• A true solution to Russell's paradox
    And your criticism is belied by the fact that the poster himself explicitly said that my answer was clear and helpful, and his followup questions do show that he basically understood my answer.TonesInDeepFreeze

    Yes but he understood the opposite of the correct answer!

    So, I really don't know what your trip is.TonesInDeepFreeze

    I've been feeling the same way about you.

    I'll quit while I'm behind here.
  • A true solution to Russell's paradox
    And my rigorous, mathematical and standard use and explanations are not refuted (or whatever your disagreement is supposed to be) by your own informal usage.TonesInDeepFreeze

    As I'm sure I've agreed several times. If you don't want to call quantification over a proper class a domain of discourse, I'm fine with that. We frequently quantify over proper classes, however you call it.

    The original question was informal. The original question was in invitation to explain a seeming contradiction. That merits a response that is rigorous and definitive, in order to appreciate that mathematics indeed does not tolerate such a contradiction, not just by informal hand waving, so that when we look at the matter with exactness, we do show that the seeming contradiction actually is avoided.TonesInDeepFreeze

    I thought your responses to the recent OP @Sunner were too detailed and technical to be of use at the level the question was being asked. And, frankly -- not really wanting to get back into this -- wrong. The Russell class DOES define a perfectly good collection. That was the question. @Sunner had the impression (rightly or wrongly) that you said it didn't. I pointed out that it did. Perhaps OP misinterpreted what you said. In that case you were right, and I added clarity. So everyone can be happy, yes?

    Not sure what contradiction "mathematics indeed does not tolerate." The referent of this paragraph is unclear.
  • A true solution to Russell's paradox
    I can't comment on that quote without a link to it.TonesInDeepFreeze

    I hotlinked it. Here's the link.

    I guess you mean "question of semantics" in the sense of how we use words.TonesInDeepFreeze


    "Then what are your rigorous, mathematical definitions of 'domain of discourse'TonesInDeepFreeze

    As I stated up front earlier, my usage is informal.
  • A true solution to Russell's paradox
    Note the quantifier ranges over the universe, but it happens that the formula is a conditional in which being x being a group is the antecedent.

    So that is a relativization of P to groups.

    Well yes, we agree on that. But there is no set of all groups! The class of groups is a proper class. So you seem to be conceding my point.

    In any event, much of the rest of your post is pretty technical and I'm not sure how it bears on the question. I did find this MathSE thread:

    How do I quantify over the proper class of all the cardinal numbers? where there's a lot of learned back and forth about quantifying over proper classes.

    The consensus seems to be that when we quantify over all sets or all cardinals or whatever, we are really restricting to the particular predicate that defines the object in question. Which doesn't help us, since the extension of a predicate is a class and not necessarily a set.

    And then there is a comment from Asaf Karagila, a professional set theorist and prolific SE contributor. He says:

    No, quantifiers quantify over everything. All sets. Bounded quantifiers are shorthands to make it clearer that we are only interested in a particular set. But you can bound them to a class just as easily.

    As far as I'm concerned, if Asaf says we can quantify over classes, we can quantify over classes. That is good enough for me.

    As far as whether a domain of discourse must necessarily be a set, this seems like a matter of which definition we choose. When we make general statements about sets or groups or cardinals, we are quantifying over proper classes. Whether you call that a domain of discourse or not seems like a question of semantics.

    It is only by models that the domain of discourse is definite, and so it is only by models that it is definite what the quantifier ranges over.TonesInDeepFreeze

    This post crossed paths with mine. As far as I'm concerned if I have Asaf on my side I'm happy. And when we say "every set has a powerset," we are quantifying over the proper class of sets. And again, as far as what a domain of discourse is, that seems like a matter of semantics. If you say it has to be a set, then so be it. When we say every set has a powerset, we are quantifying over a proper class; but if you don't want to call that a domain of discourse, that's ok by me.
  • A true solution to Russell's paradox
    The official, formal definition of a 'model' is that the domain of discourse is a set.TonesInDeepFreeze

    Agreed. We're not talking about models here, though. We're talking about domains of discourse.

    I'll admit that for me, domain of discourse is an informal phrase meaning, "the collection over which we are quantifying," rather than a formal or technical definition. So there may be subtleties I'm missing.

    But model theory is not relevant to this conversation as I understand it.

    My main point in posting was to address this concern of @Sunner:

    What I mean is just that the sentence «there is no set of which every set is a member» clearly says something about every set. But how does this not define some kind of collection or set?Sunner

    I wanted to assure @Sunner that indeed, there is a collection defined by the phrase, "the collection of all sets." The collection is just not a set.

    The unrestricted complement of a set always exists. It just may not be a set.
    — fishfry

    The context of my remark was set theory. In that context, there is no operation of absolute complement but only relative complement.

    Fair enough. But in general, there is an operation of unrestricted complement. I pointed that out and gave an example.

    Even within set theory, there are unrestricted complements. The complement of the set {1,2,3}, within set theory, is the collection of all sets that are not {1,2,3}. That complement is a well-defined collection, but it's not a set. Of course the relativized complement of {1,2,3} in the powerset of the integers is a set. But the unelativized complement is NOT a set, even in the context of set theory. It's a proper class.

    In mathematical logic, a domain of discourse is a set. You may look it up anywhere.TonesInDeepFreeze

    I semi-agree. In prepping my post I looked up Domain of Discourse on Wikipedia, and they did indeed say a domain of discourse is a set. I assumed they were mistaken, and were simply using "set" in its everyday, casual meaning, without regard for the issues of set-hood versus proper classes.

    So I agree that if I looked it up, I'd find at least one source, namely Wiki, that claims a domain is a set. I just think they're wrong, and gave many examples to show why.

    Well, we are quantifying over the collection of all sets.
    — fishfry

    Not formally. Formally, any model of set theory has as a set, not a proper class, as its domain of discourse. For any model, the universal quantifier ranges over the members of the domain of discourse, which is a set.

    Oh my goodness. I do see your point, but I can't agree with it.

    You are saying that when I make a statement such as, "Every set has a powerset," I am really saying:

    1. I assume ZF is consistent.

    2. By Gödel's completeness theorem, if ZF is consistent it has a model, which is a set.

    3. The powerset axiom is implicitly quantifying over that set.

    I simply can not believe that this is the implicit chain of logic behind every universal statement about sets. Indeed, the assumption of consistency is NOT part of set theory. Set theory can not prove its own consistency. The claim that every set has a powerset is true whether or not set theory has a model. All that is required is the axiom of powersets.

    Indeed, "Every set has a powerset" is NOT a semantic claim; it's a syntactic one. It follows from the axiom of powersets. There are models lacking the axiom of powersets where the claim is false.

    Perhaps we're arguing about syntactic versus semantic domains. "Every set has a powerset" is a purely formal statement in the language of set theory. It does not talk about models at all. And it does quantify over the universe of sets, which we know is not a set.

    Likewise it can not possibly be the case that when we say, "The binary operation of every group is associative," we are implicitly quantifying over a "set" of all groups. There is no such set, and I can not believe there's a group theorist living who would agree with your point of view. Of course I have not asked them. But nobody carries around this implicit belief that universal statements about groups quantify over a mythical set of all groups, which provably does not exist. There is no set of all groups and we are not quantifying over it when we make general statements of groups.

    Rather, we are quantifying over a proper class. From where I sit, you are being a bit unreasonable in your claim that there's a set of all groups that we're implicitly quantifying over. That's just not true.

    There are set theories in which classes are formalized
    — fishfry

    Right. But even with those theories, the domain of discourse for a model for the language of the theory is a set.

    Even a class theory such as NBG has only models that have a domain of discourse that is a set.

    Nice to know. Not relevant to our discussion here IMO.

    Moreover, if we tried to allow a proper class to be a domain of discourse, we'd get a contradiction:

    For example, suppose we are doing model theory in a class theory in which there are proper classes. Okay, so far. Now suppose U is a proper class and, for simplicity, we have a language with just one nonlogical symbol. And let R be the relation on U that, per the model, is assigned to the nonlogical symbol.

    Then we have the structure <U R>. But then, unpacking the ordered tuple by the definition of tuples (such as Kuratowski), we get that U is a member of a class, which contradicts that U is a proper class.

    I haven't sufficient technical knowledge, but for sake of discussion I'll concede your point that if we work in NBG or Morse-Kelley set theory, domains are sets. But that's beyond the scope of the discussion. In everyday math, domains of discourse frequently are proper classes. "Every set has a powerset" quantifies over the proper class of sets; NOT, as you seem to claim, over a set model of sets whose existence depends on assuming the consistency of ZF. That's just not right.

    There also is the notion of proper classes as models, or more specifically, inner models. However, I think (I am rusty on this) that when we state this formally, it actually reduces to the syntactical method of relativization, so that when we say L is an inner model of set theory, we mean something different from the plain notion of a model. If I recall correctly, roughly speaking, relative to a theory T, saying 'sentence P is true in "class model" M' reduces to: In the language for T, we define a unary predicate symbol 'M', and P relativized to M is provable in T. So, for example, when considering the consistency of the axiom of choice relative to ZF, we find that the axiom of choice is true in the constructible universe L ("L is a model of AC"), which, in one way of doing this, reduces to: In the language of set theory, define a unary predicate symbol 'L', then we show that AC relativized to L is a theorem of ZF. So if we have a model D of ZF, then the submodel that is D intersected with L (the intersection of a set with a proper class is a set) is
    a model of ZFC. That entails the consistency of ZFC relative to the consistency of ZF. But I am rusty here, so I may be corrected.

    I was never well-oiled enough to even aspire to being rusty in these subjects. I'm sure we're far beyond any considerations relevant to @Sunner. And in that impressively buzzword-compliant paragraph, there is no mention of the domain of discourse. So again, none of this is relevant. You're perfectly correct that ZFC is consistent if ZF is, but what has that got to do with the conversation?
  • A true solution to Russell's paradox
    As a layman, it is interesting to hear that no domain of discourse is truly universal.Sunner

    On the contrary, domains of discourse are often truly universal. They're just not always sets.

    Let's take as an example the simplest and most important thing we can say about sets.

    Two sets are equal if and only if they have exactly the same elements.

    That is, the sets and are exactly the same set. Sets are characterized by their membership, without regard to order.

    What domain are we quantifying over when we make this statement? We are saying, "For all sets X, and for all sets Y, if X and Y have the same elements, then X = Y."

    Well, we are quantifying over the collection of all sets. Twice, for that matter. And as Russell showed us, the collection of all sets is not a set. It is a perfectly well-defined collection. It just isn't a set.

    So in fact we can, and commonly do, quantify over domains that are not sets.

    What are these collections that are "too big" to be sets? They're called proper classes.

    First, a class is any collection defined by a property, or predicate. So, "Is a set" is a property that's true or false about any given individual. The collection of all things for which the property is true, is a class.

    From the Wikipedia page on Classes:

    In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. — Wikipedia

    Some classes are sets. Others are "too big" to be sets, such as the class of all sets. Those are the proper sets.

    Yet, we can still use a proper set as a domain of discourse. We do that every time we state a property of sets. "For every two sets, their union exists." Quantifying over the proper class of all sets. "Every set has a power set." Quantifying over a the proper class of all sets.

    It's a perfectly everyday occurence in math to use a proper class as the domain of discourse. It's so commonplace that we don't even notice ourselves doing it.

    In ZF (standard set theory), there are no official classes, so the usage is informal. There are set theories in which classes are formalized, but those set theories are not usually encountered except by specialists.

    Of interest to this thread is the Russell class, .

    Russell showed that can not be a set. But it's a perfectly well defined proper class. It's the collection of all the things that satisfy the property .

    To sum up: Sometimes domains of discourse are sets, as when we say, "All positive integers other than 1 have a unique factorization into prime powers." Here, the domain of discourse is the positive integers, because we explicitly stated that. The statement becomes false if we change the domain to the real numbers, for example.

    Other times, the domain of discourse is a proper class that is too big to be a set. For example, when we say, "Every set is completely characterized by its elements," we are quantifying over the universe of sets; which as Russell showed, is not a set. But it's still a proper class, and may be spoken of and used as a domain of discourse.

    A domain of discourse is a set.TonesInDeepFreeze

    I'm afraid I can't agree. Many obvious counterexamples come readily to mind. We literally could not do math without quantifying over domains that are not sets.

    * Every set is in bijective correspondence with itself, via the identity map. Quantifying over the proper class of all sets.

    * The identity element of every group generates a 1-element subgroup. Quantifying over the proper class of all groups

    * Every vector space has a basis. Quantifying over the proper class of all vector spaces.

    * Every topological space contains at least two open sets. Quantifying over the proper class of all topological spaces.

    Quantifying over proper classes is so common that we don't even notice it.

    And there is no set of all sets that are not in that domain of discourse.TonesInDeepFreeze

    Correct. But there is a class of all such sets. That class is not a set. It's a proper class, characterized by the property of being "not in that domain of discourse" that was being discussed.

    A final example involving complements of sets, relative and otherwise, that shows how you can annoy your teacher.

    You are asked, "What is the complement of the set of even numbers?" You answer, "The odd numbers, of course." Being a good student, you implicitly assumed that the domain of discourse is the set of integers.

    But the literally correct answer is: "Everything that is not an even number." That includes the odd numbers as well as Captain Ahab, the Andromeda galaxy, and the Mormon Tabernacle Choir. The complement of the even numbers, without any further domain restriction, is the proper class of everything in the universe, abstract entitied included, that are not even numbers.

    The unrestricted complement of a set always exists. It just may not be a set. If the teacher wanted the complement of the even numbers relativized to the set of integers, they should have said so!
  • Anti-vaccination: Is it right?
    Wow I'm really not following this "logic". The argument that a random individual is unlikely to cause harm is generally an excellent argument against separating that individual from society. Infection, unvaxxed status, serial killerhood are all reasons for separation of that person from society.hypericin

    The problem with this line of argument is that the vaxxed are every bit as contagious as the unvaxxed. The number of breakthrough infections is significant. How significant we don't know, because The CDC stopped tracking breakthrough cases in May. Wonder why.

    Now it's true that if you are vaxxed and you get covid anyway, your symptoms will be reduced. That's a good reason to be vaccinated. But if you do get a breakthrough case, you're just as contagious. And there are a lot of breakthrough cases. Israel, which has an 80% vaccination rate, reports that the Pfizer shot is only 39% effective.

    So your numbers just don't add up. The average person you meet is highly unlikely to be contagious at that moment in the first place. Of those that are, they're more likely to be vaccinated for the simple reason that most people are vaccinated, and the vaccines simply aren't that effective.

    So there's no scientific argument at all to decree that unvaccinated individuals should "forfeit the right to move freely in society," which was @Wayfarer's original quote that I objected to.

    And even if you could make such an argument, the downsides of such Othering of a group -- dirty, disease-ridden, danger to society -- would outweigh any good. I defy you to name any time in history that such an Othering came out well. And I'm sure you know all the bad instances I could name.

    The vaccinated and infected are rare. If they are identified as such, they should be restricted.
    Drunk drivers are rare. If they are identified, they should be restricted.

    Ok. Now that is entirely different than what @Wayfarer said. You agree that the vaccinated yet infected should be isolated. That's not even what we're talking about, and you are in effect conceding my point.

    Vaccination should be a requirement for entry to high risk areas such as transportation, supermarket, bars, restaurants, movie theaters, etc.hypericin

    I'm not even talking about that. @Wayfarer said that the unvaccinated should "forfeit the right to move freely in society." I'm pointing out that this is one, not scientifically supported because of the high vaccine failure rates, the fact that the vaxxed are just as contagious as the unvaxxed, and the Othering that would inevitably produce results that nobody would want to see.

    I'm not talking about any other issues, whether the unvaxxed should be allowed into bars and whatnot. I'm talking about "forfeit[ing] the right to move freely in society." That's a tremendous overreach and poorly thought out position. I wouldn't be surprised if @Wayfarer would be willing to say, "You know, I just typed that in, but I didn't really think about it, and it's wrong on many levels and totally unworkable." I don't know. You're taking up an argument on someone else's behalf but you yourself don't seem to remember what the argument was.

    The rest of your post is slippery slopehypericin

    Asking someone to drill down a level of detail is not a slippery slope. If you propose to restrict the free movement of the unvaccinated, how do you determine who they are? You have to interrogate everyone. Perhaps you discount an additional, say, one million police/citizen encounters per day. Maybe you haven't read the papers about public opinion of police encounters. It's not a slippery slope argument to challenge a highly impractical suggestion by asking the proposer to supply the details of how their idea would be implemented.


    How so? The proposal is to restrict the free movement of 75 million or so Americans. I think I understated the likely consequences of such a nonsensical and dangerous idea.

    and race baiting.hypericin

    How so? I pointed out that only 31% of blacks are vaccinated, so that if you start restricting their movement or rights in large numbers, you would create social problems that hardly need to be mentioned to be perfectly obvious to anyone who follows the news. You act like you don't follow the news.

    In any event, the New York Times made much the same point when they reported four days ago, Why Only 28 Percent of Young Black New Yorkers Are Vaccinated.

    Perhaps you could read that article in its entirety and explain whether you think the New York Times is race baiting too.

    In any event, my comments were regarding New York City's plan to require vaccinations for entry into public spaces. The New York Post reported on the details of the plan today. They said that the venues themselves would be fined, not the individuals. Seems that unlike you, New York City actually put some thought into the consequences of their policy, and enacted enforcement mechanisms intended to avoid the obvious racial consequences instead of exacerbate them.

    See how that works? People have an idea, but then they have to think through the consequences and modify and implement their policies accordingly. That's what you call "hysteria" and "race baiting." I call it basic thoughtfulness and common sense.

    This doesn't address the larger harm the unvaccinated, and the scumbag public figures that encourage them, do to society.hypericin

    Yes, you're just the person I'd pick to Other 75 million Americans and select them for special treatment. What could possibly go wrong?

    But as I pointed out, your statistical logic is flat out wrong. The vaccines aren't even 50% effective. There are huge numbers of breakthrough cases, so many that the CDC won't even report them. And while the vaccines keep you from getting as sick as you would without them, you are just as contagious. So there is no scientific argument to be made that the unvaxxed are any more dangerous to society than the vaxxed.

    What's dangerous is people letting their fear cloud their common sense. And don't you think the lockdowns themselves are harming society? Children born during pandemic have lower IQs, for one thing. The increase in alcoholism, domestic violence, and substance abuse are noteworthy. Every action has consequences both good and bad. There's no thoughtfulness and balance in the simplistic "lock everyone down and shoot the unvaxxed" kind of thinking.

    What, am I being hysterical again? UP AGAINST THE WALL: California Congressional Candidate Says Anti-Vaxxers Should Be Shot. I'm not hysterical, I'm just someone who follows the news from a variety of sources.

    If everyone was vaccinated, and diligently performed basic social distancing and hygiene during local outbreaks, we might be done with the pandemic, at least in the US.hypericin

    Might. And might not. The adverse reactions to the vaccine are off the chart, and nobody knows the long term consequences because there haven't been any studies. You're making a claim that can't be backed up by science. You're letting your lizard brain flood you with fear. Take a step back and try to think. Who's violating hygiene and social distancing? Who's arguing against it?

    I'm arguing against the thoughtless and mindless claim that the unvaxxed should "forfeit the right to move freely in society." That's what I'm arguing against and that's ALL I'm arguing against. Are you sure you're not the one who's hysterical?

    Instead, hospitals and morgues are filling up again, and actual freedom, the freedom to enjoy life without risk of death or mutilation, has slipped away.hypericin

    Well that's just terrible, I agree. I'm disagreeing that it's practical to selectively restrict the freedom of movement of 75 million Americans without a lot of unexpected and highly negative consequences. Why don't you stick to the actual topic of what I said, and try to think the issue through?

    Really, from that perspective the restriction of freedom of movement is too mild. Vaccination should be mandatory, full stop.hypericin

    The vaccines don't even work all that well. The adverse reactions, including death, are off the charts. Nobody knows if they're safe long term because the studies haven't been done. They have no FDA approval. I think you are so panicked by the media hysteria you'd send Jews to the ovens if someone told you they carried disease, which is exactly what the German media told people. You're just that kind of person. I hope you'll step back and get a grip on your own hysteria.

    I haven't considered any government enforced denial of freedom of movement, so any disagreement I might raise isn't to that effect.Cheshire

    Ok, and I appreciate your saying that. Because other than that one point, I haven't said or advocated anything. Except to push back on my hysterical and propaganda-addled friend @hypericin.

    My issue is with the pronouncement that the possibility of a vaccinated person spreading a virus and the possibility of an unvaccinated person spreading the virus are treated as equal. Or the first makes the latter not matter. It seems to me a strong argument could acknowledge that one is taking place regularly and the other is somewhere between rare and not impossible. You disagree above, but maybe I missed something.Cheshire

    The numbers don't bear you out. As I posted, the Israelis, who are 80% vaccintaed, report that the Pfizer shot is only 39% effective. The numbers for the other shots are in that range. And now everyone is supposed to get a booster shot. So in terms of effectiveness, the vaccines are essentially a bust. Yes they do make you less sick than you'd be otherwise; but you are just as contagious.

    And since most people are vaxxed, the chances that the next person you run into is contagious and vaxxed or contagious and unvaxxed are more or less the same. So there's no statistical argument to be made about treating one class differently than the other on the basis of contagion.

    The Wall Street Journal is a Murdoch paper, is it not?Wayfarer

    Yes, but are you denying their factual claim that only 31% of black are vaccinated? As I posted above, the New York Times reported that only 28% of young blacks are vaccinated. If the WSJ prints a fact, then it's a verifiable fact no matter how you feel about their editorial stances. Right? If you don't like the WSJ's 31% number, then just take the NYT's 28% if you prefer that.

    More likely crying crocodile tears over the poor benighted black population to feed meat to their civil-libertarian right-wing audience than out of any genuine concern for the former.Wayfarer

    Well I don't much care either, I just want to see the hilarity ensue. A bit of sarcasm, don't get excited.

    But as I also linked above, the New York Post reported that the enforcement actions in New York City will be against the venues and not against individuals. Meaning that they took my point to heart and realized that the optics of arresting or ticketing or shooting (as one California congressional candidate wants to do) unvaccinated black people would not look too good in heavily black NYC.

    You see once again that I am trying to get people to be thoughtful about what they're saying; and as support for my position, New York City itself was thoughtful about this point. Whether they are genuinely concerned about black people or whether they just don't want the bad optics; they're only fining venues and not individuals.

    Murdoch media worldwide are probably alone responsible for tens hundreds of thousands of infections by spreading their anti-vaccination nonsense along with all the many other lies and propaganda they peddle around the world every day.Wayfarer

    LOL. Don't hold back, tell us how you really feel.

    You're right, I actually quoted too much from the WSJ article. All I cared about was the 31% number. If I'd known it would trigger you I wouldn't have bothered.

    But it's a fascinating point. The stanard mainstream belief is that the unvaxxed are MAGA-hat wearing racist deplorables. But it turns out that the real unvaxxed are blacks and Latinos. And Ph.D.s. That's right, the single group with the greatest degree of vax avoidance is people who hold PhDs. The article didn't say why, but my guess is that people who have done actual scientific research can recognize the sham, politicized pseudoscience for what it is.

    I would never cite or refer to any articles published by any Murdoch outlet in support of any point whatever.Wayfarer

    I gave you a New York Times link reporting much the same information. You seem to have forgotten to argue your own point, you got so triggered by the WSJ.

    I acknowledge that all forms of lockdown and restriction of movement are an infringement on civll liberties, but in light of the severity of this illness, I believe that imposing a lockdown is a lesser of two evils. I mean, giving up some freedom of movement and even income, is generally preferable to getting a life-threatening illness, in my opinion.Wayfarer

    Now that is an entirely different matter that your original suggestion that the unvaxxed should forfeit the right of free movement.

    I might (or might not) argue against a general lockdown, but I'm not discussing lockdowns here. Lockdowns affect everyone equally. To implement a lockdown you don't have to Other 75 million people (in the US), subject everyone to demands to show their papers, and add a few million or so daily police/citizen interactions. Those are the issues I'm concerned about.

    Lockdowns, regardless of their merits, apply to everyone equally, and therefore don't have the problems I'm concerned about regarding your earlier idea.

    Australia generally has succeeded in controlling the infection, although the Delta variant outbreak that started in Sydney June 16 has well and truly escaped the net.Wayfarer

    I looked this up. Australia has some 25M people versus 300M in the US. And the US has only 1.27 times the area. So Australia has a much smaller population and much much sparser population density. You have no idea what it's like to get 300 million crabby Americans to do anything.

    And besides, having just discovered that the US government has been lying to us for 20 years about the progress of the war in Afghanistan, which we are even as we speak losing in a majorly humiliating fashion, I doubt that American are inclined to believe anything the government says. I remember the anti-government sentiment of the 1970's after our loss in Vietnam, and I expect the same to happen now. So you can't lock down the US. Can't be done even if was the black plague.

    And from what I hear, Australia has surrendered its civil liberties in ways that. to this American, are truly frightening. I always thought of Austrlians as liberty lovers, Crocodile Dundee kinds of folks. Guess that was only a movie.

    I'm not saying lockdowns wouldn't be effective. Only that American is an unruly country full of unruly people not currently inclined to believe anything the government says. It's just a practical matter.

    There is a lot of commentary that the mistake the NSW Govt made was in not locking down faster and harder - there was a super-spreader event on June 26th that transmitted the virus from Sydney’s East to the vast Western Suburbs, which is when it really began to escape, as there are many more large households and a high degree of geographic mobility. That’s where it remains - yesterday’s case numbers were 344, two deaths, and also cases appearing in regional centres.Wayfarer

    I don't disagree that locking everyone down works in a pandemic. China was apparently successful doing that. But they're a majory authoritarian regime. And like I say, Australia has a much smaller population.

    But mainly, why are we talking about lockdowns? I'm not talking about lockdowns. Lockdowns are imposed on everyone equally. To lock down a country you just patrol the streets and shoot or fine or chastize everyone who's out without a good excuse. Your original idea, to restrict the free movement of only the unvaxxed, involved interrogating everyone, necessitating millions of cop/citizen interactions every day, many of which, if you read the US papers, don't go particularly well, especially when there are minority groups involved. Black people are not interested in being accosted by the police in the US and frankly I can't say I blame them.

    And by the way, where are you getting all these extra cops? As a result of the anti-cop sentiment in the US, cops are quitting in droves. You can't find enough cops to enforce a selective lockdown that involves asking everyone for their papers.

    So a lockdown for all, whether it's a good or bad thing, is completely different than a lockdown for some.

    As I think I said earlier, community attitudes to vaccination have dramatically shifted in the last month, due to the insidious nature of this variant, and the fact that there’s a lot of younger people in ICU, with two otherwise healthy and comparatively young people dying. I think everyone now realises that getting a severe case of COVID-19 is a life-changing event even if it doesn’t kill you. So vaccination rates have ticked up enormously, supply problems are being overcome, the Moderna vaccine has now been approved and the country is on track to be around 80-90% vaccinated by year’s end.Wayfarer

    Not disagreeing. Only pushing back hard on the idea that the unvaxxed should have their freedom restricted; especially because there are a lot of breakthrough infections, and that the vaxxed are just as contagious as the unvaxxed. So the statistical argument for restricting only the unvaxxed is false. Let alone the problems of asking for everyone's papers in a country like the US that is in the midst of both an anti-cop hysteria and a crime wave.

    As to whether lockdowns have to be enforced, I still don’t see any other option.Wayfarer

    Why did you so radically change the subject? You can enforce a lockdown easily, just shoot/arrest/fine/shame anyone you see on the street.

    A selective lockdown, on the other hand, entails interrogating EVERYONE and asking for their papers. Which entails a lot of cop/citizen interactions; which, in the US, often go south in terrible ways. So that's a bad idea.

    AND it's statistically unsound, since your chances of meeting a contagious vaxxed or a contagious unvaxxed person are about the same, and they are equally contagious. So you haven't got a case, and you have a very poorly thought out position.

    Have you backed off your earlier proposal, or just changed the subject to universal lockdowns?

    The laissez faire approach of some of the US GOP governors simply results in higher rates of infections and more deaths.Wayfarer

    Statistics are mixed. Some red states are doing better. But I am not discussing laissez faire approaches. I'm pointing out that restricting the free movement of ONLY the unvaxxed would one, be a complete policing disaster; two, would in fact fall heavily on minorities, as I've documented; and three, is flat out wrong anyway since the vaxxed are just as contagious and there are a lot of breaktrhough infections, which in the US the CDC will not even report.

    So your idea is a non-starter. Is that why you changed the subject?

    Some US states with comparable populations to NSW are having thousands of cases and hundreds of deaths every day, which NSW might easily be matching, had not the lockdowns been enforced.Wayfarer

    Well we're not talking about lockdowns at all. You proposed selective lockdowns against a population that can't be distinguished from the vaxxed and therefore needs to be challenged for their papers; would mostly consist of harassing minorities; would be an unmitigated policing disaster; and wouldn't help anyway, since the vaxxed are potentially just as contagious.

    And that's the only point I was making. A selective lockdown against the unvaxxed wouldn't work and wouldn't help.
  • Square Circles, Contradictions, & Higher Dimensions
    Square circle as a genuinely contradictory object would look like a square and like a circle from the same perspective (and at the same time and under all other same circumstances). Such an object cannot existlitewave

    In the taxicab metric the unit circle is a square. There's a picture of a square circle on that page. A circle is the set of points equidistant from a given point. If you choose your distance function appropriately, a circle can be a square.

    Note that this is very different than an unmarried bachelor. A bachelor by definition is a male who is not married. so that a married bachelor is indeed a contradiction.

    But a circle and a square are NOT defined as each other's opposites, nor are they mutually exclusive at all. People should stop using square circles as an example of a contradiction, because in fact there are square circles.

    Note that if you define the distance between two points to be the sum of the horizontal and vertical distance between them, then the distance of each red point from the blue point is the same in each case, and that these are therefore square circles. (This is the Wiki image).


    A square circle would be a regular polygon with four sides, the perimeter of which is equidistant from a given point on the same plane.

    Draw me one of those.

    Done. It all depends on how you define distance. Standard Euclidean distance (square root of the sum of the squares of the respective differences of the coordinates) is only one way of defining distance. Even in physics, Euclidean distance is only a special case of a more general way of defining distance.
  • Anti-vaccination: Is it right?
    Whether or not they are as contagious once infected, they are infected at lesser rates. As continual testing of everyone is impractical, they therefore present less danger to the public than the unvaccinated.hypericin

    I have a small point and a large point to make. I'll start with the small. First, to recap. @Wayfarer posted:

    Perhaps their freedom of movement may also be curtailed, though less so. Perhaps 'social distancing', the wearing of masks, and other hygeine measures, will henceforth remain as part of civil society.Wayfarer

    I pointed out that when the vaccinated acqire a breakththrough infection, they are just as infectious as the unvaxxed. Therefore their movement should be restricted too. And that's when you pointed out that they're infected at lesser rates.

    This point is easily refuted. The fact that the average vaxxed person is statistically unlikely to infect you means nothing. After all, the average person is not a serial killer, but we endeavor to take serial killers out of society to protect the public. The argument that a random individual is unlikely to cause harm is no argument against separating that indvidual from society.

    Likewise drunk drivers, which you mentioned.

    The unvaccinated are making this choice to (in their mind) improve their well being, at the expense of the public well being. It is therefore rational public policy to restrict their freedom of movement, to both protect the public well being, and to discourage this selfish choice.hypericin

    In terms of protecting the public well being, you need to restrict the movement of the vaccinated as well, since they are just as contagious as the vaccinated, even if perhaps fewer in number.

    So in the end, your point is purely punitive and unrelated to public health.

    The situation is rather similar to driving. Everyone on the road presents some danger. But drunk drivers, as a result of their selfish decision to be drunk drivers, present a greater danger. Therefore their freedom of movement is restricted, to protect the public and to discourage drunk driving.hypericin

    But by your logic the contrary conclusion is forced on us. The average driver is statistically rare, even if all too common. Since contagious vaxxed people and drunk drivers alike are statistically rare, they should both be free to travel. After all, your likelihood of encountering either one is relatively low.

    So your statistical argument is wrong, and all you have left is your feelings that the unvaxxed should be punished for their "selfishness," as you put it. How about people who don't get their flu shots? People who don't contribute enough to charity? Those with unpopular political opinions? If punishment is your only argument, you yourself wouldn't want to live in the world you wish for.

    Now to the larger point. @Wayfarer suggests,"Perhaps their freedom of movement may also be curtailed ..."

    Ok. Let's think that through. I can think of two extremes. One is what is done by the a grocery store near me. They have a sign out front that non-vaccinated people must wear masks. They don't check, and rely on the honor system. Then again I live in a relatively small, laid-back town with a relatively low infection rate.

    The other alternative is full on police-enforced compliance. You're walking down the street, and the police may ask to see your papers. If you can't produce a vax card, you're arrested on the spot.

    Those are the extremes. Perhaps you and @Wayfarer would like to say, specifically, how you think the restriction of free movement in the US (or your country, whatever it may be) should be implemented.

    I well remember a few years back when the US state of Arizona wanted to implement a "show your papers" law to challenge brown-skinned people on their immigration status. Decent people across the country were rightfully outraged. Most people think of "show me your papers" as something said in a German accent in a late-night black and white movie from the 1940's. In the US, at least, we don't "show our papers" to the authorities without the police having probable cause or a damn good reason.

    So perhaps you think this is a good reason, and that American citizens should be required to show their papers on demand. Can you see how this would quickly go south? Did you get your flu shot? Have any unapproved political opinions? Maybe you tweeted that "All lives matter," or that you believe in rationality and hard work. Those ideas are racist, according to the Smithsonian.

    Can you look at history and give me an example of when "show your papers" ever came out well for a society and didn't quickly get abused?

    How about when you're driving? Surely if freedom of movement is to be constrained, we need highway checkpoints. That's not so farfetched; there are already interior immigration checkpoints as far as 75 miles inside the US border, where travelers staying entirely within the US may be stopped, interrogated, and searched. Of course if they happen to find a joint or some other contraband, that's your bad luck. What, the Fourth Amendment to the Constitution forbids such an abomination? Sadly, courts have repeatedly allowed these interior checkpoints. It would be easy to set up a lot more of them to check people's vaccination status.

    Do you think that's a good idea? Is that the country you want to live in?

    Let me point out one more "inconvenient truth," as Al Gore once put it. Who in fact are the unvaxxed in the US? In the popular imagination they're white, MAGA hat-wearing deplorables with unapproved ideas.

    In fact, the unvaxxed are blacks and Latinos. Don't believe me?

    So what are you going to do? Start pulling over or checkpointing black drivers, accosting blacks and Latinos on the streets and demanding their papers, refusing access to great numbers of blacks and Latinos to restaurants and movie theaters? Can't wait to see how that works out.

    We have a real-life datapoint coming up. In New York City, restaurants and other indoor venues will soon require proof of vaccination for entry.

    But it turns out that only a fraction, one third or so, of NYC blacks are vaccinated.

    The policy takes effect this Monday, August 16, and enforcement begins in September. It will be administered by the health department and not the police. So can you imagine what it's going to be like when two thirds of the black people in New York City are banned from restaurants?

    The WSJ has the summary. Most of the article is paywalled but the free part says plenty.

    The modern progressive speaks the language of high-minded purpose but always ends with coercion. Witness New York Mayor Bill de Blasio, the uber progressive, who announced Tuesday that New Yorkers will soon need proof of vaccination to do everything from dining out to working out at a gym. He’s proud that New York is the first U.S. city to impose such a mandate.

    “It’s time for people to see vaccination as literally necessary to living a good and full and healthy life,” he said at his press conference. You gotta love Mr. de Blasio telling you what is necessary for a good and full life. According to the data, roughly 55% of the city’s residents are fully vaccinated, ranging from 46% in the Bronx to 67% for Manhattan.

    His response is to exclude the unvaccinated from many of the functions of daily life. He doesn’t seem to care that this burden will fall heaviest on the city’s black population, which is only 31% fully vaccinated (versus 71% for Asian Americans, 42% for Hispanics and 46% for whites).
    — WSJ

    @Wayfarer and @hypericin, is this what you want? 69% of black people in NYC excluded from public life? And if not, then what DO you mean when you talk about restricting people's movement?

    Feedback appreciated. You disagree with my facts? My reasoning? Or are you you all in on "show me your papers" to every non-white face in New York City? You want to bring back stop-and-frisk but for vax cards instead of guns and knives?? And if you did implement nationwide walking and driving checkpoints, how long do you think it would be before the inevitable scope expansion and mission creep set in? Check for your vax card, check your wants and warrants. Behind on your child support? Carrying any unapproved contraband? Tweet any unapproved thoughts recently?

    You serious? Anyone thinking this thing through? Or do you all want to live under the Chinese social credit system and can't wait till it's implemented here? I'm afraid that's exactly what some people want.
  • Five different calculuses
    Some second hand books that I have leafed through, seemingly of the period when I was at school, averred that calculus was all about areas and speeds, though that had never had anything to do with the lessons I had "had". Is this a class thing?Fine Doubter

    Velocities and areas are applications of calculus, as in Newton's fluxions and fluents, corresponding to today's derivatives and definite integrals. In pure math, one only uses velocity and area as illustrations to help students understand, but they're irrelevant to the mathematical content.

    Find Doubter, nice pun on "Find Outer."

    reforms of the calculus in the 19th C? In terms of limits, as explained copiously hereabouts by fishfrybongo fury

    Thanks for remembering :-)
  • A patent for computing, can someone help out?
    Well if you want a perfectly unhackable root, this idea might suffice.Shawn

    Do you know that in normal operations, the OS needs to constantly make changes to the kernel in privileged mode? How would you determine what's a legitimate change versus a malicious one? If you disallow all kernel changes the computer won't boot and won't run.
  • A patent for computing, can someone help out?
    Well, as a direct outcome of having TimeShift running on the go, it would be a safer system by default. The root folder would restore itself once any alteration would be attempted on it by monitoring any attempted change to values on the kernel.Shawn

    How would necessary privileged mode (aka kernel mode or supervisor mode) operations be done? Such as adding a device, adding a new process to the process table, and so forth?

    See Kernel.

    The kernel is a computer program at the core of a computer's operating system and has complete control over everything in the system.[1] It is the "portion of the operating system code that is always resident in memory",[2] and facilitates interactions between hardware and software components. A full kernel controls all hardware resources (e.g. I/O, memory, Cryptography) via device drivers, arbitrates conflicts between processes concerning such resources, and optimizes the utilization of common resources e.g. CPU & cache usage, file systems, and network sockets. On most systems, the kernel is one of the first programs loaded on startup (after the bootloader). It handles the rest of startup as well as memory, peripherals, and input/output (I/O) requests from software, translating them into data-processing instructions for the central processing unit.

    So the point is, how would the computer function if every time the OS needed to modify the kernel's data structures, you backed those changes out? How would you distinguish between legitimate and malicious alterations to the kernel?

    What if a friendly actor needs to make a change to this root folder for legitimate reasons?darthbarracuda

    @Shawn, What he said.
  • A patent for computing, can someone help out?
    I've had this idea of making computing fasterShawn

    For example, when a hacker tries to alter the systemShawn

    These two are in direct conflict.

    After all, any intrusion detection scheme must necessarily slow down a computer. It takes extra cpu cycles to detect intrusions. It's like the extra time it takes you to get in your front door because you need to use a key. Any security measure always takes extra time.

    When a piece of code attempts to alter memory, how does your system know when it's legitimate or not? After all doesn't a computer operate by way of software constantly making changes to memory?

    Or (perhaps?) you are saying that when a piece of code attempts to alter the OS code it's automatically restored to its default state. The problem is that you often have to alter the state of the OS in memory. That's why computers operate in either privileged mode or user mode. In user mode you're not allowed to change the OS, in priv mode you are. I don't see how a computer could function if you disallowed priv access.

    As a simple use case, suppose you buy a new printer and connect it to your computer. You have to install a driver and register the computer with your OS. In Windows there's a clunky user procedure and in Mac it's automatic, but the same things are being done either way. In your scheme you would reject all attempts to add new hardware.
  • A patent for computing, can someone help out?
    The CPU would simple directly input integer values into the OS by having mounted on the RAM and bypassing slow hard drives through an application like TimeShift.Shawn

    I'm afraid I can't comment. But (for sake of discussion) how does the CPU input anything anywhere? It has to be instructed to do so by instructions stores in memory. And all instructions are in hardware at the time they're being executed. If the program (OS or application) is on disk, the page containing the currently executing instruction must be loaded into memory and into a CPU register for execution. So I am honestly not following your idea. But it could be me, my technical knowledge on operating systems and hardware is not current.
  • A patent for computing, can someone help out?
    fishfry, may I ask for your opinion?Shawn

    I'm honored. I could not parse the following:

    utilizing the ECC-RAM utilized nowadays in servers to be able to predetermine the state of a computer through directly interacting with the system OS itself.Shawn

    ECC Ram is just error-correcting memory. It wouldn't offer any functional difference from any other kind of RAM. So I didn't understand that part. How would it be able to "predetermine the state of a computer?" Are you talking about branch prediction? This is a 20 or 30 year old idea as far as I know.

    What does it mean to directly interact with the OS? Of course the software directly interacts with the hardware, especially the privileged kernel. So I'm afraid I couldn't make sense of this line and kind of got stuck here.

    Booting off RAM? Is this like a RAM disk?

    I looked up TimeShift, it's a backup thingie, creates and restores snapshots.

    I'm afraid I couldn't understand exactly what you're getting at. There's always Stackexchange or some of the Reddit groups for finding computer experts.
  • Anti-vaccination: Is it right?
    I get a lot if insights from your posts on maths.Wayfarer

    LOL. Can you comment on my point here? You said the unvaccinated should have their freedom of movement restricted because they may spread covid. I linked an article showing that the vaccinated spread covid at the same rate as the unvaccinated. In view of that, shouldn't the vaccinated be prohibited from free movement as well?

    Have you a response?
  • Anti-vaccination: Is it right?
    But then, the person that might end up paying the price for that might be someone they infect. So perhaps the compromise is, anyone has a right to refuse to be vaccinated, but by so doing they forfeit the right to move freely in society.Wayfarer

    Excellent point. And by the same token, I assume you favor restrictions on the free movement of the vaccinated, since they too may infect others.

    Vaccinated People May Spread the Virus, Though Rarely, C.D.C. Reports

    You agree? If not, why not? If perhaps you're going to invoke the word "rarely," what's your standard for restricting free movement? Have you hard data on how many people are being infected by the unvaccinated? Is that rare, or common? What are the actual numbers? What is the science? I think people on all sides of these issues would like to see the data. Why is the Biden administration itself growing frustrated with the lack of CDC transparency?

    From that latter article: "Public Health England published data collected through the end of July showing that vaccinated people are less likely than the unvaccinated to become infected with Delta, but once infected, they may be equally contagious."

    I guess we SHOULD restrict the vaccinated individuals' freedom to move through society after all. If you have hard data on any of this I think we'd all be grateful, particularly the Biden administration. Of course you don't have data, because the CDC won't release it.
  • Dating Intelligent Women
    Smart men never get married lol. It's a bad deal.hope

    Newton either. LOL. This was an old thread!
  • The Creativity of AI (an exerpt from recent writings)
    not von Neumann but of neural networksD2OTSSUMMERBUG

    Neural nets run on conventional computers. They're a clever way of organizing data mining, but they are not a new paradigm of computation. They are physical implementations of Turing machines, in fact finite state machines. The hardware they run on is perfectly ordinary, off-the-shelf. There is no computational difference between a neural net and your laptop; or, for that matter, the 386 machines of the 1990's.

    There is so much AI hype out there it needs to be countered. This subject is on my mind because just last night I heard this guy on the radio.

    The Myth of Artificial Intelligence
    Why Computers Can’t Think the Way We Do
    Erik J. Larson
  • The Creativity of AI (an exerpt from recent writings)
    Where does the creativity of AI come from?D2OTSSUMMERBUG

    An algorithm can never be creative. In particular, there is no hope whatsoever that the current big data and machine learning approach to AI can ever achieve creativity. The current approach is entirely based on datamining and statistical analysis of things that have already happened.
  • Golden Rule, Morality and BDSM
    So … can we say that this rule is where all morality should stem from?Deus

    Masochist: Beat me.

    Sadist: No!
  • Simone Biles and the Appeal to “Mental Health”
    I begin this thread in response to the backlash I receivedLeghorn

    Never helps.

    In fine, courage used to be overcoming fear. Now it is succumbing to it.Leghorn

    Agree. I'm reminded of the blitz, when Hitler's Luftwaffe bombed London every night for eight months from 1940-41. If it happened today, the Brits would have surrendered to Germany "to prioritize their mental health." What they did instead was show incredible courage, huddling in the underground subway stations and stiffening their spines, till Hitler gave up and went off to attack Russia. In retrospect it was the bravery of those Londoners who turned the course of the war.
  • Moods are neurotransmitter levels working in the brain.
    Is there anything wrong in stating that neurotransmitters are scientifically assumed to play a role in the regulation and experience of affective behavior?Shawn

    Yes. Here's one striking example. There's no proof that serotonin insufficiency causes depression and mental illness, yet millions are on SSRIs. These drugs have major side effects Literally every single 20-something mass murderer and school or movie theater shooter turns out to be on SSRIs. Are the drugs causing the bad behavior? We're not allowed to ask. The politicians blame the guns and nobody ever asks about the drugs. Of course the drug companies say that well, the kids were emotionally disturbed to start with which is why they were on psych drugs. Nobody ever looks into the high correlation between psycho killers and SSRIs.

    A casual Google search on the phrase, "psycho killers and SSRIs' turned up a bunch of links.


    Do Antidepressants Increase Violent Behavior?

    Antidepressants are supposed to make people feel happier and more at ease, but a study has linked several prescription antidepressants to an increased risk of violent behavior, including physical assault and homicide.

    This is a major problem with the claim that in order to lead a happy life, all we need is a little pill to balance out our neurotransmitters. Not only is there no scientific evidence for the proposition; there is mounting evidence that these drugs cause harm.
  • On Gödel's Philosophy of Mathematics
    The first video (I didn't watch the second video) is stupid nonsense and disinformation.

    In this context, infinite summation is defined only for converging sequences. If the rules of definition are violated, then, of course, contradictions may be derived. There is no mystery or even problem about that.

    The person at the blackboard says, "The problem with infinity is all sorts of weird things happen when you're dealing with infinity". First, that doesn't even mean anything. Second, instead of explaining that the fallacy is in using an undefined notion (infinite summation on a sequence that does not converge), the person at the blackboard doesn't even suggest how we may investigate further to see that there is not an actual conundrum.

    The video is yet another example of Internet ignorance and disinformation. That person seems to be teaching a classroom. He should be told by the school administrators to clean up his act: If he wants to present mathematical challenges, then he should provide his students with the benefit of techniques and information for solving the challenges rather than obfuscate with "weird things happen".

    :up: :up: :up: :up: :up:
  • Banno's game
    Reminds me of Calvinball.
  • On Gödel's Philosophy of Mathematics
    Contra that, in the item I cited, there are mentions of empiricism and the example of the applicability of the mathematics of elasticity to engineering a bridge; see (3) here.Banno

    Yes but your (3) is under the heading that says: "Gödel apparently characterises the syntactic view as consisting of three requirements:"

    So he is characterizing the syntactic view. But we've already agreed that he's describing the syntactic view in order to disagree with it.

    I would be extremely surprised to find that Gödel advocated Platonism because of the use of math in building bridges or whatever. On the contrary, Gödel's Platonism argued for the existence of large cardinals and other highly abstract and decidedly un-physical entities of set theory.

    Again, your (3) is under the heading of the view that Gödel is trying to refute, not advocate for. Am I missing something?

    so again:
    It seems that for
    , being consistent and being true are inseparable; that mathematical truth is displayed in mathematic's empirical applicability; and that hence the consistency of mathematics shows that it is empirically applicable and hence incompatible with the syntactic view.
    — Banno

    So if we trust this secondary source - I have no reason not to - Gödel held something like that we can speak of mathematical propositions being true because they are empirically applicable;

    Two things may be true here:

    1) It's perfectly possible that Gödel said that, or believed that; or at the very least, that some third-party interpreted his beliefs that way; and

    2) I totally do not believe Gödel himself justified his Platonism on day-to-day physical grounds. Gödel believed in large cardinals, and that's one reason he did not believe we should adopt the axiom of constructability.

    In other words: You may be right but I still don't believe it. Both those can be true. I believe many false things.

    that truth is at a meta-level to the mathematics itself; and that together these show that maths is not just stuff we make up.Banno

    I just can not believe that Gödel paid much attention to the construction of bridges as a justification of large cardinal axioms. Like I say: even if I'm wrong, I'm sticking to my sense of the matter.

    The exact argument remains opaque, but that is the implication.Banno

    But wasn't the use of the word empiricism listed under Gödel's description of the syntactic view? The very view that he's describing in order to refute?

    I am feeling uneasy expressing strong opinions on things I know nothing about. I don't know what Gödel thought. I do know that he (later in life, at least) came to reject V = L (the axiom that says that the constructible universe is the true universe of sets) because L doesn't have enough sets. And the sets that it's missing are large cardinals, transfinite quantities so large they can't be proven to exist in ZFC and that are as far from empirical concerns as can possibly be.

    That's really all I know about it; and I sincerely agree that I could well be completely wrong.
  • On Gödel's Philosophy of Mathematics
    From what I could glean from Wikipedia, a constructible set is one which can be, well, constructed via set theoretic operationsTheMadFool

    Wrong article. The Wiki page on "constructible sets" has something to do with topology. Different usage.

    This is the relevant page.

    The constructible sets are built out of first-order formulas in stages. I don't know enough about this to give a simplified explanation. Basically each stage is built from first-order statements with parameters and quantifiers that range only over the previous stage. It's very logic-y. Wish I could say more but I don't know too much about it. Only that Gödel cooked it up to prove the consistency of AC and CH.
  • On Gödel's Philosophy of Mathematics
    Way above my paygrade! Thanks though. I hope to advance my knowledge in math ASAP.TheMadFool

    Above mine too actually. My point was that Gödel apparently believed in an expansive view of the set-theoretic universe, and that his Platonism was probably motived by that and not by practical considerations such as its use in physics.

    FWIW Gödel cooked up a model of set theory called the constructible universe, in which the axiom of choice and the continuum hypothesis are true. That shows that they're consistent with ZF.

    So why not just adopt the axiom that the constructible universe is the true universe of sets? If you did that, AC and CH would be theorems and we'd be done. The reason this assumption is not made is that most set theorists believe that the true universe of sets (if there even is such a thing) has way more sets in it than just the constructible ones. Gödel apparently first believed that the constructible universe was the true universe, and later came to not believe that.

    It's trivial stuff, but may not have been around beforejgill

    Or it has been waiting around since the big bang for you to come along and discover it.

    Probably there are different versions of him to be consider of: Platonism, realism, relativism, etc...javi2541997

    Agreed, his thoughts were probably a lot more complex than the articles about him can capture.
  • On Gödel's Philosophy of Mathematics
    One of the reasons why some, like Gödel I suppose, believe math is discovered is how math seems to,

    1. Describe nature (math models e.g. Minkowski spacetime)

    2. Describe nature accurately (we can make very precise predictions to, say, the 15th decimal place)

    My sense is that these mundane physical considerations were not on Gödel's mind. He believed in the Platonic existence of abstract sets including large cardinals, sets far too large to be of any conceivable interest to the real world. See
    2.4.4 Gödel’s view of the Axiom of Constructibility

    I really can't say what Gödel thought about or believed, since apparently he initially thought the axiom of constructability (the claim that the constructible universe includes all sets) was true, then came to doubt it. But my sense is that he was thinking of the Platonic reality of a very large universe of sets, and was not thinking about the utility of set theory in physics. On the other hand he did do some work in relativity, so who knows.
  • On Gödel's Philosophy of Mathematics
    I so mean.

    But we call them assumptions.

    Ok, I think we're all clear and in agreement then.

    We can start with the inference rules of ND, and, one-by-one, introduce as assumptions the standard axioms of ZF set theory. Of course there are infinitely many axioms because Specification and Replacement are actually axiom schemas, meaning that they each represent one axiom for each of infinitely many predicates. But presumably we can do that. [Specification can be derived from Replacement so technically we only need Replacement].

    So far so good. Now as soon as you add the axiom of Infinity, you will have a model of the Peano axioms, and you'll introduce incompleteness. So in the end this is all exactly the same as standard set theory.

    I did look this up in the Wiki article on first-order logic, which is the logic used for set theory. It says, in the section, "Hilbert-style systems and natural deduction,"

    A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a logical axiom, a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.

    Natural deduction systems resemble Hilbert-style systems in that a deduction is a finite list of formulas. However, natural deduction systems have no logical axioms; they compensate by adding additional rules of inference that can be used to manipulate the logical connectives in formulas in the proof.

    So as far as I can understand, you get the same set theory either way.
  • On Gödel's Philosophy of Mathematics
    Quick question: why can't we throw out self reference with regard to Gödel like they did with Russell's paradox?Gregory

    I found an answer on math.SE which you may or may not find satisfactory. See the checked answer by Asaf Karagila. I'm repeating it verbatim here:

    Self-reference has a problem, if you want to think about it in terms of "I am not provable" sort of approach. A well-formed formula cannot refer to itself. Moreover, a formula cannot refer to the meta-theory (which is where proofs exist).

    What Gödel did was two things:

    1. Internalize the meta-theory into the natural numbers via coding, and show that this internalization is very robust.

    2. Showed that there is a sentence with Gödel number n, whose content is exactly "the sentence coded by n is not provable".

    The importance is in both points. They allow us both (limited) access to the meta-theory and the proofs; as well circumvent the problem of being a well-formed formula while still referring to itself. And while the importance of the incompleteness theorem is mainly in the fact that it shows there is no reasonable way to have a finitary proof-verification process to mathematics, and also prove or disprove every sentence; the proof itself is also important because it gives us the internalization of the meta-theory into the natural numbers.

    FWIW we don't "throw out self reference" to fix Russell's paradox. Rather, we outlaw unrestricted comprehension and require restricted comprehension.

    That is, if is a predicate, we outlaw set specifications of the form , which says we can form a set out of all the things that satisfy the predicate.

    Instead we require that the predicate is used to cut down an existing set. So we have some set that already exists, we say that we can form a new set . That makes all the difference.

    For example if we form the set of everything that's not a member of itself, as in , we get Russell's paradox.

    But if we start with, say, the natural numbers, we may legally form the set , we do NOT get any contradiction.

    Let's walk through it Is 0 an element of itself? No, so 0 is in . Is 1 an element of itself? No, so 1 is in . Continuing like this, we see that is just the set of natural numbers. The paradox goes away.
  • On Gödel's Philosophy of Mathematics
    No axioms. That's pivotal. Instead there is a rule of assumption: one can introduce any proposition on chooses at any time, and rules for deriving more theorems from those assumptionsBanno

    Yes but there are no axioms in any deduction system. First you have the rules of deduction, then you add in some rules of set theory, say, and you crank out your theorems. It's like saying gasoline doesn't have a steering wheel. Gasoline is the stuff you put in your car, and the gas makes the car go. In order to apply the deduction system to something you have to write down some axioms. The axioms aren't part of the deduction system. So it's not a pivotal aspect of ND that there are no axioms. To do math, you write down some axioms and then use the rules of deduction to derive theorems. If you have no axioms you have rules of deduction but you can't prove anything.

    Unless you mean that we can introduce our axioms in an ad hoc manner, for example, saying "if we assume the axiom of unions and we have sets X and Y then we can conclude that there a set X union Y. Is that what you mean by not having axioms?
  • On Gödel's Philosophy of Mathematics
    SO the lesson might be that when the love of axioms tried to tighten up mathematics, it ended up toppling the axioms.Banno

    Indeed. It was Hilbert who said, "Wir müssen wissen – wir werden wissen ("We must know — we will know."). And as I recall. it was either a few days before or after that, that Gödel announced his incompleteness theorems. The search for mathematical certainty ended in the proof of uncertainty. Be careful what you wish for, or something like that.

    I suspect that in mathematics any true formula can serve as an axiom form which to develop more cool mathematics. That's quite a different thing to an axiom in logic.Banno

    We don't know what the true formulas are. But it's true that math and logic are very different.

    Can you tell me, from your knowledge of ND, is my Google-ish description that "To prove P implies Q we assume P and derive Q," is a fair summary of what it's about? I've seen natural deduction mentioned many times but never knew what it is.
  • On Gödel's Philosophy of Mathematics
    Need to take a break so we don't post over each other. Yes, ND is just deduction. But in ND, any theorem can be taken as axiomatic, to be discharged as the deduction proceeds. As I say, they are functionally equivalent.Banno

    Ok I waited a bit. I did not understand ND from its Wiki page. But I did completely understand it from the Google description: To prove P implies Q, I assume P and derive Q. This is basic, everyday mathematical practice.

    I should point something out. Mathematicians don't use formal logic. They use this kind of casual, everyday reasoning. In formal logic, the things people talk about are foreign to working mathematicians. By the standards of formal logic, no working mathematician has ever seen a proof, if you look at it that way.

    So whatever ND is, if it's just "To prove P implies Q assume P and derive Q," then that's what I've been doing all my life and that's what everyone else does. Any distinction between that and some other kind of formal logic is "inside baseball" for logicians and apparently of little interest to mathematicians.

    Bottom line, I don't see how ND can add anything to mathematical practice, nor relieve us of the need to start somewhere by writing down our axioms. If you add axioms one at a time making sure they are complete and consistent, you will never get to the arithmetic of the natural numbers, which is known to be incomplete.

    Don't know if this was helpful to you, but it sure was to me. "Today I learned" that natural deduction is just the normal type of reasoning done by mathematicians. So in the end, thanks for mentioning it!
  • On Gödel's Philosophy of Mathematics
    Too fast. Why?Banno

    Ok. I Googled "natural deduction as basis for math" and the following popped up right at the top:

    What can you assume in natural deduction?
    In natural deduction, to prove an implication of the form P ⇒ Q, we assume P, then reason under that assumption to try to derive Q. If we are successful, then we can conclude that P ⇒ Q. In a proof, we are always allowed to introduce a new assumption P, then reason under that assumption.

    Now this is just what I call perfectly normal mathematical deduction. That's how mathematicians do proofs. To prove P implies Q we assume P and derive Q. If that's natural deduction, I've been using it all my life! Like the literary character who discovered he'd always been speaking prose.

    So if that's all ND is, then it appears to be perfectly standard everyday mathematical reasoning.

    I still don't understand the technical distinction being made by the Wiki article on ND, but based on this example, ND is just how everyone does math.

    But I can see how we might have different pictures of mathematics, which are despite that functionally equivalent: one of maths as a series of axiomatic systems, the other as a language that becomes increasingly complex as new formation rules are added.Banno

    I don't think this is what's being said by ND. As the Google example shows, ND is basically normal everyday mathematical reasoning. To show P implies Q we assume P and derive Q.
  • On Gödel's Philosophy of Mathematics
    Is there a reason to think that natural deduction is not as powerful as axiomatisation?Banno

    I don't know, I'm asking you. I don't know anything about it other than that it comes up when students are discussing logic. I glanced at the Wiki article on ND and perhaps you are making a valid point that I'm too ignorant to understand. I never studied formal logic, just picked up the basics from math classes.
  • On Gödel's Philosophy of Mathematics
    Hmm. Your puzzlement has me puzzled,Banno

    I no longer have any idea where you are going with this. I am sure the fault is all on my side. I can't respond intelligibly because I just don't know what you are saying relative to what the thread is about.

    Or we might treat it as a series of rules for derivation, never complete but always consistent, to which we add new rules of derivation in a similarly asymptote fashion.Banno

    Are you saying you can do modern math like that? I'm sure somewhere somebody's making the effort. I can't relate to what you are saying. My apologies.
  • On Gödel's Philosophy of Mathematics
    Did you use an axiomatic system?Banno

    Math uses axiomatic systems, period. I didn't study much formal logic.

    I studied logic forty years ago.Banno

    Yes but the subject of the OP is axiomatic systems in math. Not that threads don't wander, but I'm no longer understanding your point. Is ND somehow not subject to incompleteness? I tend to doubt that. Underlying set theory is first-order predicate logic. If there's some other basis for doing set theory I am unfamiliar with it. I've heard of ND used in logic, but I can't imagine it gives you different math.
  • On Gödel's Philosophy of Mathematics

    I thought we were talking about math, not logic. At least the OP was about math FWIW. Is ND offering an alternative foundation of math? Is it somehow not subject to incompleteness? I'm afraid you've lost me again but I don't know anything about ND.