Mathematics - 2024-06-30 (page 1 of 2)

Thorgott

00:22

I'd be concerned if you had, but the name is funny

(it's a book about ring theory and radical ideals, for those who haven't looked it up)

nickbros123

02:30

its kinda annoying to remember the matrix representation of linear transformations, and ccoordinates (the part that i always have to "rederive" is how the invertible matrix looks like when I change the basis, and how the matrix of the transformation T looks like), is there a way to remember this so that I can just simply, looking at how the bases of V map to W, write out the linear transformation in terms of the basis of W?

leslie townes

02:53

nick i honestly think it helps to think that through every time, if only because textbook authors and paper writers do not and sometimes there are typos where "of course anyone will see what i meant." i do not know of a purely symbolic way of letting notation take the wheel

it's one of those things where e.g. if it's the same space on both domain and codomain and there's a change of basis, the thing to remember is "the new matrix will be S^{-1}MS or SMS^{-1}, depending on what S is" or the more general thing that you will be changing the matrix by pre or post multiplying by things depending on what basis you are changing, and then you think it through to work out what it is exactly

also worth remembering (at least outside of a context where software aided computation is going to be involved) that depending on what you are trying to learn about a linear transformation, you might not need any particular matrix representation, or even any matrix representation :)

user 85795

03:22

@Thorgott why the concern?

user 85795

04:05

[@onepotatotwopotato ](youtu.be/cadx9KepuV8?si=5oOq8uSyPfK6AA7R)

nickbros123

05:14

@leslietownes I get that. One small shortcut might be that, the columns of the matrix are the coordinates of the image of the individual bases which are m dimensioned, and there are n such bases' images, hence the m x n matrix. For coordinates, I think the same principle can be applied, coordinates of bases 1 in terms of the other bases gives the columns of the change of basis formula. Still, quite a few stuff to keep track of

leslie townes

05:26

nick: it's only a 'shortcut' if you have that data somehow already given in an actual problem you want to solve, otherwise it's just another thing you might have to solve for

nickbros123

thats a fair point

leslie townes

at a higher level, the point i was trying to get at above, kind of vaguely, is that, the fact that specific formulas can always been written down and computed is sometimes a distraction from whether that's the easiest thing to do, or even thing that it's at all helpful to do, vs. just something that you could do if you want. and if it all has to go into matlab because you really need the 1,1 entry of some matrix that's one thing, but often even applied problems are less concrete than that

just what you regard as "the" transformation of interest vs. the "inverse" will affect all of those formulas - in a simple way, but still affect them to the point that i wouldn't place too much emphasis on any one example of what those formulas look like

nickbros123

I was just ranting over a few examples and problems given in Hoffman Kunze to be fair. This topic is starkly different from the rest of the chapter, as in , working through it has been a bit annoying

leslie townes

05:43

yeah i dunno. i generally get why books tend to lean pretty heavily into only one of the 'theory' vs 'matrix calculation' directions, and it's kind of annoying to switch perspectives just because a textbook wants to tell both sides of the story, and not for any specific reason

i once taught linear algebra out of a book that had been designed on the assumption that students would be learning matlab in parallel (or maybe already knew it, at least as a kind of 'matrix calculator'), which was a cool idea in the abstract because it allows you to discuss tons more examples without even considering, let alone spending time on, how any calculations would go by hand

but because of where the class sat in a curriculum i could not in fact assume that anybody knew matlab or assign those exercises

so it was just an inexplicably matrixy book

Jakobian

12:24

0

Q: Countable zero-sets are $C$-embedded?

I was browsing Gillman and Jerison for known relations between zero-sets, $C$-embedded sets and so on. The spaces I'm considering are $T_{3.5}$. There are two properties that pseudocompact spaces have All countable $C$-embedded sets are compact All countable zero-sets are compact Where (1) is...

gn.general-topology

@AlessandroCodenotti

Jakobian

13:59

never mind, I've answered the question in the positive

Jakobian

14:10

the curse of answering my own questions continues

Thorgott

14:28

I also did that recently

Jakobian

I've answered like half of the questions I had on mathoverflow, and probably a similar amount for my math.stackexchange questions in recent years

okay, not half, 3/9

Pizza

17:46

hi

robjohn

hello

Sahaj

18:11

hello sires and madams

Binky McSquigglebottom

18:38

Bellingoooooooooooooooooooooal

Obliv

18:59

would it make any sense to say $\exists \varnothing$

nvm found the better statement en.wikipedia.org/wiki/Axiom_of_empty_set

do you need to explicitly assert the existence of the variable you are quantifying over in a given set theory?

for example in that axiom of empty set, would there be an underlying axiom $\exists x$ (where x is the primitive notion which you develop your theory with)

Thorgott

19:20

a variable is a variable, you do not assert its existence

Obliv

you don't have to, but you can. $\exists x$ -> now you can quantify over $x$ with $\forall$

but yeah it's usually just stated in english

i'm just confused as to what we're quantifying over in set theories sometimes. It seems like capital letters A,B,C,... are all sets and the lowercase ones are elements. How do we distinguish elements from sets?

since we can have treat an element of a set as a set but saying a set $\in$ another set is wrong apparently.

good timing @jakobian if you get a chance can you help me with the distinction

Joe

19:38

@Obliv: In naive set theory, it is common for lower case letters to represent objects which are not necessarily sets (e.g. numbers), and appear as elements of sets (which are usually denoted by upper case letters). But in axiomatic set theory, everything is a set, and so the situation is a bit more complicated.

First of all, it doesn't make sense to speak of "elements" in the abstract – elements of what set? Moreover, every set $x$ appears as an element of another set, e.g. $\{x\}$. However, given a set denoted with an upper case letter, it is conventional to denote its elements by lower case letters.

Obliv

but then in the axiom of extensionality isn't it $\forall x(x\in A \iff x \in B)\implies A = B$ I thought it was inappropriate to write $x \in A$ if $x$ is a set

Joe

It's perfectly fine to write $x\in A$ even if $x$ is a set.

For example, $\varnothing\in\{\varnothing\}$.

Thorgott

everything is a set

Jakobian

@Obliv why are you assuming I even looked

Thorgott

quantifiers quantify over the domain of discourse, which is all sets

Obliv

19:44

@Obliv I don't get why this is wrong then

Maybe because of the way $\mathbb{N},\mathbb{Z}$ are defined?

Jakobian

@Obliv no

Thorgott

being an element of and being a subset of simply mean different things

Jakobian

@Obliv no

@Obliv not in the axiom of the empty set, but there might be an underlying assumption that a set exists

@Obliv that's not how it works

@Obliv this is language of mathematics, you are not quantifying over anything, formally, you are using text to define some formula

$x$ is ultimately a letter and you need not to assert its existence

the logical formulas follow certain rules that you need to obey, and thats it

theoretically even a finitist could interpret set theory, although there doesn't seem any reason to do that

you don't need to prove that a "$1$" or anything else exists, like in platonistic view

Obliv

You can't prove existence, hence why I asked if you have to include it in your formal system as an axiom. Like how we assert the existence of the empty set

Joe

The fact that there exists a set follows from the axiom of infinity. (Some authors also include a separate axiom asserting the existence of the empty set, and some authors axiomatise first order logic in such a way that the domain of discourse must be nonempty. But regardless, we can deduce that $\exists x(x=x)$ from the axiom of infinity.)

Jakobian

19:52

@Obliv in ZFC, there is no distinction

in other set theories, who knows, I don't care, I use ZFC only

Obliv

@Joe ok i will read that thanks.

Jakobian

@Obliv its objectively correct

Obliv

yeah those remarks were for my mistake about equating $\in$ with $\subset$. That's the distinction I should have asked about

(i'm still not totally sure what it is)

Jakobian

$x\in y$ means "$x$ belongs to $y$" but other than this informal meaning, its really just a symbol we base all ZFC on

there is nothing recursive about $\subseteq $

$x\subseteq y$ simply means that all elements that belong to $x$, belong also to $y$

i.e. for all $z$, if $z\in x$ then $z\in y$

in set theory, everything you are quantifying over, is called a set

Obliv

ok for concrete example, $A = \{1\}$ and $B = \{1,2\}$. We have $A \subset B$ is true, since $1 \in \{1,2\}$ but $A \in B$ is not true since $\{1\} \notin \{1,2\}$?

Jakobian

19:58

sets are just objects in ZFC

Joe

@Obliv: The symbol $\subseteq$ is not in the language of ZFC. It is informal abbreviation, just as $\varnothing$, $\bigcup$, $\mathbb N$ are. Every time we write $x\subseteq y$, this is simply an abbreviation of the longer formula $\forall z(z\in X\implies z\in y)$. (The symbol $\subset$ either means the same thing as $\subseteq$, or it means proper subset, depending on the author. I would avoid using it without defining what it means first.)

Obliv

in other words, $\in$ is used strictly for sets in ZFC and $\subset$ applies to the sets inside of a set

Jakobian

so when you write $x\in y$, here both $x$ and $y$ are sets, this is valid to ask for truth value for sets, and only for sets

Obliv

@Joe ok that makes sense. Thanks

Jakobian

aside from when $y$ is a class, since then informally we are interpreting $y$ as some proposition $\varphi$ and then $x\in y$ just means $\varphi(x)$ is true

Yeah. In set theory the only symbols are $\in$

everything else is a definition i.e. its a symbol which we added to set theory

Joe

20:00

@Obliv: I think you might have slightly misinterpreted my comment. People doing set theory in ZFC still use the symbol $\subseteq$ all the time. The point is that it is officially just an abbreviation, and in principle, every proof in set theory could be unwound to a sequence of logical formulas where the only non-logical symbol used is $\in$.

Obliv

@Jakobian What does predicate formulae stand for? can $\varphi(x)$ be any statement in first order logic?

Jakobian

I don't like deciphering through the logic jargon, to be honest

Predicate (mathematical logic)

In logic, a predicate is a symbol that represents a property or a relation. For instance, in the first-order formula P ( a ) {\displaystyle P(a)} , the symbol P {\displaystyle P} is a predicate that applies to the individual constant a {\displaystyle a} . Similarly, in the formula R ( a , b ) {\displaystyle R(a,b)} , the symbol R...

some set theories allow for the so called urelements, to have elements which are not sets but which can belong to sets

but in ZFC, there is no urelements, and literally everything you are considering, as in, internally in your theory, is a set

so $0, 1, 2, ...$ in ZFC must be sets

similarly for everything else

I'm not a logic expert, I won't explain it like a logician would, I also don't care to explain it that way

Thorgott

that would only make it less comprehensible anyway

Joe

I feel that logicians get a bad rap, to be honest.

Obliv

didn't they found computer science and computing logic?

It's just not something mathematicians like working with I guess, I'm sure there is respect between the professions.

Joe

20:12

People studying mathematical logic would consider themselves to be mathematicians, though.

And there are many people who consider set theory to be a branch of mathematical logic.

Jakobian

20:49

@AlessandroCodenotti sorry, actually, I might postpone on reading Sakai and read Bredon instead (I forgot about it). I want to learn some basic algebraic topology, since it certainly might help me understand topology at large later

Obliv

21:00

@Joe en.wikipedia.org/wiki/Axiom_of_infinity#Formal_statement I don't quite understand the formal statement here

the "$\neg\exists n\text{ } n \in o$" part

Jakobian

there doesn't exists $n$ such that $n$ belongs to $o$

basically they're describing the empty set

Obliv

the english translation below seemingly ascribes the empty set to this part

@Jakobian but how does that translate to the empty set? it seems like that only describes one $n$?

Jakobian

what do you mean by "translate to the empty set"

The empty set is defined to be the set $o$ satisfying $\lnot \exists n\ n\in o$

From axiom of extensionality its justified to call it "the" empty set

Obliv

is $\neg\exists n$ shorthand for $\varnothing$ or something

Jakobian

$\lnot \exists n$ reads: there doesn't exist $n$

$\lnot \exists n \ \varphi(n)$ is a statement that there doesn't exist $n$ such that $\varphi(n)$ holds

Obliv

21:07

Ohh so $o$ isn't the empty set, it just has it inside of it.

$I$ (the postulated infinite set) contains all sets that also contain $\varnothing$

Jakobian

No

the set $o$ is the set satisfying property that the empty set has to satisfy i.e. it is by definition the empty set

Obliv

But it says $\exists I(\exists o(o \in I \land \neg\exists n \text{ } n \in o)...$ and doesn't every set contain the empty set?

Jakobian

@Obliv no

Obliv

I think $n$ is the empty set here

Jakobian

@Obliv the word contains doesn't really distinguish between $\subseteq$ and $\in$

$\emptyset \subseteq x$ for any set $x$, but $\emptyset\in x$ is not always true

@Obliv no $n$ is a variable

Obliv

21:11

@Jakobian For $\varphi(n)$ to hold, $\neg\exists n$ must be true so $n$ must not exist?

imo this makes it seem like $n$ is the empty set

Jakobian

$\lnot \exists n$, by itself, doesn't hold a truth value

Obliv

oh

so if it doesn't have a truth value, what is $\neg\exists$ saying?

Jakobian

that there doesn't exist

Obliv

Oh I see what you meant about axiom of extensionality now maybe.

because I was thinking what if $m \in o$ and $m \neq n$

so we can have $o$ nonempty

Jakobian

combining symbols is part of the language

Obliv

21:14

like "there doesn't exist n, such that n is in o" doesn't guarantee something that isn't $n$ is in $o$?

Jakobian

@Obliv it means that any two sets which satisfy the empty set property must be equal

Obliv

so $\neg\exists n$ such that $n \in o$ IS the empty set property.. my bad

thank you @jakobian for your patience.

en.wikipedia.org/wiki/Axiom_of_empty_set I didn't read the "alternatively:"

Jakobian

@Obliv you're welcome, I am being patient

@Obliv yes, if we were to describe what does "$x$ is an empty set" means as a proposition, we'd write that it means "$\lnot \exists n \ n\in x$"

@Obliv there is no fixed $n$

perhaps the property $\lnot \exists$ being the same as $\forall\lnot$ helps, so that $\lnot \exists n \ n\in x$ means $\forall n \ \lnot n\in x$

in other words, all $n$ don't belong to $x$

Joe

@Obliv: In natural language, the axiom of infinity asserts the existence of an inductive set. An inductive set is a set $S$ such that (i) $\varnothing\in S$ and (ii) for all $m$, if $m\in S$ then $m\cup\{m\}\in S$. Statements that use the symbol $\varnothing$ can be translated into logical formulas that just use the symbol $\in$. Here, $\varnothing\in S$ means $\exists x(\forall y(y\not\in x)\land x\in S)$.

Obliv

@Jakobian yes this is helpful as well

@Joe does $\exists x(...)$ kind of mean the same thing as $x = \{...\}$ in set builder notation?

Joe

21:23

(By the way, even the expression "$y\not\in x$" is an abbreviation. Essentially nobody uses the logical notation solely, except to convince themselves that they could in principle do so.)

Jakobian

$x = \{...\}$ doesn't have a formal meaning

you can say $x = \{z\in A : \varphi(z)\}$ for some set $A$ and a proposition $\varphi$. However, this is a specific case. You can think of sets $x$ as bags, yes, but this is an intuition rather than anything explicit

Joe

@Obliv: Writing $x=\{y\}$ is again an abbreviation, not a part of the official syntax of ZFC. You can think of it as short for $(y\in x)\land(\forall z(z\in x\rightarrow z=y))$. Similarly, we can think of $x=\{a_1,...,a_n\}$ as an abbreviation for a much longer formula.

Obliv

can you use quantifiers inside of set builder notation? I remember using logical operators like $\land,\lor$ in my intro to math reasoning class last year and I don't know if that was appropriate or not

Joe

Can you give an example?

Jakobian

@Obliv yes, you can insert propositions in there, which are built out of logical symbols, including quantifiers

Obliv

21:28

@Joe Oh that's pretty neat actually. I knew $x = y$ implied $\forall a (a \in x \implies a \in y)$ but seeing it written out is nice.

Jakobian

of course, they can't be built arbitrarily, say, $\lnot \exists n$ above doesn't have any logical meaning

things have to "make sense"

just like in natural language, you can't just say "there exists" and expect people to not look at you weird

Obliv

@Joe like for example $x = \{a \mid (a \in B) \land (a\in C)\}$ I know better now than to do something like this

but would it be feasible to mix the syntax like that

Joe

@Obliv: By the way, while it is technically true that all statements in set theory can boil down to statements just involving $\in$, I have to admit that this is rarely done in practice, and also that if you open a set theory book, the proofs you will see there will be natural language proofs, using all kinds of symbols. While understanding this hyper-formal stuff does help to an extent, I wouldn't begin my study of set theory with it.

Jakobian

yes. In logic you have such thing as "definition" which as Joe is calling them are abbreviations

Joe

@Obliv: Yes, what you wrote is fine, although if you were writing a proof or a definition or whatever, I would expect to see a definition of what $B$ and $C$ mean.

Jakobian

21:33

for example $\emptyset$ is a definition, justified by axiom of extensionality

you can go backwards, and write those definitions purely in terms of logic and symbol $\in$. No one does that

Joe

I like what Kenneth Kunen has to say about definitions in set theory.

Maybe reading this will help.

Jakobian

Maybe, or it will just confuse Obliv. The problem I see they're struggling with, is that of reading logical statements

but perhaps it will be a good exercise

Obliv

@Joe how is $\forall x$ abbreviated version of $\neg\exists x\neg$?

Joe

Saying that "all apples are red" is the same thing as saying "it is not true that there exists an apple which is not red".

Jakobian

@Obliv it can be an abbreviation, as in logical statements they mean the same thing. But that's not to say that people always define $\forall$ this way in logic. For example, I'd be against this type of definition. Its more natural to go with both $\forall$ and $\exists$

Obliv

21:43

oh okay so like instead of $\forall x \text{ }\varphi(x)$, we can also say $\neg\exists x\neg\varphi(x)$ which can be read "there doesn't exist $x$ such that $\varphi(x)$ isn't true"?

Joe

Yes, as Jakobian says, this is just the abbreviation that Kunen makes. It's not one that all logicians make. And it's definitely not how I read the symbol $\forall$ in practice.

Obliv

@Jakobian agreed

Jakobian

@Obliv yes and this can be thought of as a particular case of $\lnot \forall_x$ and $\exists_x \lnot$ being the same in formulas (and $\lnot\lnot \varphi(x)$ being the same as $\varphi(x)$, I suppose)

Obliv

is that called a logical equivalence? I vaguely recall demorgan's laws being relevant to that?

Jakobian

Its not just a logical equivalence

its really more like a "law" which describes logical equivalences of all sorts

its more like an axiom schema than an axiom, by comparison with set theory

it just says that you can always replace $\lnot \forall x$ with $\exists x \lnot$

or $\forall x$ with $\lnot \exists x \lnot$

and obtain a logically equivalent statement

Obliv

21:49

maybe it's called axiom schema of negation or something idk

Jakobian

again, I'm not a logician, and this is not important for mathematics

nomenclature that logicians use can be left for them

Obliv

@Joe so then $y \notin x$ is an abbreviation for some messing around with quantifiers? Like $\forall x(y \notin x)$ is the same thing as $\forall y(x \in y)$ or something?

Jakobian

no, this has nothing to do with quantifiers. Instead, it has to do with negation

$y\notin x$ means $\lnot y\in x$

Obliv

I thought you said before there is no truth value to expressions like $\neg\exists n$

Jakobian

yes, and?

Joe

21:55

In the notation $\neg(y\in x)$, neither $x$ nor $y$ are quantified. Variables that appear in a formula that are not introduced by quantifiers are called "free variables".

Jakobian

Yes. $\lnot y\in x$ has no logical value either, but for slightly different reasons

$\lnot \exists n$ is just not a valid sentence in this language of math

Obliv

@Joe does "quantified" essentially mean being taken into account for the following expression $\varphi(x,y)$ or whatever

Jakobian

$\lnot y\in x$ is sort of valid, but the variables $x, y$ are free, which means that they either need to be quantified over, or some concrete sets have to be put in their place for the truth value to exist

Joe

A quantified variable is simply a variable in a logical statement that is "attached" to a quantifier. So in the formula $\exists x(x=y)$, the symbol $x$ is quantified but $y$ is unquantified. The technical term for a variable that is being quantified over is a "bound variable" and an unquantified variable is a "free variable".

(It is possible to give a 100% precise definition of "free/bound variable", by recursion on formulas, but I don't think that will be helpful to you now.)

Jakobian

basically, it means that somewhere to the left there exists a quantifier which involves the variable $x$, then $x$ is a bound variable. If not, then its free

Obliv

22:03

Ok that makes sense. What about uniqueness $\exists !$? Is uniqueness an axiom or something that is derived

Jakobian

I don't know what some standard axioms of logic would look like, I'd be more inclined to talk about axioms of set theory, and my mind jumps to that when someone mentions axioms in that setting.

$\exists! x \varphi(x, ...)$ could be represented as $\exists x (\varphi(x, ...) \land \forall y (\varphi(y, ...) \implies x = y))$

this is the same issue as with the empty set above

we could describe things in terms of logic and $\in$ above, but no one does that

formulas would just get too complicated, and another point, they're supposed to be read by human beings

where $y$ is some variable not appearing in the formula $\varphi(x, ...)$

Joe

Depending on how you set your formal system up, there might be "logical axioms" as well as the "axioms" of a particular first-order theory like ZFC. But this is quite a technical point, and not all formal systems have logical axioms (e.g. natural deduction systems tend to have no logical axioms, replacing them with inference rules). I don't think learning mathematical logic to this level will be helpful until you have mastered the more basic stuff that every mathematician should know.

Jakobian

@Obliv Or you can keep on going this route and end up as a great logician one day

I suspect you don't actually need to learn those things, even though right now you think that you should

Obliv

I just seek answers to questions that I have :P nothing to do with what I want to be

Jakobian

it doesn't matter what you want to be. If you're learning quantum physics, eventually you'll become something related to that

I was just pointing out that you're overdoing the logic. No mathematician really thinks about those things in that much depth

If your goal is to learn math, you're not doing it right

think about symbols intuitively, understand them by translating them to English

but don't translate $x\in y$ as "$y$ contains $x$" but rather, "$x$ is an element of $y$"

former is ambiguous while the latter is not

also the order of sentence is preserved this way

most important is to not confuse yourself (you're your own enemy here)

Obliv

22:19

btw what was the consensus before on what $y \notin x$ can be abbreviated for?

Jakobian

$y\notin x$ is short for $\lnot y\in x$

Obliv

so we can define the empty set as $\exists A\forall x(\neg x \in A)$ instead of $\exists A\forall x(x \notin A)$?

Jakobian

this is not a definition of the empty set, but a claim that the empty set exists

Obliv

there was also the other form $\exists x \neg\exists y(y \in x)$. Which can be turned into $\exists x\exists y (y\notin x)$?

Jakobian

i.e. this is the axiom of the empty set

@Obliv the former is correct, the latter is not

when swapping negation with a quantifier, you need to swap the quantifier

Obliv

22:23

oh it should be $\exists x \forall y(y\notin x)$ I guess

yea

Jakobian

yes

Obliv

so for $y\notin x$ and $\neg y \in x$ we don't have to do a swap since nothing is being quantified except for $y$

but if it were $\forall y(y\notin x)$ then it'd be $\neg\exists y(y\in x)$

Jakobian

this is the definition of symbol $\notin $

robjohn

@Jakobian so, is the room easier to read with LaTeX on your phone?

Jakobian

whenever $y\notin x$ occurs in a formula, you can swap it with $\lnot y\in x$

@robjohn I'm on pc right now, but yes, on the phone its much easier

Obliv

22:26

right, but if it's quantified then you have to take the negation of the complement of whatever you quantified

Jakobian

@Obliv I don't understand you here

by definition, $\forall y (y\notin x)$ means $\forall y (\lnot y\in x)$

Obliv

yes you are right

I put the negation on the outside for some reason

Jakobian

it just happens that this is logically equivalent to $\lnot \exists y (y\in x)$ by one of the laws I stated somewhere above

Obliv

oh wait so that is also true?

$\forall y(y\notin x)$ means $\forall y(\neg y\in x)$ but also $\neg\exists y(y\in x)$

Jakobian

yes, if we were to quantify over variable $x$, or plug some value for it, those two things will be either both true or both false

of course, as a string of symbols the two are different

but when interpreted, they mean the same thing

3 mins ago, by Jakobian

by definition, $\forall y (y\notin x)$ means $\forall y (\lnot y\in x)$

here its just "as a string of symbols they mean the same thing"

Obliv

22:32

Not sure what $\neg y$ means in this context.

since it's a bound variable, what are we negating?

Jakobian

while equivalence with $\lnot \exists y (y\in x)$ says that the two, while being different strings of symbols, mean the same thing

@Obliv $\lnot y\in x$, while I thought it was obvious, since you can't negate $y$ as its not a statement of logic, means $\lnot (y\in x)$

the usual practice is to omit brackets when they are unnecessary, and sometimes when some convention on order of operations is used

Obliv

@Joe I misunderstood this. I thought Joe was saying that statement translated to x and y not being quantified

@Jakobian so $y\in x$ has a truth value?

Jakobian

Joe meant that in this example there is no quantifier to the left of the $x, y$ which means those variables are not quantified over

Obliv

OH it does.

it 100% does.

Jakobian

@Obliv not by itself

Obliv

22:36

@Jakobian yes I get it now

Jakobian

$\varphi(x, y) = "y\in x"$ is just a statement of logic, with free variables, which means you need to either quantify over them, or substitute $x$ or $y$ into the statement for it to have a logical value

Say, $\forall x \exists y \varphi(x, y)$ or $\varphi(\emptyset, \{\emptyset\})$

its a bit like a function

Obliv

@Jakobian yea you've said that before. What do you mean by either quantify over them or sub in $x$ or $y$ into the statement to have a logical value?

@Jakobian so this example the left side is true if for all $x$ $\varphi(x,y)$ is true and there exists $y$ to make $\varphi(x,y)$ true

not sure what the right side means

Jakobian

@Obliv no

first, this says nothing about if those sentences are true or not

Obliv

Hm.. i thought the point of this was binding the variables $x,y$ to the statement $\varphi(x,y)$

Jakobian

second, what does it mean for all $x$ to $\varphi(x, y)$ if you don't know what $y$ is?

> so this example the left side is true

okay sorry I didn't read that part, but you're still wrong about your interpretation

left side of that means in English: For all $x$ there exists $y$ such that $\varphi(x, y)$

what you said above, is just wrong, and I don't really know how you came up with it

it does not make sense to interpret it this way in English, nor does it make sense as a sentence in English

Obliv

22:47

Maybe it's the way I worded it. I just meant that $\varphi(x,y)$ is true for all $x$ and at least one $y$

is that not correct?

Jakobian

@Obliv its not correct

You can't say that $\varphi(x, y)$ is true for all $x$, since it depends on $y$ as well

Moreover, if you mean to bulk in the two quantifiers together, you can't do that either

the order of quantifiers is extremely important

Obliv

I see what you mean

Jakobian

$\forall x \forall y$ can be swapped for $\forall y \forall x$

but the same is not true for existential and universal quantifiers

for example, in definition of something like continuity

if you swap the quantifiers you might end up with uniform continuity

and if you swap the quantifiers again, even the property of being a constant function (if I recall correctly)

order of them is important

Obliv

for example: $\forall a \in \mathbb{Z}\exists b\in \mathbb{Z}(a+b=0)$ I can read this as, every integer has an additive inverse. Or in a more formal logic way, for all $a \in \mathbb{Z}$, there exists $b \in \mathbb{Z}$ such that $a+b = 0$ is true. I can't say "for all $a \in \mathbb{Z}$, $a+b=0$ is true and there exists $b$ to make $a+b=0$ true"

Jakobian

yes. The last sentence is just nonsense

Obliv

22:53

I was breaking up the quantifiers before to distinguish their purpose/nature of how they are quantified but it looks bad

Jakobian

@Obliv the right side means simply that if that $\varphi(x, y)$ holds true for $x = \emptyset$ and $y = \{\emptyset\}$

in other words, as per definition of $\varphi$, that $\emptyset\in \{\emptyset\}$

Obliv

Hmm I see

nvm its $\exists y$ not for all $y$

so yeah that left side is true, probably by some axiom

Jakobian

saying things like $x = \emptyset$ can only mean that $x$ satisfies the definition of the empty set

but this is meaningless of a thing to say

$x$ is a variable thats being quantified over, intuitively it goes over a lot of sets

you can say, if $x$ has property this and this, it needs to be $x = \emptyset$

that would be acceptable

saying $x = \emptyset$ however, refering to a variable thats being quantified in this way, is not acceptable

$\forall x \exists y \varphi(x, y)$ means the same as for example $\forall z \exists y \varphi(z, y)$

the question is, what are you referring to by $x$?

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