6:28 AM
@Zanna Logical in the sense of being "of logic," that is, of having their meanings determined ahead of time by one's logic.
I should've linked to that but also this message which appears to be where I first mentioned logical symbols.
When you say, xkcd 435 is totally wrong, you remind me of the last time I looked up that xkcd having remembered it for some reason. At that time, I was surprised by it, because I misremembered its tone, which seems to affirm that there is a hierarchy of fields in terms of "purity". I remembered the tone as satirizing that claim. So I must have seen another cartoon or heard a joke, in which that claim was made fun of
@Zanna Might you have remembered rightly? I think it is satirizing one version of the claim but arguing for another, and that both versions are wrong. It's satirizing the idea that physics is is more laudable or important than other sciences on the grounds that other sciences reduce to it, or that it is more fundamental than them, or that it has greater rigor or certainty. But it's doing so by claiming that this fails on account of how math has those virtues, but more so.
6:43 AM
I also remember originally reading a criticism of this reductionism in a book about astrology (a collection of essays by astrology experts) when I was in secondary school. Probably that argument also went in some unhelpful direction eventually. But still it helped me to see why we need all these fields and keep refusing to allow one discipline to be put above another
@Zanna Well, from context, it seems to mean pure in the sense of "pure or applied." Another criticism of the claims the characters in the comic make is that none of them are true. Sociology is not applied psychology, psychology is not applied biology, biology is not applied chemistry, chemistry is not applied physics, and none of those things is applied mathematics. For example, applied physics is an actual thing, and chemistry is not, in general, a kind of applied physics.
7:06 AM
@EliahKagan Oh, I don't remember, but I suspect that the essay was suggesting that science isn't as great as people make out. If science is not that great, maybe there's some value in astrology even though it has no scientific basis and attempts to check any of its claims using scientific methodology have all failed. I don't agree that science is not that great or that astrology is useful for finding out things about the world (it might be useful for making money by tricking people).
I'm just talking about how several characters in the comic use the adjective "applied," leading me to think that the intended meaning of "purity" is one that relates to the pure vs. applied distinction, between studying a field to advance the field and studying the field to advance some other thing that the field is useful for.
Oh. I think you're not talking about the same thing but that what you are talking about is closely related to what I mean when I say it's wrong. What i mean when I say it's wrong is that there's not a linear chain of dependencies between the fields like that, though a circular chain, with mathematics connected up to one or more social sciences, might be a reasonable (if still inaccurate) model.
I can see that this strays from formal logic and set theory... but I don't think that's what you mean, so I'm not sure if I understand you. What you, and I, have recently said, does seem closely related to what I was saying about fields of study:
7:24 AM
@EliahKagan I read this book from the library when I was in secondary school (then I borrowed it and read it twice more and eventually bought it, and then I think I gave it away as I do not seem to have it); it has a story about infinite sets... that is, it has an intergalactic hotel which has an infinite number of rooms
7:37 AM
@EliahKagan I am not actually sure I quite agree with either of those claims, though. As for the first, the breakdown of different fields of study is an artifact of culture, which responds both to what has been learned so far and to other factors. It's not obvious that the current divisions are the best ones. I do agree in spirit, though, in that think that what each of these fields study is important.
@EliahKagan Yet one could, er, imagine a world in which great resources were put into various fields, in recognition of how those fields are needed by everyone, but in which not everyone benefits from all of them, and how even some people are terribly exploited by some of them.
Some people, who regard those fields to be important, might be unaware of, or even defend, these problems, by regarding them as essential to a field. If shown a future in which a field they care about has the same name but is far better, they might happily admit they were wrong but regard it to have become an entirely different field.
8:17 AM
As for the second, I find this quite interesting. I think it is possible for one field of study to be better or worse than another is some specific respect, and I also think that what things we should most care about is not arbitrary. I think one should be careful that one's judgments of one field compared to another are based in actual critiques of them, rather than serving the purpose of demonstrating one's loyalty.
But just like, even though people are complicated, it sometimes make sense to judge some historical figures as better than others in ways that matter, that this also applies to institutions, and thus to fields of study.
@EliahKagan (Edited to fix wording that seemed to claim that good-faith critique is less performative than bad-faith critique. That's not true, and I didn't mean to imply it.)
@Zanna No, I'm supposed to be talking about those not talking about other things so much that it gets in the way of those. :)
@Zanna Yeah, to create infinitely many vacancies without anyone checking out, move each guest from their room $k$ to room $2k$. Then the odd-numbered rooms are available. This illustrates that there are exactly as many positive integers as there are even positive integers. (One can similarly show that there are exactly as many positive integers as integers.) That is, $2 \aleph_0 = \aleph_0$.
@EliahKagan I am trying to remember why I thought it was valuable for me to talk about that. There was some reason...
Oh, right. I think a lot of influential ideas about how fields of study relate or compare to one another are based on idealizations of what those fields study or of how they fit into a broad epistemological project or into civilization. I think this undervalues the approach of looking at what is actually happening.
As applied to the specific question of how to understand or justify the foundations of a field, I think that the relationships that one finds when one looks are often quite different from widely repeated cliches. In particular, logic is at the foundation of mathematics†, and logic is part of philosophy, and takes its justification from considering how people think and communicate. So that puts us in the social sciences.
† It may be worth examining if I am committing the fallacy of equivocation by using "foundations" in two different ways.
I won't try to do this now, since it is bad enough that I'm off on this tangent, but my argument would be a lot more persuasive if I were to demonstrate that what is known as "foundations of mathematics" is actually at the foundation of mathematics as a field, in the sense of being a source of rigor, telling us why we should believe mathematics, or explaining its significance as a cultural project. It is not obvious that this is the case.
8:54 AM
@EliahKagan So the connection is: putting one field over another, in the sense of trying to look at what is actually happening in them and being willing to make judgments based on that, can make sense. But again maybe I agree with the spirit of what you're saying--or maybe just agree with what you're saying, depending on what you mean.
1 hour later…
9:55 AM
@EliahKagan thank you for making me think better about it. I hadn't thought about it much. Here I was only thinking that, maths is not the only thing we need... but in education I thought more about this. Like the distinction between physics, chemistry and biology didn't seem very helpful for kids and it was interesting to think about organising the curriculum around topics (like energy!) instead, as was done in primary schools before the 2010 coalition government ruined it
I think that would require something in the way people who are going to teach are educated to also change (not that it isn't totally inadequate already)
but at the moment there is a shortage of physics teachers and non-physics specialists are teaching physics everywhere in secondary schools
when I was in teacher training I was in a school with a fellow trainee whose background was in pharmacy. We all taught all the science subjects as trainees. Once she was taking a physics class and a kid asked her "miss, what is speed?". I don't know what I would have answered to that. I'm not good at thinking on my feet and I don't like to let good questions go to waste. I might have given everyone a homework to find a definition of speed and try to prepare something for the next lesson.
when I was studying psychology, I felt that a lot of it seemed to be sort of quite bad and wrong or misguided
but what I realised later was that I was greatly helped to feel that by the large extent to which evaluating the various approaches of psychology and their claims was part of the syllabus, part of what we were assessed on (if it's not assessed, it won't be taught). Every theory that was presented came with its criticisms and counter criticisms
the reason I realised that later was that I was reading criticisms of education and specifically science education: there was a school of thought that secondary science ought to be citizenship-oriented, i.e. it should equip people to understand and reason about and evaluate science-related things they might read or hear about in the media and suchlike
we almost never learn about how such and such a thing was found out or established, or why anyone should care about it. The only exception I can think of is the Geiger-Marsden experiment, and I'm not even sure whether I learned about that in physics in secondary school (I think I did) or in philosophy of science in college, or both. But I do know that, outside of teaching, I've told that story more than any other sciency thing
when I was living in London I did some work for an agency that specifically taught kids who were planning to go to US colleges. A lot of them were actually USian and they were going to the American School in London, so they were studying USian syllabuses... occasionally I got to help those kids with their homework and I noted they had a lot of stuff about famous figures, like Newton, Einstein, or whoever was associated with the topic. But I don't know if that was assessed...
(I was mainly helping with the SAT - the agency owner had come up with a technique to teach the Writing test... I could get 800 on Reading easily, Writing after learning the technique, Math never no way not by miles)
11:07 AM
@EliahKagan I don't think I'm really getting these concepts of validity, equivalence etc properly. Need to practice somehow or read again.
@EliahKagan yes, interesting, need to read it again (these days I'm even reading the book I'm reading twice; I read what I read for the first time yesterday and a bit more, tomorrow I'll read what I read for the first time today and a bit more...)
@EliahKagan I am confused by this. It seems like the conclusion of an argument with zero premises must be true...
@EliahKagan no you have not put me off at all... only unfortunately I don't seem to have anything to communicate that requires it. Hopefully that will change
@Kulfy :) fresh bread is a good thing! When I was a teenager I went on holiday to Italy with my family and my brother and I went to a bakery in the morning to buy croissants.
Then in my 20s I went on holiday with my family again to Kefalonia which is one of the Greek islands, and my brother and I again would go to a bakery to get things for breakfast. We would buy bread to make sandwiches for lunch and biscuits for breakfasts. I think Greek bakeries are a well kept secret. They have got the most amazing biscuits and their bread is better than Italian, better than French (shhh!) and the best sweets and cakes. Sesame and honey! <3<3<3
But in the UK most people eat this factory sliced bread which is mostly tasteless but at least remains edible for a few days. I have not bought bread in the UK for very many years. If I want bread, I bake it, with special flour because all the flavour has been bred out of the ordinary wheat in favour of yield...
@Kulfy my parents are both committed vegans now. I became vegetarian when I was 13 or 14. My parents were accepting rather than supportive, but my mum's lack of enthusiasm for cooking spurred me to try to learn it myself, I suppose. I became vegan around 2013. About one year after I became vegan, my mum became vegetarian. About one year after that, my dad became vegetarian. Gradually they became vegan. Now they wear vegan tshirts and go to vegan festivals and shout vegany things at the telly
(I find it socially super difficult to be completely vegan here. This month I ate sugar pongal (because of pongal festival) in my house owner's place, and they gave me some prasad (a piece of besan laddu) from Tirumala also. Ghee will be there. I daren't even tell my parents XD)
@Kulfy I have heard from other Indian people the view that the Chinese and Japanese eat insects... I never saw any insects being eaten or on any menus when I was in China (in Zhongshan) but there is a joke, which I think and hope is a Chinese joke, that in China we eat anything with four legs except a table... certainly their cuisines relish a wider variety of meat than seems to be the case here or in UK!
11:53 AM
I was given an apartment to stay in by the company I was working for and it had a wok... the wok was absolutely magical. Whatever I put into it came out perfectly cooked.
On workday mornings while making my breakfast, I would concurrently make myself some noodles with dried mushroom stock, a few vegetables and tofu or a boiled egg (at that time I still ate eggs, cheese, yogurt and honey, but not milk) for packed lunch.
The lunch was effortless... but to make my soymilk porridge I had to fight a strong stove and a flimsy little saucepan... it was a daily struggle to get it done.
Similarly, once a month, instead of the noodles and fried rice and other Chinese style dishes I was mostly cooking for dinner in the magic wok, I would make myself a "Western" meal of baked (microwaved) big potato, canned baked beans with herbs, and cheese. I ate this with a knife and fork (instead of the usual chopsticks). Here, the difficulty was not so much in preparing the meal, but in obtaining the costly imported ingredients.
It's always so much easier and nicer to go with the flow of the food culture of a place! Yet the migrant always brings something...
@Kulfy well, yes, definitely, and I wanted to stop eating it every day because I wanted to try out more things, but it was not the same every day because all the elements varied.
Like, for the raw version, it was oat-based, but I would use different mixtures of plain oats, mueslis and granolas, or add different grains like millet flakes, puffed rice etc, I would use different yogurts, mixtures of soy yogurt and coconut yogurt, sometimes chocolate or caramel or fruit flavoured...
if I had biscuits or cake I'd baked or some chocolate I'd crumble it in, or I'd add a bit of maple syrup, or jam (fig, blueberry, raspberry, rhubarb, cherry) or nut butter (peanut, almond, cashew, hazelnut pumpkin seed) or lucuma or maca powder.
I'd add cinnamon or vanilla or rosewater or dry ginger or cocoa powder or "mixed spice" (a common UK blend of sweet spices including clove, cardamom, cinnamon, coriander, nutmeg), I'd use different dry fruits (fig, apricot, dates, raisins, goji berries, banana, pineapple) and nuts (pecans, Brazil nuts). Then the fresh fruits would also vary by season. So it was satisfying, it varied enough for me.
I require variation and complexity in food. I once had a job on a cruise ship, and it was not a very nice job but I could have lived with that, and I liked my cosy shared cabin and I liked being at sea, but the food was unbearable, so I left after 4 months, couldn't stick it
I can have idli, dosa, poori, paratha, rava upma, wheat upma, semya upma, poha upma, besan cheela, adai, pongal, idiyappam... all with many variations, with chutney, sambar, pickle, aviyal and all sorts of other side dishes, and a hundred other things I haven't tried...
I can totally understand not liking sambar or getting tired of it, but sambar should also vary like my oat breakfast... different vegetables used, sometimes adding coconut or poppy seeds, using sesame oil, making a fresh ground masala instead of using sambar powder, etc
12:42 PM
1:19 PM
2:06 PM
That it follows deductively from the premises, yes. That is, the conclusion follows from the premises due to the form taken by the premises and conclusion.
So for example, this argument is not valid: $$\text{Jones is safe.}$$ $$\therefore \text{Jones is not trapped at the bottom of a giant vat of tomato juice.}$$
Yet informally, when presented with such an argument, it might be rational to accept that, if its premise is true, its conclusion is true.
Furthermore, the argument $$\text{Jones is safe.}$$ $$\therefore \text{Jones is safe.}$$ is valid. However, in practice we should not usually be persuaded by it in the course of informal discourse, as it is an instance of the begging the question fallacy.
To see why "Jones is not trapped at the bottom of a giant vat of tomato juice" does not follow deductively from "Jones is safe," note that it relies on knowledge or assumptions about what it means to be safe and trapped, what a giant vat is, and what tomato juice is. It's also perhaps not an idea example since one can imagine someone who is trapped at the bottom of a giant vat of tomato juice yet is safe, even without assigning unusual meanings to those words.
Although this is probably not the best general way to represent our knowledge of these topics, consider this argument, which is valid: $$\text{Only very careful tomato juice divers are ever safe when trapped at the bottom of a giant vat of tomato juice.}$$ $$\text{Jones is not a very careful tomato juice diver.}$$ $$\text{Jones is safe.}$$ $$\therefore \text{Jones is not trapped at the bottom of a giant vat of tomato juice.}$$
2:25 PM
Truth-functional logic is insufficient to determine the validity of that argument. That argument, while valid, is not truth-functionally valid. However, quantification theory (which as you may recall contains truth-functional logic but also the logic of atomic sentences, and quantifiers "there exists" and "for all") is sufficient to determine its validity.
$$\forall x [(\mathrm{Trapped}(x) \wedge \mathrm{Safe}(x)) \rightarrow \mathrm{CarefulDiver}(x)]$$ $$\neg \mathrm{CarefulDiver}(\mathrm{jones})$$ $$\mathrm{Safe}(\mathrm{jones})$$ $$\therefore \neg \mathrm{Trapped}(\mathrm{jones})$$
Notice that, in making a language of first-order logic, and translating my sentences into them, I have deliberately avoided capturing the full syntactic complexity and semantic depth of the sentences. I have only captured what I need in order to make a valid argument. I could do it more intricately, though. Like, I could use a compound sentence, with several predicates, to express "$a$ is a very careful tomato juice diver." The argument would still work, but it would rely on those details.
Although truth-functional logic is not sufficient to determine the validity of that argument, it is almost sufficient; all we need to do is to convert the general sentence $\forall x [(\mathrm{Trapped}(x) \wedge \mathrm{Safe}(x)) \rightarrow \mathrm{CarefulDiver}(x)]$ into a more specific, weaker sentence specifically about $\mathrm{jones}$. Once quantification theory is used to do that, via a rule called universal instantiation, the rest of the argument is truth-functional.
That is, this argument is valid in quantification theory: $$\forall x [(\mathrm{Trapped}(x) \wedge \mathrm{Safe}(x)) \rightarrow \mathrm{CarefulDiver}(x)]$$ $$\therefore (\mathrm{Trapped}(\mathrm{jones}) \wedge \mathrm{Safe}(\mathrm{jones})) \rightarrow \mathrm{CarefulDiver}(\mathrm{jones})$$
And this argument is valid in just truth-functional logic (and thus also in quantification theory, of which truth-functional logic is a fragment, but we don't need the rest of quantification theory for it): $$(\mathrm{Trapped}(\mathrm{jones}) \wedge \mathrm{Safe}(\mathrm{jones})) \rightarrow \mathrm{CarefulDiver}(\mathrm{jones})$$ $$\neg \mathrm{CarefulDiver}(\mathrm{jones})$$ $$\mathrm{Safe}(\mathrm{jones})$$ $$\therefore \neg \mathrm{Trapped}(\mathrm{jones})$$
Its truth-functional validity can perhaps more easily be seen by abstracting away the structure of the atomic sentences by representing:
$\neg p \vee \neg q$ and $q$, so $\neg p$. (I haven't named the rule that permits this, but one of the names for it is disjunctive syllogism. One might pedantically insist that I first infer $\neg \neg q$ from $q$, and then say that $\neg p \vee \neg q$ and $\neg \neg q$ so $\neg p$.)
You can verify with a truth table that in all rows in which a $\top$ appears for $(p \wedge q) \rightarrow r$, $\neg r$, and $q$, a $\top$ appears for $\neg p$.
Though for arguments if only moderate complexity, such as this one, putting it into words in a natural language may make it intuitively clear. Abbreviations are handy for facilitating this technique.
Then we have: $$\text{If Jones is both trapped and safe, then Jones is a careful diver.}$$ $$\text{Jones is not a careful diver.}$$ $$\text{Jones is safe.}$$ $$\therefore \text{Jones is not trapped.}$$
So, that argument is valid because, regardless of how you interpret the non-logical symbols -- that is, the predicates "$\mathrm{Trapped}$", "$\mathrm{Safe}$", and "$\mathrm{CarefulDiver}$" and the constant "$\mathrm{jones}$" -- if the premises are all true then the conclusion is true.
@EliahKagan That is, the argument's corresponding conditional, whose antecedent is a conjunction of the premises and whose consequent is the conclusion, is true under all interpretations of those symbols.
Truth-functional logic doesn't recognize the structure of atomic sentences. Given an atomic sentence, the most specific truth-functional schema that matches it is a single sentence letter (which is the most general truth-functional schema). Truth-functional logic also doesn't recognize quantifiers, so any quantified formula, no matter how complex, also cannot be represented as more than a single sentence-letter.
So if we were to try to use truth-functional logic alone to assess the validity of that argument, the best we could do would be to represent it in the form: $$s$$ $$\neg r$$ $$q$$ $$\therefore \neg p$$ And that's not good enough, as that form is not valid: there are interpretations of those sentence letters that make all the premises come out true but the conclusion come out false.
But once you make that inference (i.e., perform the universal instantiation), what's left to do is verify the validity of that argument, which is truth-functionally valid.
No matter how you interpret the sentence letters appearing there, either at least one premise is false or the conclusion is true.
3:22 PM
I had understood what you were saying there as primarily being about not getting what means for a sentence to be valid, or for a schema to be valid, or for two sentences to be equivalent, or for two schemata to be equivalent. I realize I'm not sure if I understood you correctly, though.
@Zanna So I don't forget: the conclusion of an argument with zero premises need not be true, since I can make the argument $$\therefore \text{I am the pope.}$$ and that argument has zero premises, yet its conclusion is false. However, a valid argument with zero premises will always have a true conclusion.
3:44 PM
A sentence is said to be valid when it is true, and its truth only depends on its structure and the meanings of words like "and" and "for all."
For example, suppose we have a theory of the integers, with a binary function symbol "$+$" (recall that an $n$-ary function symbol attaches to $n$ terms to form a term). Suppose we regard our universe of discourse to be the integers. Suppose further that our underlying logic supplies "$=$" with its usual semantics, by which I mean, $\forall x\, (x = x)$ and how, if $a$ is true of $b$, anything true of $a$ is also true of $b$.
Then the sentence $$\forall x\, \forall y\, (x + y = y + x)$$ is true. I say it is true based on what I said about our universe of discourse and the meaning of "$+$". It may also be a theorem of our theory; that happens if our theory can prove it. However, whether or not our theory can prove it, it is not valid. That is, it is not a logical truth.
This can be seen by considering that our confidence in its truth is based on a particular interpretation of the non-logical symbol "$+$" (and what I said about the universe of discourse being integers). More concretely, it can be seen by picking a different interpretation of "$+$" that makes the sentence come out false. For this purpose, we might interpret "$+$" so that it always evaluates to its first argument. That is, so that $a + b = a$, no matter the value of $b$.
I should perhaps have said, "...suppose we have a theory of the integers, with a binary function symbol '$+$', to which we give the usual meaning."
Validity is strictly stronger than truth; a sentence that is valid is a sentence that is true merely due to logic, so that its truth does not depend on anything it says corresponding with a feature of the world.
One of the points Quine makes about validity in Methods of Logic is that, to say of a claim that it is valid is not, generally, to praise it. A valid sentence is a sentence that gives no information whatsoever about the world. (Likewise, an inconsistent sentence may be regarded as giving so much information about the world that it is unsatisfiable.)
no, I have not read that book, unless I have forgotten doing so. The only thing I've read by Asimov (I think) is this collection, Gold (because it was in my college library and I tried to read all the books that interested me from A-Z hahaha).
Depending how it is presented, the rooms in the hotel are numbered $0, 1, 2, \ldots$ or $1, 2, 3, \ldots$. You can start with a symbol, $\mathbb{N}$ for the set of those numbers.
Unfortunately, there really are two conventions for whether or not $0 \in \mathbb{N}$. This is a matter of what $\mathbb{N}$ is taken to mean, of course; it's not an actual mathematical disagreement. I suggest using the convention $0 \in \mathbb{N}$, in general. I also suggest regarding the hotel to have a room $0$.
4:25 PM
Regarding the above stuff... I based everything on the concept of validity of arguments. So once that fully makes sense, I think the rest may too. For this, there are many good sources. I'll think about that and perhaps recommend something.
For syllogisms, Smullyan presents validity in the first chapter of A Beginner's Guide to Mathematical Logic. I suggest that not because I would ordinarily send people there to learn about the general concept of validity, but because I know you have that book.
I thought well of the explanation of validity and soundness of arguments given in The Power of Logic by Howard-Snyder et al., which covers both informal and formal logic, at the expense of some depth but still with enough detail to be useful. I'm thinking of what I read in the 4th edition. But I presume that book is expensive, and I'm not sure if the book as a whole would be of interest to you.
As for other kinds of validity... For validity of truth-functional schemata--and actually, for anything involving truth-functional schemata--Methods of Logic by Quine cannot be beat. It is also by far the least expensive of all the popular formal logic books, even though it is one of the best.
(If you do seek to check out Methods from a library or buy it, then based on your interest in set theory and also, or perhaps more importantly, on the details of how you described becoming interested in it, I would recommend the 4th edition of Methods over previous editions. None cover set theory except briefly at the end, so you may wonder why I am saying this. I'm going AFK soon so I can't go into detail about it now. You can remind me though.)
While I'm dropping the names of various logic books, I would be remiss if I did not mention Language, Proof and Logic by Barker-Plummer, Barwise, and Etchemendy. Note that I have only read and use the first edition, of which Plummer was not an author, which is why I often refer to the book as "Barwise and Etchemendy." Also, this means I cannot speak definitively to the quality of the second edition.
So you might be surprised to hear me proclaim that anyone who wishes to use the book seriously should use the second edition. However, the book is integrated with software, and the software is much improved in later versions.
Note that I'm not specifically recommending Lanugage, Proof and Logic in connection with explaining validity.
Anyway, the Hilbert's hotel starter example I was going to give was, suppose the Hotel has rooms numbered $0, 1, 2, \ldots$, and take $\mathbb{N} := \{0, 1, 2, \ldots\}$. Suppose further that the hotel has no guests checked in, until $76$ guests check in. The clerk puts them in the lowest-numbered available rooms. Then the set of room numbers of occupied rooms is: $$\{x \in \mathbb{N} \mid x < 76\}$$
Consider also, for the purposes of completing later scenarios, how to write a term for the set of numbers that are multiples of some number. There is more than one way to do this but I'll show one; specifically, I'll show one that uses the other major form of set-builder notation. Suppose the haunted rooms are those rooms whose numbers are multiple of $5$. Then the set of room numbers of haunted rooms is: $$\{5x \mid x \in \mathbb{N}\}$$
Then one way to write a term for the set of room numbers of rooms that are unoccupied but haunted is: $$\{5x \mid x \in \mathbb{N} \wedge 5x \geq 76\}$$
I suggest writing terms for the set of strictly positive room numbers, the set of odd room numbers and the set of even room numbers. This will show how the clerk an solve the problem of admitting $1$ guest when all rooms are occupied, and two ways the clerk can solve the problem of admitting infinitely many guests (specifically, $\aleph_0$ guests) when all rooms are occupied.
5:15 PM
$\aleph_0$ is the smallest infinite cardinal. For a set $S$, the cardinality of $S$ is denoted $\lvert S \rvert$. $$\aleph_0 = \lvert \mathbb{N} \rvert$$ Recall we had been talking about relations. The next step, in that line of development, is to talk about functions. In the context of Hilbert's hotel, you put original room numbers into a function and get out the new room numbers for moving the guests.
There are many different functions, of course, and not all are fit for this purpose. For example, some would put guests that started out in different rooms into the same room.
Even before the precise meaning of "the cardinality of" is formalized, one can formally and precisely explain the conditions under which $\lvert S \rvert \leq \lvert T \rvert$ and under which $\lvert S \rvert = \lvert T \rvert$, once equipped with the apparatus of functions. I mention this to connect Hilbert's hotel up to what we were doing before. :)
@Zanna Yes, exactly as weak as any other valid sentence. The valid sentences are the weakest sentences. After all, they don't say anything that we couldn't already know.
@Zanna You're welcome--though if you've managed to benefit from it, I think that may speak well of you rather than of me. I think what I was saying about that was correct, and that it can be said in a way that is illuminating... but I was fairly horrified at how poorly and unclearly I stated it, and upon rereading what I had written about that, I realized I was too tired, and went to sleep immediately. Looking at what I said with fresh eyes, I remain unimpressed with myself. :)
@EliahKagan Notice, by the way, that set-builder notation, like definite descriptions, is syntactically like quantification and might even be considered a form of quantification (and its meaning can be stated in terms of quantification). It involves specifying a variable, and binding all free occurrences of that variable in a subformula.
To make this concrete, this is how we claim $Fx$ holds for all $x$: $$\forall x\, Fx$$ This is how we claim $Fx$ holds for some $x$: $$\exists x\, Fx$$ This is a way to claim $Fx$ holds for exactly one $x$: $$\exists! x\, Fx$$ This denotes the $x$ whose existence the above sentence asserts i.e. this definite description is a term for that unique $x$ such that $Fx$: $$ɿx\, Fx$$ Remember, that definite description may fail to refer to anything. It succeeds if $\exists! x\, Fx$ and fails otherwise.
And this denotes the set of all and only those $x$ for which $Fx$ holds: $$\{x \mid Fx\}$$ Remember, that form of set builder notation may fail to refer to anything. For example, if $Fx$ means $x \notin x$, then $\{x \mid Fx\}$ would be the Russell set $\{x \mid x \notin x\}$ and there cannot be any such thing (due to Russell's paradox).
However, given an existing set $S$, this denotes the set of all and only those $x \in S$ (i.e., those $x$ that are members of $S$) for which $Fx$ holds: $$\{x \in S \mid Fx\}$$ ...which, under axioms of modern set theories such as ZFC, is guaranteed to exist. As far as is known, modern set theories are consistent. That is, unlike with Frege's system, nobody has found a way to prove a contradiction from axioms of modern set theories yet.
6:33 PM
4 hours later…
10:58 PM
I'm curious if ChatJax++ will render tables. $$\begin{array}{c||c||c}p& q& p \wedge q& p \vee q& p \rightarrow q& p \leftrightarrow q\\ \hline \top& \top& \top& \top& \top& \top\\ \top& \bot& \bot& \top& \bot& \bot\\ \bot& \top& \bot& \top& \top& \bot\\ \bot& \bot& \bot& \bot& \top& \top \end{array}$$ It does! :)
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