How can the Continuum Hypothesis be independent of the ZFC axioms? Why does the lack of “explicit” examples of sets with a cardinality between that of the naturals and that of the reals not guarantee that there are no examples at all? What would an “implicit” example even mean?
It means that you can’t reach a contradiction by starting with “Let S be a set of intermediate cardinality” and following axioms of ZFC.
All the things you know and love doing with sets —intersection, union, choice, comprehension, Cartesian product, power set — you can do those things with S and nothing will go wrong. S “behaves like a set”, you’ll never catch it doing something unsetlike.
Another way to say this is: There is a model of ZFC that contains a set S of intermediate cardinality. (There is also a model of ZFC that doesn’t. And I’m sympathetic to the view that - since there’s no explicit construction of S -we’ll never encounter an S in the wild and so the model not including S is simpler and better.)
Caveat: All of the above rests on the usual unstated assumption that ZFC is consistent! Because it’s so common to leave it unstated, this assumption is questioned less than maybe it should be, given that ZFC can’t prove its own consistency.
At least one mathematician (I forget his name) considers V=L to be a reasonable axiom to add. Informally put, it says that nothing exists except the things that are required to exist by the axioms. ZF + V=L implies choice, the generalised continuum hypothesis, and many other things. His argument is that just as we consider the natural numbers to be the numbers intended to be generated by the Peano axioms, i.e. the smallest model, so we should consider the constructible universe L to be the sets intended to be generated by the ZF axioms. The axioms amount to an inductive definition, and the least fixed point is the thing they are intended to define. One can think about larger models of ZF, just as one can think about non-standard natural numbers, but L and N are respectively the natural models. I don't know how popular this view is.
In Peano arithmetic, the induction axiom (not axiom schema) basically says "... and nothing else is a natural number". It can only be properly formulated in second-order logic, and the result is that Peano arithmetic becomes "categorical", which means it has only one (the intended) model up to isomorphism. The real or complex number systems and geometry also have categorical axiomatizations. Standard (first-order) ZFC is not categorical, since it allows both for models that are larger than intended (like first-order Peano arithmetic) and smaller than intended (unlike first-order Peano arithmetic). However, second-order ZFC is also not categorical, although I think it rules out some part of the non-standard models. But your description of the theory ZF+V=L sounds like this theory (i.e. a second-order version) would indeed be categorical. Though presumably this would be somewhat of a big deal but is nowhere mentioned in the Wikipedia article. So I guess the theory probably is still not categorical.
Your "simpler is better" is hard to apply. One way of thinking about models where there are no intermediate cardinals isn't that S doesn't exist. But that T, a mapping from S to either the naturals or the reals, does exist.
And T will also be something you can't explicitly construct.
Also, the axiom of choice basically says "there exists loads of sets that can't be explicitly constructed".
Very good, fundamental questions.. I don’t understand question 85 though. Here are two more good questions.
Basic idea of 85 is that we generally agree there have been moral catastrophes in the past, such as widespread slavery. Are there ongoing moral catastrophes? I think factory farming is a pretty obvious one. There's a philosophy paper called "The Possibility of an Ongoing Moral Catastrophe" that gives more context.
I thought that was what was meant. The question is probably the easiest one to answer affirmatively with a high degree of confidence. I can think of several ongoing ”moral catastrophs”.
9. Is consciousness fundamental?
10. What effects does consciousness have?
11. How was consciousness incentivized by evolution, if at all? (I now think it probably was.)
16. What evidence can we get for physical systems, such as AI’s, being conscious?
I have a post speculating on that, taking on the "follow physicalism off a cliff" perspective! Nobody liked it, though.
You may also find the last chapter of Hoffman's The Case Against Reality interesting, here; also in relation to your Question 2. I don't personally buy it, and think it introduces a bunch of redundant, will-obviously-be-proven-wrong details, but the fundamental approach there is interesting.
20. What is the most reasonable response to the Fermi paradox?
The Grabby Aliens model + Dissolving the Fermi paradox both seem like plausible, and not mutually exclusive, answers.
86. Does marginal reasoning plus the view that individuals cannot significantly shift the margins on large issues imply that individuals trying to maximize their impact should focus all their efforts towards one issue rather than diversifying? Is there any principled counterargument to this?
You may want to look at geometric rationality, and this post specifically.
What does quantum entanglement mean for causality? Due to entanglement, there can be spacelike separated measurements such that there exists a reference frame > where it looks like measurement A precedes and has a causal influence on the outcomes of measurement B, and > also a reference frame where it looks like measurement B precedes and has a causal influence on the outcomes of measurement A.
"Causality" is already a somewhat fraught notion in fundamental physics irrespective of quantum mechanics; it's not clear that one needs to have some sort of notion of causality in order to do physics, nor that the universe necessarily obeys some underlying causal law. To the extent that quantum mechanics breaks our common-sense notions of causality, it's only in this very particular sense (where it seems like Alice measuring first "causes" Bob's measurement to take a certain value, or vice versa), and since neither party can use a measurement scheme like this to send information, the breakage doesn't invite paradoxes or any sort of other weirdness.
Outside of philosophical musings about causality (which, to be clear, I think are perfectly valid and interesting) it suffices to say that entangled systems exhibit correlations without a common cause, and leave it at that.
If you're interested in a recent technical discussion of some of these ideas, I recommend the following paper: https://arxiv.org/pdf/2208.02721.pdf
Could playing a well-designed video game (to the point of “knowing the meta”?) allow a human to have thorough intuitions about quantum mechanics? About 4D space?
4D space, I think yes. Especially if you used 3D glasses, and then represented the 4th dimension by color.
Quantum mechanics in general, probably no, because it would require tracking an exponential number of states along with their complex amplitudes. That's just too much data to track.
What are the benefits of expanding one’s comfort zone?
It unlocks the options you otherwise wouldn't have. Imagine some things you did in your life that led to a good outcome... and now imagine someone else in your position, for whom doing that thing would be outside their comfort zone. They would be deprived of that good outcome. Now probably the same is true also for you. (You may underestimate how much, because we often do not even notice the opportunities we wouldn't take anyway.)
As an answer to #43, find some people who achieved something you would like to achieve, and observe them. What is inside their comfort zone that is outside of yours? Could it have contributed to their success?
How can a person kickstart themselves during a period of low motivation?
This is tricky, because any advice in form "do X" will turn into "but how do I make myself do X when my motivation is low?". I think a better way is to surround yourself with people who motivate you. (Easier said than done.)
Maybe make some simple checklist, such as taking a short walk outside every morning, and then hire some online assistant from a cheap country to call you every day and ask you if you have completed your checklist for today. (This may be a generalization from one example, but for me, it is easier to make myself doing things, if they have social consequences, however trivial. On the other hand, it is too easy to give up on promises made to myself, if no one else cares.)
What are unconventional social circumstances that would be highly beneficial to at least some people if they were to enter into them?
Possible example: several couples with kids living together, sharing some responsibilities of raising the children.
Note the "conventional" in some (sub)culture may be "unconventional" in some other. So let's simply talk about things that some people never tried, and probably never even thought about them.
Many people are working from home these days, but didn't try coworking. A few people working for different companies remotely can take their notebooks and meet in the same room for the day. Either rent a space, or at someone's home.
Polyamory?
How can people (like me) who spend lots of time thinking about meta-level ideas keep from frequently sabotaging their own ability to "enjoy the moment?"
Some kind of meditation? Stop thinking, and then either (a) do the loving-kindness thing, or (b) focus on your sensory inputs -- especially when you are in nature.
Thinking can be turned off, you just need to learn how. Just like for some kids it is almost impossible to learn to shut up; this is similar.
How important is it to maintain a diversity of cultures in the world?
I think that diversity is good (in the sense that having a choice between N cultures of comparable goodness is better than only having one such culture), but some cultures are better than others. Of course, cultures can also evolve. So my advice would be approximately: first try to fix all cultures, then destroy the ones that cannot be fixed, and keep the diversity of those that are okay.
What major insights about the world do other people have that I do not have? How can I acquire as many of these as possible?
Meet different kinds of people and talk to them.
How can a person intentionally optimize their social environment to have as positive an influence on their life as possible?
I suspect that at some moment this becomes a group effort, so maybe the first step is to find a group of people who want to do the same?
Are there ongoing moral catastrophes?
All the wars, famines, racism, sexism, and random interpersonal violence, all around the world. Factory farming. Aging and death. Probably missed a lot, but the answer is clearly yes.
What factors lead to variation in values across humans?
I suspect that some values arise in a way "if you keep practicing X, you start valuing X". For this category, the answer is: previous experience. Which was probably shaped by upbringing, or talent.
How should the notion and inevitability of death inform the way I live?
You should adjust your risk-taking accordingly. On one hand, you don't want to needlessly die too soon. On the other hand, death puts a limit on long-term thinking -- at some moment you need to actually spend the resources you saved, or you will die without ever spending them.
Are there opportunities for having a massive positive impact on the world that no one has thought of yet?
I suggest to widen this category to also include "things that people have thought of, but didn't actually do". Or "they did a half-assed job, and doing this properly requires 3× more effort, but has 100× larger impact".
My example would be Khan Academy. It's not like no one had an idea to make an amateur educational video before Salman Khan. Probably thousands did. Yet there was an extra value possible to make by making those videos (a) good, (b) free, (c) covering a large part of the curriculum, and (d) putting them all in one place, rather than let people google each of them individually.
Digital stuff seems to scale better. This includes websites, applications, videos, books, etc.
How good is the average conscious experience on factory farms today?
Horrible, I suppose.
Interesting questions!
I will try to answer #3, but full disclosure, in set theory, I am just a mathematically gifted autodidact.
How can the Continuum Hypothesis be independent of the ZFC axioms? Why does the lack of “explicit” examples of sets with a cardinality between that of the naturals and that of the reals not guarantee that there are no examples at all? What would an “implicit” example even mean?
It seems to me that the literature on set theory focuses on the technical details and ignores the meaning of what they are doing. Which makes sense from certain perspective: it is easy to bullshit when you talk about the meaning of stuff, so you can signal seriousness by only doing the equations. Also, for the experts, the meaning is probably so obvious that it doesn't seem to them like it needs to be communicated.
Unconcerned with expert status, here is my attempt to communicate what the set theory seems to be about, to me.
First, we have some intuition of set as a collection. Set of all natural numbers. Set of 4, 8, 17. Set of sets, as in "all possible pairs of natural numbers between 1 and 10". So it seems like we talk about the same thing. Yet I bet that if you interrogated hundred people about their intuitions, you would get many different answers.
Can a set contain objects of a different type, such as the number 5, but also the set of 7, 9, and 11? The official ZF answer is "yes", but probably many mathematicians do not actually have a practical use for such sets, so they would not mind a negative answer. Can a set contain itself? Some people would say "why not", others would object that in their intuition, a set is an afterwards created collection of pre-existing objects, therefore no (ZF also says no). But it is also possible to answer positively. Can there be an infinity so insanely large that it cannot even be expressed in the terms of smaller infinities? (Technically: Can there be K which is larger than a limit of any sequence of numbers smaller than K where the length of the sequence is also smaller than K? Well, there is the omega, which is larger than a "limit" of any finite sequence of finite numbers, but other than that?) Some people would go like "why the heck would I need such infinity", while others would be like "yeah, if we can describe them consistently, the richer the set theory, the better". You can keep inventing questions like this, so you can have dozens of mutually different definitions of "a set" what comply with someone's intuition.
Second, there is an ambition to express the axioms in the language of first-order logic. There are some good reasons to do that, but the problem is that first-order logic cannot even define what "a natural number" means. The ZF set theory is supposed to have well-founded ordinals, but you can't express "well-founded" in first-order logic either! The Axiom of Foundation/Regularity, translated to first-order logic, is just not the same thing.
Thus, if we stick to the first-order logic, we are not actually talking about "sets" (in the sense of: collections compatible with some sane person's intuition of a set), but about "mathematical structures that technically follow the axioms of the set theory, even if no sane person would actually call them sets". The latter is usually abbreviated to "sets". But it includes (mostly) monstrosities beyond imagination which technically happen to obey the dozen axioms that we chose.
Generally, "X independent of ABC axioms" means that the axioms are ambiguous, there are many possible structures that obey them, in some of them X is true, in others X is false. Among the monstrosities that follow the ZFC axioms, and no sane person would call most of them sets, in some of them the technical interpretation of Continuum Hypothesis is true, in others it is false.
The idea of "cardinality" itself does not mean what you might naively expect it to mean. Consider Skolem's paradox: "there are countable models of uncountable sets". In other words, sets that are obviously countable, but also technically uncountable. That's because, technically, "same cardinality" is defined as "there is a bijection", and "bijection" is defined as "a set of pairs"... so if you cleverly redefine "set" to exclude those inconvenient sets of pairs -- tada! -- you have an obviously countable set which, technically, now does not have a bijection to the natural numbers, and therefore, technically, is uncountable.
So the "set with a cardinality between that of the naturals and that of the reals" could technically be some set of reals defined by a really good lawyer. (Unfortunately, I am not that good lawyer, yet, so I cannot give you the exact definition.)
Actually, this is also related to #35:
Does truth transcend proof? In what sense?
I think "proof" is a concept from first-order logic, and "truth" is a concept from second-order logic. But I am confused about this, so this is just a vague pointer towards the answer, not the answer itself.
I'm a hardcore consciousness and metaphysics nerd, so some of your questions fall within my epistemic wheelhouse. Others, I am simply interested in as you are, and can only respond with opinion or conjecture. I will take a stab at a selection of them below:
4: "Easy" is up in the air, but one of my favorite instrumental practices is to identify lines of preprogrammed "code" in my cognition that do me absolutely no good (grief, for instance), and simply hack into them to make them execute different emotional and behavioral outputs. I think the best way to stay happy is just to manually edit out negative thought tendencies, and having some intellectual knowledge that none of it's a big deal anyways always helps.
8: I would define it as "existing in its minimally reduced, indivisible state". For instance, an electron is a fundamental particle, but a proton is not because it's composed of quarks.
12 (and 9): I think you're on the best track with B. Consciousness is clearly individuated. Is it fundamental? That's a multifaceted issue. It's pretty clear to me that it can be reduced to something that is fundamental. At minimum, the state of being a "reference point" for external reality is something that really cannot be gotten beneath. On the other hand, a lot of what we think of as consciousness and experience is actually information: thought, sensation, memory, identity, etc. I couldn't tell you what of any of this is irreducible - I suspect the capacities for at least some of them are. Your chosen stance here seems to approximate a clean-cut interactionism, which is at least a serviceable proxy.
13: I think this is the wrong question. We don't know anything yet about how physics at the lowest level ultimately intersects and possibly unifies with the "metaphysics" of consciousness. At our current state of progress, no matter what theory of consciousness proves accurate, it will inevitably lean on some as-yet-undiscovered principle of physics that we in 2023 would find incomprehensible.
16: This will be controversial here, but is a settled issue in my field: You'd be looking for phenomenological evidence that AIs can participate in metaphysics the same ways conscious entities can. The easiest proof to the affirmative would be if they persist in a discarnate state after they "die". I sure don't expect it, but I'd be glad to be wrong.
19: I think a more likely idea along the general lines of the simulation hypothesis, due to the latter's implications about computers and consciousness that, as I said above, I do not expect to hold up, is that an ultra-advanced civilization could just create a genuine microcosm where life evolved naturally. Not to say it's likely.
20: Total speculation, of course - my personal pet hypothesis is that all civilizations discover everything they need to know about universal metaphysics way before they develop interstellar travel (we're firmly on that track), and at some point just decide they're tired of living in bodies. I personally hope we do not take such an easy way out.
21: I can buy into a sort of quantum-informed anthropic principle. Observers seem to be necessary to hold non-observer reality in a stable state. So that may in fact be the universe's most basic dichotomy.
33: In my experience, the most important thing is to love what you're learning about. Optimal learning is when you learn so quickly that you perpetually can't wait to learn the next thing. I don't think there's any way to make "studying just to pass the test" effective long-term. You'll just forget it all afterwards. You can probably imagine my thoughts on the western educational system.
43-44: Speaking to one's intellectual comfort zone, Litany of Tarski-type affirmations are very effective at that. The benefit, of course, is better epistemics due to shedding ill-conceived discomfort with unfamiliar ideas.
45: I've actually never experienced this, and was shocked to learn it's a thing in college. Science will typically blame neurochemistry, but in normal cognition, thought is the prime mover there. So all I can think of is an associative mechanism whereby people relate the presence of a certain chemical with a certain mood, because the emotion had previously caused the chemical release. When transmitters are released abnormally (i.e. not by willed thought), these associations activate. Again, never happened to me.
56: I'd consider myself mostly aligned with both, so I'd personally say yes. I'm also a diehard metaphysics nerd who's fully aware I'm not going anywhere, so I'd better fricking prioritize the far future because there's a lot of it waiting for me. For someone who's not that, I'd actually say no, because it's much more rational to care most about the period of time you get to live in.
58: As someone who's also constantly scheming about things indefinitely far in the future, I feel you on this one. I find that building and maintaining an extreme amount of confidence in those matters enriches my experience of the present.
71-73: For me, studying empirical metaphysics has fulfilled the first two (rejecting materialism makes anyone happier, and there's no limit of possible discovery) and eventually will the third (it'll rise to prominence in my lifetime). I can't say I wouldn't recommend.
78: Same as 71-73, for an obvious example. I can definitely set you in the right direction.
81: Following the scientific method, a hypothesis must be formed as an attempt to explain an observation. It must then be testable, and present a means of supporting or rejecting it by the results of the test. I've certainly dealt with theories that seem equally well supported by evidence but can't both be true, but I have no reason to think better science couldn't tease them apart.
89: Definitely space travel, AI, VR, aging reversal, genetic engineering. I really think metaphysical science will outstrip all of the above in utility, though...
96: ...by making this cease to be relevant.
98: Of course there are, because there's so much we know nothing about when it comes to what the heck we even are. I'd almost argue we have very little idea how to truly have the biggest positive impact on the future we can at this stage. We'll figure it out.
Lots of interesting thoughts, thanks for sharing!
You seem to have an unconventional view about death informed by your metaphysics (suggested by your responses to 56, 89, and 96), but I don’t fully see what it is. Can you elaborate?
Yes, I am a developing empirical researcher of metaphysical phenomena. My primary item of study is past-life memory cases of young children, because I think this line of research is both the strongest evidentially (hard verifications of such claims, to the satisfaction of any impartial arbiter, are quite routine), as well as the most practical for longtermist world-optimizing purposes (it quickly becomes obvious we're literally studying people who've successfully overcome death). I don't want to undercut the fact that scientific metaphysics is a much larger field than just one set of data, but elsewhere, you get into phenomena that are much harder to verify and really only make sense in the context of the ones that are readily demonstrable.
I think the most unorthodox view I hold about death is that we can rise above it without resorting to biological immortality (which I'd actually argue might be counterproductive), but having seen the things I've seen, it's not a far leap. Some of the best documented cases really put the empowerment potential on very glaring display; an attitude of near complete nonchalance toward death is not terribly infrequent among the elite ones. And these are, like, 4-year-olds we're talking about. Who have absolutely no business being such badasses unless they're telling the truth about their feats, which can usually be readily verified by a thorough investigation. Not all are quite so unflappable, naturally, but being able to recall and explain how they died, often in some violent manner, while keeping a straight face is a fairly standard characteristic of these guys.
To summarize the transhumanist application I'm getting at, I think that if you took the best child reincarnation case subject on record and gave everyone living currently and in the future their power, we'd already have an almost perfect world. And, like, we hardly know anything about this yet. Future users ought to become far more proficient than modern ones.
1 Due to entanglement, there can be spacelike separated measurements such that there exists a reference frame where it looks like measurement A precedes and has a causal influence on the outcomes of measurement B, and also a reference frame where it looks like measurement B precedes and has a causal influence on the outcomes of measurement A.
If the traditional idea of causality is an asymmetric a->B relationship , then entanglement doesn't look like causality.
In important ways, it isn't: a mathematical truth is not per se a physical truth.
- How can the Continuum Hypothesis be independent of the ZFC axioms?
Why not? There's no guarantee that any set of axioms should solve every problem.
- Is there a really good reason to believe or not believe one of the following theories of consciousness? (I think I find b most likely.) (I do not consider epiphenomenalism a serious option anymore, basically due to the arguments described here.) (I don’t really consider c a very serious option either.)
Everything is physical—consciousness isn’t fundamental. (Basic response—I feel like any attempt to describe consciousness in terms of physical phenomena will be missing something.)
Arguments: phsyicalism is generally succesful. Counterarguments: hard problem, irreducubillity, Mary's room.
Consciousness is fundamental. Conscious experiences are caused by physical processes, and have effects on physical processes in return. (Basic response—it seems a little absurd to believe that the Standard Model makes incorrect predictions in the brain because consciousness intervenes. It feels even more absurd to imagine there is any reasonable answer to the question, “How fast can physical processes affect consciousness, and how fast can consciousness affect physical processes?”)
Arguments: same as the counterarguments to physicalism. Counterarguments: parsimony, physical closure, interaction.
- What are we really doing when we talk about counterfactuals? Is there any actually principled way to consider them? If not, why does nothing go wrong in our standard use-cases for counterfactuals?
Rationalists think of counterfactuals in terms of the behaviour of agents and Newcomb's paradox. Whereas, the mainstream view is that counterfactual is a "what if", or path not taken -- not necessarily involving agents at all. On the mainstream view, the five numbers that did no come up on the die are counterfactuals.
Rationalists have problems with counterfactuals that the mainstream does not. This immediately suggests that rationalists can solve their problems by adopting the mainstream view.
In the mainstream view, counterfactuals are ott defined in terms of free will, only probability. Which is to say, that as far as everyone who is not a Yudowskian rationalist is concerned,counterfactuals aren't defined in terms of free will, only probability.
Counterfactuals are defined in terms of probability, but not of objective probability. Subjective probability is always available because subjects have limited knowledge..so subjective counterfactuals are always available.
Whether there is a "principled" way of handling them depends on your principles. Assume determinism and omniscience, and you'll have problems.
An excellent exercise! There's also a meta-question which shows up in how you choose to frame the questions. There's an implication underneath "should", "appropriate", "proper", "good", and other glosses you chose for that archetype that they reference -- the thing they point at tends to make new handles for itself if you taboo the existing ones, and tends to resist rigorous formal definition. However, trying on potential definitions for it can nevertheless un-ask or reframe most questions that rely on it.
I saw the application for SPARC, a rationality summer program for high school students, earlier this year. It included a prompt I found super exciting: “List ~100 questions you have that interest you.” This is my response. (It's not very organized - I go in no particular order and I sometimes interject some of my current thoughts about the questions.) I would love to hear people’s thoughts on any of these questions, and I would love to see other people post their responses.
Bonus
The value of the universe: