged 3+ will play together a lot. One child will constantly seek <i>your</i> attention.) In a playground or a community garden, one adult is enough to supervise a group of children; especially if there is an option to call a parent in case of problem. Going anywhere else, there is the problem with safely crossing the street; plus the risk that multiple kids will simultaneously throw a tantrum, or decide to run away in opposite directions… so it’s like 90% of time it is okay, but you want to have some help for the remaining 10%. Essentially, you can have a small informal kindergarten.</p><p>No experience with older kids, yet.</p>ViliamGi2cxQbGvtLLzKc6yWed, 03 Mar 2021 23:42:33 +0000Comment by Viliam on Naïve Set Theory - Part 1: Construction of Sets
https://www.greaterwrong.com/posts/qGAYpZRHkbvKRDkqc/naive-set-theory-part-1-construction-of-sets#comment-FcE6xuhDgBBjBvjpe
<blockquote><p>consider the set A = {a,b,c,...}</p></blockquote><p>What if I am an asshole mathematician, and I insist that there is no set <i>{a,b,c...}</i>?</p><p>I mean, I know that it exists, you know that it exists, but I propose a <i>model</i> of set theory where individual sets <i>a, b, c…</i> exist, but there is no set <i>{a, b, c...}</i>. Can you <i>prove</i> there is one?</p><p>Wikipedia—if I understand it correctly—assumes that there is a function <i>0->a, 1->b, 2->c…</i> and uses axiom of replacement. Following my passive-aggressive strategy, I insist that there is no such function in my model either. (We see the function from outside, but there is no in-universe set <i>{{0, a}, {1, b}, {2, c}… }</i>.) Can you construct one, if you are only given the sets in the chain with no “metadata”?</p><p>Specifically: V(0) = an infinite descending chain <i>{a, b, c...}</i> such that <i>a ∋ b ∋ c...</i>; V(1) = everything you can directly build from V(0) using ZF axioms (all subsets, pairs, unions, replacements...); and so on, as usual.</p><p>This is probably my Dunning–Kruger moment, but I’d appreciate if you’d play along. Does this attempt to create a set universe contradict the ZF axioms somewhere (where exactly?), or is this a model containing an infinitely descending chain where the axiom of foundation is still <i>technically</i> true.</p><p>(Meta: I probably have no further questions. Except perhaps if I wouldn’t understand some part of your answer. This is the most complicated thing I was able to think about set theory so far.)</p><p>EDIT:</p><p>Tried to do my homework, here is the part where I got stuck:</p><p>Imagine that you have a countably infinite amount of ur-elements. Is it possible to have a universe that would be a model of ZF with these ur-elements, but <i>wouldn’t</i> contain a “set of all ur-elements” as a set?</p><p>(For example, because it would only contain sets that contain, however indirectly, only a finite number of the ur-elements. That is, the set of ur-elements is countable from our perspective, but not necessarily in-universe. As <a href="https://en.wikipedia.org/wiki/Skolem%27s_paradox">Wikipedia says</a>: “there is no absolute notion of countability”.)</p><p>Because if that is impossible, then if you take the infinite descending chain <i>a, b, c...</i>, the universe would also have to contain the set <i>{a, b, c...}</i>, which contradicts the Axiom of Foundation. So, what I tried, is really impossible.</p><p>On the other hand, if it is possible, then let’s take the model of ZF that contains the ur-elements <i>a, b, c...</i>, but does not contain the set <i>{a, b, c...}</i>, and now replace the ur-elements with sets from the infinitely descending chain. I believe that what you would get after replacement, would satisfy all the ZF axioms. (I could try to prove it, but if the answer to the question above is “no”, it would probably be a waste of time.)</p>ViliamFcE6xuhDgBBjBvjpeWed, 03 Mar 2021 19:38:29 +0000Comment by Viliam on Good, Evil and (?)Indifferent God Challanges
https://www.greaterwrong.com/posts/NbNNnuZAcDvpLyBKx/good-evil-and-indifferent-god-challanges#comment-tTozDKtiv86WBzSdP
<p>In that case, I think there is no reason to believe that God’s idea of “good” is the same as ours. (Especially if our ideas of “good” depend on culture.) But not the same doesn’t necessarily imply opposite or orthogonal. It could be e.g. that God shares a <i>part</i> of our concept of goodness, but not all of it.</p><p>It might help to know if the “creator of universe” is particularly interested in Earth and specifically humans, or not. If yes, that would increase the probability that there is a relation between God’s values and human values.</p>ViliamtTozDKtiv86WBzSdPTue, 02 Mar 2021 22:39:24 +0000Comment by Viliam on Good, Evil and (?)Indifferent God Challanges
https://www.greaterwrong.com/posts/NbNNnuZAcDvpLyBKx/good-evil-and-indifferent-god-challanges#comment-BdbeqCBh9fxjtjBKr
<p>This is all built on the assumption that some kind of god exists. If you don’t buy that premise, the entire debate is nonsensical.</p>ViliamBdbeqCBh9fxjtjBKrTue, 02 Mar 2021 20:37:52 +0000Comment by Viliam on Naïve Set Theory - Part 1: Construction of Sets
https://www.greaterwrong.com/posts/qGAYpZRHkbvKRDkqc/naive-set-theory-part-1-construction-of-sets#comment-i3XT5FucCrJ3GupWJ
<p>The <a href="https://en.wikipedia.org/wiki/Axiom_of_regularity">version from Wikipedia</a> seems okay with an infinite descending chain of sets, as long as each of them would also contain e.g. an empty set. In combination with <i>other</i> axioms, though… well, that’s where my knowledge of set theory becomes insufficient.</p><p>Okay, here is one of the questions I couldn’t answer by self-study; let me use this opportunity to ask you: The Wikipedia page on <a href="https://en.wikipedia.org/wiki/Axiom_schema_of_replacement">Axiom schema of replacement</a> says:</p><blockquote><p>Suppose P is a definable binary relation (which may be a proper class) such that...</p></blockquote><p>What <i>exactly</i> does “definable” mean in this context?</p><p>I am asking precisely because I want to play a “this is <i>not</i> a set in my model of set theory” card with regards to the set of all sets in the infinite descending chain, and I am not sure whether there is or isn’t a loophole I could use.</p>Viliami3XT5FucCrJ3GupWJTue, 02 Mar 2021 20:23:28 +0000Comment by Viliam on If the cosmos isn't inherently 3d, then...
https://www.greaterwrong.com/posts/j2JmXxEBiPSF5JFTg/if-the-cosmos-isn-t-inherently-3d-then#comment-AbBFpegrZAhKbc65F
<p>It’s not just our <i>senses</i>. It also e.g. when you calculate how much water you need to fill a barrel—using the assumption of the 3D space—and you use that amount of water, it fits.</p><p>Could an observer with a different biology use a different amount of water? I suppose we would have noticed if e.g. bees were able to store more honey in their hives than seems physically possible to us.</p>ViliamAbBFpegrZAhKbc65FMon, 01 Mar 2021 22:30:05 +0000Comment by Viliam on Naïve Set Theory - Part 1: Construction of Sets
https://www.greaterwrong.com/posts/qGAYpZRHkbvKRDkqc/naive-set-theory-part-1-construction-of-sets#comment-GfYxi5wDxLK4h6vwJ
<p>In this context, by “monstrosity” I meant some technically-a-set that <i>couldn’t</i> be constructed. Like the <i>a = {a} </i>example, only I suspect that you can create an example of that type that <i>would</i> technically satisfy the Axiom of Foundation. I am probably wrong here, but I need to think about this more until I see <i>how exactly</i> I am wrong. (That will probably take weeks.)</p><p>But that “every connected directed graph is a picture of a set” is the kind of perspective I have in mind. Being given the entire <i>structure of sets</i>, interconnected, at once. Now the question is whether the structure can be designed in a way that technically satisfies the Axiom of Foundation, while somehow is <i>not</i> construable, e.g. because it is infinite in both directions. So no set contains <i>itself</i>, directly nor indirectly, it’s just a line of sets infinite in both directions, like the integers, where each set contains an infinite amount of “previous” sets; and probably also something else, to make the Axiom of Foundation happy. -- Chances are, I am just talking complete nonsense here.</p>ViliamGfYxi5wDxLK4h6vwJMon, 01 Mar 2021 22:22:08 +0000Comment by Viliam on Naïve Set Theory - Part 1: Construction of Sets
https://www.greaterwrong.com/posts/qGAYpZRHkbvKRDkqc/naive-set-theory-part-1-construction-of-sets#comment-ioG8L97y7PoekDQ3H
<p>Thanks for the explanation, especially the part about different set theories. My knowledge of set theory is very unsystematic—I have picked up and read a few books, but never attended actual lessons. So there are moment where I read some words and wonder “did they <i>actually mean </i>X or Y?” but I have no one to ask. I wonder whether reading a few more books would fix the problem. Looking at random comments e.g. at StackExchange sometimes fills a gap, but this happens unpredictably.</p><p>So far I am only familiar with ZF. I have a strong suspicion that there are models of ZF which <i>cannot</i> be created by the infinite iterative construction you described. Like, models that do not even correspond to anything meaningful; monstrosities that technically follow the letter of the ZF axioms but have nothing in common with “sets” and “elements” as any sane person might imagine them. (Because of the first-order logic, which is unable to keep the monstrosities out… for reasons that I still don’t grok, but hopefully one day I will.)</p><p>I wish I could give this topic more time and attention, unfortunately, real life gets in the way.</p>ViliamioG8L97y7PoekDQ3HMon, 01 Mar 2021 18:03:44 +0000Comment by Viliam on Naïve Set Theory - Part 1: Construction of Sets
https://www.greaterwrong.com/posts/qGAYpZRHkbvKRDkqc/naive-set-theory-part-1-construction-of-sets#comment-t4eE4MiRfuLpjjeRi
<p>Yeah, the surreal numbers are somewhat similar to my intuition, but not the same. Kolmogorov complexity is probably closer. But I don’t have anything precise in mind. Just a feeling that “first numbers, then sets of numbers” makes sense on some level, but there is no way to make the same sense about (all) sets of sets.</p><p>I just started reading the reviewed book. Thanks for inspiration!</p>Viliamt4eE4MiRfuLpjjeRiSun, 28 Feb 2021 21:21:26 +0000Comment by Viliam on "If You're Not a Holy Madman, You're Not Trying"
https://www.greaterwrong.com/posts/s3rAKTkdSHb6Hwwoz/if-you-re-not-a-holy-madman-you-re-not-trying#comment-3fttLSxrtzgzytFjZ
<blockquote><p>For example, when he describes altruists selling all their worldly possessions, it doesn’t <i>sound</i> like he intends it as an example of Goodhart; it sounds like he intends it as a legit example of altruists maximizing altruist values.</p></blockquote><p>Goodharting is one thing, another thing is short-term (first-order) consequences vs long-term (second-order) consequences.</p><p>Imagine that you are the only altruist ever existing in the universe. You cannot reproduce or make your copy or spread your values. Furthermore, you are terminally ill and you know for sure that you will die in a week.</p><p>From that perspective, it would make sense to sell all your worldly possessions, spend the money to create as much good as you can, and die knowing you created the most good possible, and while it is sad that you couldn’t do more, it cannot be helped.</p><p>(Note that this thought experiment does <i>not</i> require you to be <i>perfectly</i> altruistic. Not only are you allowed to care about yourself, you are even allowed to care about yourself more than about the others. Suppose you value yourself as much as the rest of universe together. That still makes it simple: spend 50% of your money to make the remaining week as pleasurable for yourself as possible, and the remaining 50% to improve the world as much as possible.)</p><p>We do not live in such situation though. There are <i>many</i> people who feel altruistic to smaller or greater degree, and what any specific one of them does is most likely just a drop in the ocean. The drop may be even smaller than the waves it creates. Maybe instead of becoming e.g. a lawyer and donating your entire salary to charity, you could become e.g. a teacher or a writer, and influence many <i>other</i> people, so that <i>they</i> become lawyers and donate <i>their</i> salaries to charity… thus indirectly contributing to charities much more than you could do alone.</p><p>Of course this approach contains its own risk of going <i>too meta</i>—if literally <i>everyone</i> who ever feels altruistic becomes a teacher or a writer, and spends their whole salary on flyers promoting effective altruism, that would mean that the charity actually gets nothing at all. (Especially if it becomes common belief that being a meta-altruist is much better—i.e. higher status—than being a mere object-level altruist.)</p><p>The effect Scott probably worries about is the following: Should it become known that altruists generally live happy lives, or should it become known that altruists generally suffer a lot in order to maximize the global good? In short term, the latter creates more good—optimizing for charity gives more to charity than optimizing for a combination of charity and self-preservation. But in long term, don’t be surprised if people who are generally willing to help others, but have a strong self-preservation instict, decide that this altruism thing is not for them. A suffering altruist is an <i>anti-advertisement</i> for altruism. Therefore, in the name of maximizing the global good (as opposed to maximizing the good created personally by themselves) an effective altruist should strive to live a happy life! Because that attracts more people to become affective altruists, and more altruists can together create more good. But you should still donate <i>some</i> money, otherwise you are not an altruist.</p><p>So we have a collective problem of finding a function <i>f</i> such that if we make it a social norm that each altruist <i>x</i> should donate <i>f(x)</i>, the <i>total</i> number donated to charities is maximized. It should be sufficiently high so that money actually is donated, and sufficiently low so that people are not discouraged to become altruists. And it seems like “donate 10% of your income” is a very good rule from this perspective.</p>Viliam3fttLSxrtzgzytFjZSun, 28 Feb 2021 20:34:25 +0000Comment by Viliam on Naïve Set Theory - Part 1: Construction of Sets
https://www.greaterwrong.com/posts/qGAYpZRHkbvKRDkqc/naive-set-theory-part-1-construction-of-sets#comment-Tf9xDZE9JiKaB7aWg
<p>Just to make sure, by “books recommended by MIRI” you mean <a href="https://www.goodreads.com/list/show/101757.A_Guide_to_MIRI_s_Research">this list</a> or is there a better resource?</p><p>My impression is that the concept of “set” seems quite straightforward when we talk about sets of <i>other objects</i>. Saying “a set of all natural numbers” seems like a wannabe-academic way of saying “natural numbers”. The problems seem to start when we try to apply the concept of “set” to itself; when we talk about “sets of sets”, “sets of sets of sets”, and worse.</p><p>(Although, I am not completely sure about this part: the <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">Banach-Tarski paradox</a> is about sets of <i>points</i>.)</p><p>The problems are related to the concept of “infinity”. Korzybski (the “<a href="/tag/the-map-is-not-the-territory">map is not the territory</a>” guy) would probably dismiss this whole line of thinking as confused—actual infinities do <i>not</i> exist; there are only processes that can go on <i>indefinitely</i>, but in finite time they obviously <i>can’t</i> arrive to infinity literally. For every natural number <i>n</i>, we can imagine a natural number <i>n+1</i>, but we can never complete the entire set of <i>all</i> natural numbers. So if you pretend to have a <i>complete</i> set of all natural numbers… plus some even more unreal sets… and then you arrive at paradox, that’s entirely your fault. You should have stopped at the first moment you started talking nonsense.</p><p>(Though with this approach, we would probably have to give up the concept of <i>real</i> numbers, too; at least the <a href="https://en.wikipedia.org/wiki/Computable_number">non-computable</a> ones. Because the concept of the real number involves the infinite amount of digits, infinite precision, and sometimes even infinite complexity.)</p><p>Even with infinite sets, the problems seems to be not their infinite <i>size</i>, but rather their infinite <i>complexity</i>. A set containing natural numbers, like “a set of all prime numbers”? No problem. A set containing sets of natural numbers? Oh, you mean like “all pairs of natural numbers”? Yeah, I guess that’s still okay. A set containing sets <i>and</i> natural numbers? Uhm, what does that even <i>mean</i>? If comparing apples to oranges is wrong, surely comparing apples to sets of apples is not-even-wrong. And when we get the infinitely nested hierarchies of sets, it’s time to admit this has no relation to the actual apples.</p><p>From that perspective, it’s kind of a relief when the pretense of apples is removed completely, and we start talking about sets containing sets containing… ultimately empty sets and nothing else. Just a long maze of parentheses that would make even a devoted Lisp programmer admit that this has gone <i>too</i> far.</p><p>I think that at that moment it would be even better to completely drop the pretense that the words “set” and “contains” in set theory have <i>any correspondence whatsoever</i> to our usual intuition of a set as a collection of something, and containing as… the notion that there was something, and some other thing, and someone labeled all these things as beloning to the same whole. (Specifically, I suspect, although I do not really understand the set theory that far yet, that the whole approach of “<a href="https://en.wikipedia.org/wiki/Forcing_(mathematics)">forcing</a>” is just some kind of trolling; creating arbitrary structures that technically satisfy the axioms of set theory, without having anything in common with sets qua collections of things.)</p><p>.</p><p>When I think about mathematical objects, I naturally picture them as constructed in some (partial) order. Like the number 5 comes before number 10 not just the sense of “the ‘less than’ operator”, but fundamentally; that it is possible to imagine a universe where “5 apples” exist but “10 apples” do not, but impossible to do the other way round. From that perspective, minus 5 comes before minus 10; integers come before rationals (at least of comparable size; I am not really sure that 10^^^10 comes before <span class="frac"><sup>1</sup>⁄<sub>2</sub></span>); and rationals come before reals.</p><p>From that perspective, integers come before sets of integers, because first you need to create the integers, and only then you can start collecting them to sets. The entire process can be done in two clearly separated steps. Step 1: create integers. When you are done, step 2: create sets of integers.</p><p>Except with sets of sets this process doesn’t work, because sets of sets are of the same type as sets. Hence the paradox with “set of all sets (not containing themselves” and similar. Step 1: create sets. When you are done, step 2: create the set of all sets… oops, it turns out you were actually <i>not</i> done with the step 1. With integers, you can imagine that an infinite amount of time has passed, and you have created them all. With sets, there is no amount of time, finite or infinite, is enough. With integers, the idea of “last integer” does not make sense, but the idea of “all integers” does. With sets, the “last set” and the “set of all sets” are both the same. So even the generous assumption that we have unlimited infinities of time does not allow us to create all sets.</p><p>Mathematics does not really require the concept of gradual construction (of integers, or sets). We can just assume we have all integers, magically created at the same moment. And it will be the same as when we create the integers step by step, using an infinitely fast machine. We cannot do the same thing with sets though, lest we get a paradox. Unless we give up on some notions, such as having a “set of all sets”. (Which notions exactly do we have to give up? I guess those that would prevent the infinitely fast machine from even completing the construction of “all sets”. So at the end, even if we pretend it is not a gradual construction, it kinda behaves as if it was.)</p>ViliamTf9xDZE9JiKaB7aWgSun, 28 Feb 2021 19:49:12 +0000Comment by Viliam on Suspected reason that kids usually hate vegetables
https://www.greaterwrong.com/posts/TRvKb6st9M7Jfsnyz/suspected-reason-that-kids-usually-hate-vegetables#comment-afNtX8S9D7SjLxe9i
<p>We give our kids fruits and raw vegetables as snack—mostly apple, tangerine, kohlrabi, blueberry. When the food is conveniently cut into small pieces and brought next to the toys or computer, it is rarely refused.</p>ViliamafNtX8S9D7SjLxe9iSun, 28 Feb 2021 13:04:11 +0000Comment by Viliam on Suspected reason that kids usually hate vegetables
https://www.greaterwrong.com/posts/TRvKb6st9M7Jfsnyz/suspected-reason-that-kids-usually-hate-vegetables#comment-WuGBHfmmiGGyGASP5
<p>I live in Slovakia, but I noticed that Asian restaurants are the only ones here that provide significant amount of vegetables with lunch. Everyone else either gives nothing by default (hey, if you <i>really</i> want vegetables, you can order them separately in a small bowl for an extra euro), or put a microscopically thin slice of cucumber on the side of the plate (and some of my colleagues are like “ewww… why did they put <i>this</i> in my meal?”).</p>ViliamWuGBHfmmiGGyGASP5Sun, 28 Feb 2021 12:53:19 +0000Comment by Viliam on Suspected reason that kids usually hate vegetables
https://www.greaterwrong.com/posts/TRvKb6st9M7Jfsnyz/suspected-reason-that-kids-usually-hate-vegetables#comment-CzJSNXBryEHGe2sEd
<blockquote><p>We even let him help out in the kitchen, so he can give input on the preparation of the meal and have contact with each ingredient before it’s on his plate, but <i>still</i> he often won’t eat <i>what he himself chose and cooked</i>! </p></blockquote><p>Haha, exactly the same with my 5yo daughter! Sometimes she even <i>invents</i> the food, I am like “actually, this might taste quite good”, we cook it together, and then… only me and my wife eat it.</p><p>(The only successful invention I remember was couscous with canned fish.)</p>ViliamCzJSNXBryEHGe2sEdSun, 28 Feb 2021 12:47:07 +0000Comment by Viliam on Suspected reason that kids usually hate vegetables
https://www.greaterwrong.com/posts/TRvKb6st9M7Jfsnyz/suspected-reason-that-kids-usually-hate-vegetables#comment-Jcw8hpyufpKdujbk7
<p>In former Czechoslovakia, there was an official list of recipes that restaurants were allowed to cook, during socialism. (To legally cook anything else, you had to ask for an official exception and get it approved.) That explains why some meals were not just horrible, but <i>identically horrible</i> across restaurants. The only tasty vegetable I remember from my childhood was fried cauliflower—probably not very healthy.</p>ViliamJcw8hpyufpKdujbk7Sun, 28 Feb 2021 12:38:43 +0000