As of January 2025, I have signed no contracts or agreements whose existence I cannot mention.
French, but because my teacher tried to teach all of the days of the week at the same time, they still give me trouble.
They're named as the planets: Sun-day, Moon-day, Mars-day, Mercury-day, Jupiter-day, Venus-day, and Saturn-day.
It's easy to remember when you realize that the English names are just the equivalent Norse gods: Saturday, Sunday and Monday are obvious. Tyr's-day (god of combat, like Mars), Odin's-day (eloquent traveler god, like Mercury), Thor's-day (god of thunder and lightning, like Jupiter), and Freyja's-day (goddess of love, like Venus) are how we get the names Tuesday, Wednesday, Thursday, and Friday.
Why is Google the biggest search engine even though it wasn't the first? It's because Google has a better signal-to-noise ratio than most search engines. PageRank cut through all the affiliate cruft when other search engines couldn't, and they've only continued to refine their algorithms.
But still, haven't you noticed that when Wikipedia comes up in a Google search, you click that first? Even when it's not the top result? I do. Sometimes it's not even the article I'm after, but its external links. And then I think to myself, "Why didn't I just search Wikipedia in the first place?". Why do we do that? Because we expect to find what we're looking for there. We've learned from experience that Wikipedia has a better signal-to-noise ratio than a Google search.
If LessWrong and Wikipedia came up in the first page of a Google search, I'd click LessWrong first. Wouldn't you? Not from any sense of community obligation (I'm a lurker), but because I expect a higher probability of good information here. LessWrong has a better signal-to-noise ratio than Wikipedia.
LessWrong doesn't specialize in recipes or maps. Likewise, there's a lot you can find through Google that's not on Wikipedia (and good luck finding it if Google can't!), but we still choose Wikipedia over Google's top hit when available. What is on LessWrong is insightful, especially in normally noisy areas of inquiry.
It's also available on Android.
Finitism doesn't reject the existence of any given natural number (although ultrafinitism might), nor the validity of the successor function (counting), nor even the notion of a "potential" infinity (like time), just the idea of a completed one being an object in its own right (which can be put into a set). The Axiom of Infinity doesn't let you escape the notion of classes which can't themselves be an element of a set. Set theory runs into paradoxes if we allow it. Is it such an invalid move to disallow the class of Naturals as an element of a set, when even ZFC must disallow the Surreals for similar reasons?
Before Cantor, all mathematicians were finitists. It's not a weird position historically.
We do model physics with "real" numbers, but that doesn't mean the underlying reality is infinite or even infinitely divisible. My finitism is motivated by my understanding of physics and cosmology, not the other way around. Nature seems to cut us off from any access to any completed infinity, and it's not clear that even potential infinities are allowed (hence my sympathy with ultrafinitism). I have no need of that axiom.
Quantum Field Theory, though traditionally modeled using continuous mathematics, implies the Bekenstein bound: a finite region of space contains a finite amount of information. There are no "infinite bits" available to build the real numbers with. However densely you store information, eventually, at some point, your media collapses into a black hole, and packing in more must take up more space.
Physical space can't be a continuum like the "reals". It's not infinitely divisible. Measuring distance with increasing precision requires higher frequency waves, and thus higher energies, which eventually has enough effective mass to gravitationally distort the very space you are measuring, eventually collapsing into a black hole.
Below a certain limit, distance isn't physically meaningful. If you assume an electron is a point particle with "infinitesimal" size and you zoom in enough, you should be able to get arbitrarily high electric field strength. But at some point, high enough field strength results in vacuum polarization: virtual electron/positron pairs get pushed around and finally one of the positrons annihilates whatever you thought the real electron was, and then one of the virtual electrons doesn't have anything to pair with and becomes the real one. It's as if the electron is jumping around. You can't nail it down. It doesn't physically have a position down below a certain scale in time and space. There are no infinite bits. All the fundamental particle types are like this. There are no infinitesimal point particles. They're just waves.
There's also a cosmological horizon limiting how much of the Universe we can see. There's also a (related) past temporal horizon at the Big Bang. We can't see a completed past-temporal or spacial infinity, in any direction. We're not sure of the Ultimate Fate of the Universe, but it looks like Heat Death is probably it, given our current understanding of physics. So there's a future limit as well. The other likely candidate Fates are finite in time as well.
But even supposing finite information content in a finite region seems to be enough to make potential-infinite time not really meaningful. There's a finite number of states possible, so eventually all reachable states are reached. If physics is deterministic (it seems to be), then we get into a cycle. So time is better modeled as a finite circle, rather than an infinite line. And if it's not deterministic? Then we still saturate all reachable states, the order just gets shuffled around a bit. There's no phycial way to tell the difference.
Potential-infinite space is the same way. Any accessible region has a finite number of states, so at least some of them must repeat exactly in other regions. If there's some determinism to the pattern, then it's maybe better modeled as some curled-up finite space (although aperiodic tilings are also possible). If it's random, then we still saturate all reachable states, the order just gets shuffled around a bit. There's no physical way to tell the difference. Once all reachable states have been saturated, why does it matter if they appear only once or a googol or infinity times?
I installed Mindfulness Bell on my phone, and every time it chimes, I ask myself, "Should I be doing something else right now?" Sometimes I'm being productive and don't need to stop. When I notice I've started ignoring it, I change the chime sound so I notice it again. The interval is adjustable. If I'm stuck scrolling social media, this often gives me the opportunity to stop. Doesn't always work though. I also have it turned off at night so I can sleep. This is a problem if I get stuck on social media at night when I should be sleeping. Instead, after bed time, I progressively dim the lights and screen to the point where I can barely read it. That's usually enough to let me fall asleep.
I'm hearing intuitions, not arguments here. Do you understand Cantor's Diagonalization argument? This proves that the set of all integers is "smaller" (in a well-defined way) than the set of all real numbers, despite the set of all integers being already infinite in size. And it doesn't end there. There is no largest set.
Russell's paradox arises when a set definition refers to itself. For example, in a certain town, the barber is the one who shaves all those (and only those) who do not shave themselves. This seems to make sense its face. But who shaves the barber? Contradiction! Not all set definitions are valid, and this includes the universal one, which can be proved to not exist in many ways, at least in the usual ZFC (and similar).
There are two ways to construct a universal object. Either make it a non-set notion like a "proper class", which can't be an element of a set (and thus can't contain itself or any other proper class), or restrict the axiom of comprehension in a way which results in a non-well-founded set theory. Cantor's Theorem doesn't hold for all sets in NF. The diagonal set argument can't be constructed (in all cases) under its rules. NF has a universal set that contains itself, but it accomplishes this by restricting comprehension to stratified formulas. I'm not a set theorist, so I'm still not sure I understand this properly, but it looks like an infinite hierarchy of set types, each with its own universal set. Again, no end to the hierarchy, but in practice all the copies behave the same way. So instead of strictly two types of classes, the proper class and the small class, you have some kind of hyperset that can contain sets, but not other hypersets, and hyper-hypersets that can contain both, but not other hyper-hypersets, and so forth, ad infinitum.
Personally, I'm rather sympathetic to the ultrafinitists, and might be a finitist myself. I can accept the slope of a vertical line being "infinite" in the limit. That's just an artifact of how we chose to measure something. Measure it differently, and the infinity disappears. I can also accept a potential infinity, like not having a largest integer, because the successor function can make a bigger one. We can make an abstract algorithm run on an abstract machine that can count, and it has a finite description. But taking the "completed" set of all integers as an object itself rubs me the wrong way. That had to be tacked on as a separate axiom. It's unphysical. No operation could possibly construct a physical model of such a thing. It's an impossible object. One could try to point to a pre-existing model, but we physically cannot verify it. It would take infinite time, space, or precision, which is again unphysical.
Similarly, there is no physical way to verify an infinite God exists, because we physically cannot distinguish it from a (sufficiently large, but) finite one. I might be willing to call such an alien a small-g "god", but it's not the big-G omni-everything one in valentinslepukhin's definition. That only leaves some kind of a priori logical argument, because it can't be an empirical one, but it has to be based on axioms I can accept, doesn't it? I can entertain weird axioms for the sake of argument, but I'm not seeing one short of "God exists", which is blatant question begging.
The main idea here is that one can always derive a "greater" set (in terms of cardinality) from any given set, even if the given set is already infinite, because there are higher degrees of infinity. There is no greatest infinity, just like there is no largest number. So even if (hypothetically) a Being with infinite knowledge exists, there could be Beings with greater knowledge than that. No matter which god you choose, there could be one greater than that, meaning there are things the god you chose doesn't know (and hence He isn't "omniscient", and therefore isn't "God", because this was a required attribute.)
I don't know how to interpret "all existing objects", because I don't know what counts as an "object" in your definition. Set theory doesn't require ur-objects (although those are known variations) and just starts with the empty set, meaning all "objects" are themselves sets. The powerset operation evaluates to the set of all subsets of a set. The powerset of a set always has greater cardinality than the set you started with. That is, for any given collection of "objects", the number of possible groupings of those objects is always a greater number than the number of objects, even if the collection of objects you started with had an infinite number to begin with. So no, this doesn't prove that an infinite universe cannot exist, just that there are degrees of infinities (and no "greatest" one).
Naiive set theory leads to paradoxes when defining self-referential sets. The idea of "infinite" gods seem to have similar problems. There are various ways to resolve this. The typical one used in foundations of mathematics is the notion of a collection that is too large to be a set, a "proper class". ("Class" used to be synonymous with "set".) But later on in the discussion it was pointed out that this isn't the only possible resolution.
I don't know of any officially sanctioned way. But, hypothetically, meeting a publicly-known real human person in person and giving them your public pgp key might work. Said real human could vouch for you and your public key, and no one else could fake a message signed by you, assuming you protect your private key. It's probably sufficient to sign and post one message proving this is your account (profile bio, probably), and then we just have to trust you to keep your account password secure.
I feel like this has come up before, but I'm not finding the post. You don't need the stick-on mirrors to eliminate the blind spot. I don't know why pointing side mirrors straight back is still so popular, but that's not the only way it's taught. I have since learned to set mine much wider.
This article explains the technique. (See the video.)
In a nutshell, while in the diver's seat, tilt your head to the left until it's almost touching your window, then from that perspective point it straight back so you can just see the side of your car. (You might need a similar adjustment for the passenger's side, but those are often already wide-angle.) Now from normal position, you can see your former "blind spot". When you need to see straight back in your side mirror (like when backing out), just tilt your head again. Remember that you also have a center mirror. You should be able to see passing cars in your center mirror, and then in your side mirror, then in your peripheral vision without ever turning your head or completely losing sight of them.