Mathematics - 2024-12-15

nickbros123

05:24

@Jakobian ok, I observed this now. what next

are we trying to map a countable subset $G \subset X^{\omega}$ to a unique element in $X^{\omega} \times \mathbb{N} \cup \infty$ ?

Jakobian

06:00

@nickbros123 I wrote that while still believing that the argument has a chance of working

nickbros123

ok. ill try to think of somthing different

Jakobian

I wrote a sketch of an argument for you

nickbros123

this is an exercise in munkres, and it comes before introducing AC, zorns etc

Jakobian

12 hours ago, by Jakobian

An injection of $X^\omega$ to countable subsets of $X^\omega$ is easy, you just send $x$ to $\{x\}$

starting from here

@nickbros123 is there no context to it and this is the exercise

nickbros123

just straight up this exercise was given, shortly after the exercise of proving cantor schroder bersnstein theorem. prior to this was just basic stuff on countable and uncountable sets.

Jakobian

06:18

@nickbros123 which exercise

Mats Granvik

platonicrealms.com/quotes/Winston-Churchill

nickbros123

page 52 on munkres

Jakobian

06:45

@nickbros123 Munkres in fact did use axiom of choice before

so its not that Munkres doesn't want you to use axiom of choice, but he doesn't want you to mention it explicitly

theorem 7.5 for example uses it

you probably do need to use some form of axiom of choice here

namely, for any countable subset $A$ of $X^\omega$ lets fix an injection $f_A:A\to \mathbb{N}$ - this uses axiom of choice

however you proceed, doesn't matter. But you can use axiom of choice, for example in this form

nickbros123

09:31

@Jakobian just to be clear, yesterdays argument fails because we wont be able to pick a 1st element in the ordering, right?

for every subset, that is

yeah makes sense, sometimes we can have infinite descending sequences

my set theory knowledge is kinda suspect

nickbros123

10:01

@Jakobian ok i think we have to use AC

Pizza

Hi

Soumik Mukherjee

13:11

@nickbros123 What I suggested won't work as it won't be injective. e.g consider the three element set {(0,0,0,...),(1,0,0,0,...),(1,1,1,...)}. The corresponding element would be (0,0,1,1,0,0,0,0,1,1,...) but the same element could be obtained from the two point set {(0,0,0,0,...)(1,1,0,0,...)}

nickbros123

13:41

yeah true, it breaks for finite sets. but also @jakobian alluded to the lexicographic ordering being not well-ordered as well

I think a workaround can be done by considering simply ${X^{\omega}}^{\omega}$

Mats Granvik

15:25

Is there a third alternative to yes no questions that go beyond answering yes no both neither?

Or fifth if you will.

Xander Henderson

@MatsGranvik In what context?

Mats Granvik

@XanderHenderson Like something beyond binary logic. 1 or 0, yes or no.

Xander Henderson

Fuzzy logic

Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1. The term fuzzy logic was introduced with the 1965 proposal of fuzzy set theory by mathematician Lotfi Zadeh. Fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz and Tarski. Fuzzy logic is based...

Undecidable problem

In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether an arbitrary program eventually halts when run. == Background == A decision problem is a question which, for every input in some infinite set of inputs, answers "yes" or "no". Those inputs can be numbers (for example, the decision problem "is the input a prime number...

Non-classical logic

Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is commonly the case, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth. Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well. In addition, some parts of theoretical computer science...

Mats Granvik

Oh yes. I will have a look at that. And yes to that undecidable too. 3-d Great !

PM 2Ring

@MatsGranvik How about mu? From en.wikipedia.org/wiki/Mu_(negative)

> mu is translated as "no thing", saying that it meant "unask the question".

Mats Granvik

15:34

I have never heard of that before.

pie

16:36

It’s been a few months since I last studied math, and I’m planning to start learning complex analysis. However, I’ve completely forgotten much of linear algebra, real analysis, and multivariable calculus. Should I dive into complex analysis and review these topics as needed, or would it be better to thoroughly review them first before moving on?

Xander Henderson

If you have to go back and review everything you once knew before starting to study something new, will you ever study anything new?

And "a few months" is nothing. In the US, we would call that "summer break".

pie

@XanderHenderson Good point.

Xander Henderson

One final thought: you never learning anything by simply studying it. You actually learn something when you have to apply it to something else. Students don't really learn the idea of partial fraction decomposition until they need it to evaluate integrals in calculus (and then they actually learn it in some higher level class when you can start talking about bases for vector spaces and whatnot).

Students don't really get Taylor's theorem in a calculus class; rather, they figure it out when it is applied to PDE.

And so on.

pie

@XanderHenderson I wonder how students are supposed to remember everything they studied in the long run. How do they even remember things after the summer break?

Xander Henderson

@pie What makes you think that you are supposed to remember everything?

Jakobian

16:43

@nickbros123 you can consider it... if every countable set was well-ordered as $\omega$ then you can send it to this set

Xander Henderson

Frankly, I've probably forgotten more than you have learned at this point in my career. But I know where to go in order to remind myself of important things when they come up.

pie

@XanderHenderson If not what is the point of learning them? If I am going to forget this course so why would I study it for the first place?

Jakobian

Of course with proper modification to accomodate for finite sets

Xander Henderson

@pie Reread the last sentence of my previous comment.

nickbros123

@Jakobian I was thinking in the lines of showing $|B| \leq |{X^{\omega}}^{\omega}|=|X^{\omega}|$ where B is the collection of countable subsets of $X^{\omega}$.

pie

16:50

@XanderHenderson 'Will you ever study anything new?' Honestly, I’d be embarrassed to admit how many times I’ve reread Baby Rudin and worked through at least 15 problems from each chapter.... now looking back that was a waste of time

Xander Henderson

@pie Yes, it was.

Jakobian

@nickbros123 I know

Xander Henderson

The whole point of learning---particularly at the introductory level---is to build up a toolbox that can be used for actually solving problems.

pie

By the way, when you need to review a topic, do you revisit and redo all (or most of ) the problems?

Xander Henderson

@pie Good g-d, no.

I think that you have a very weird idea about how professionals do their jobs.

pie

16:52

@XanderHenderson Now I really know I wasted my time :(

Xander Henderson

Usually, the process is like "Hrm... it would be nice if [x] were true. I think I remember a theorem about this." [checks several books and Google] "Oh, yeah. It's true under [y] hypotheses. Groovy. Let me just add a citation and move on."

nickbros123

@pie I do the interesting problems again, or atleast try to recall the killer trick. one pill ive had to swallow is that im gonna forget things no matter what the heck i do, like I can only remember theorems that I prove myself. so I just chill now, let my mind retain what it wants

pie

@XanderHenderson Since I study alone, I honestly have no idea what I’m doing.

nickbros123

some people are born with good memory, but also keep in mind that a notebook, or a latex document etc, can be treated as an extension of your memory xD

pie

The funny part is : I was going to reread baby rudin for the forth time just to be ready for complex analysis

Xander Henderson

16:56

@pie Yeah, that's a problem. There is a reason that one has an advisor when working on a graduate degree.

Math is not a solo sport.

nickbros123

@pie are you trying to install baby rudin into your head lmao

I just remember what thing is talked about where- roughly

Thorgott

well, there also is a difference between doing math professionally and trying to learn mathematics

pie

@nickbros123 If [x] is a prerequisite I thought I really need to know it 1000% in order to move on

@XanderHenderson That really explains a lot. I wish I knew people who are interested in math in real life.

@Thorgott what is that difference?

Thorgott

it's what Xander said

nickbros123

well i also am in the same boat, I just review material once a week while im learning the thing, in a really quick pace. I just like to keep theorems in my arsenal box, so that I dont have to flip the book when solving problems.

I Basically prove stuff, solve problems etc throughout the week, and sunday is consolidation + review day, but theres no hard and fast thing like that. in a course ofc u have to review things like 3 times, for 3 exams, i think that is enough if done properly. I think feynman quote is apt: learn things in most irresponsible, irreverant and original way posibl

Jakobian

17:04

When I'm doing math professionally I'm wearing my professional glasses. That's the difference

Soumik Mukherjee

Lol

nickbros123

@Jakobian ofc. safety is number 1 priority

pie

17:15

I went through a mental breakdown for months because I was frustrated with my progress in math, but it’s reassuring to know that people don’t keep rereading the same book endlessly like how I did.

But it's also disheartening to realize I wasted so much time... truly wasted my time.

nickbros123

Dont beat urself up, I also wasted a lot of time. I re-watched all 3 Nolan batman films instead of studying topology.

also the new batman film. that was really nice

Řídící

Are comments being "moderated" (and deleted) more than before?

Sine of the Time

@Řídící no?

Xander Henderson

@SineoftheTime mu.

Sine of the Time

eta.

Xander Henderson

17:28

Why is epsilon afraid of zeta?

8

Because zeta eta theta.

8

Sine of the Time

it's so funny I forgot to laugh (joking)

Jakobian

@pie what does it mean to waste your time. That's right, nothing

nickbros123

my prof told me not to read the AC, Well ordering, zorns, maximality principle stuff in munkres and go straight to topological spaces and continuous functions.

but arent there rich counterexamples from $S_{\Omega}$

Soumik Mukherjee

Don't avoid Zorn's lemma. It's useful in many context.

Thorgott

counter-examples that aren't really relevant in topological practice

Soumik is also correct, though

nickbros123

17:47

@SoumikMukherjee I know zorns lemma, and that it's equivalent to AC, but not any deeper results.

definitely dont have enough knowledge of ac to understand Asaf Karagila's answers

Ben Steffan

18:00

ah yes, Zorn's lemma, my favourite result from commutative algebra

copper.hat

18:41

in general, i don't understand enough to follow Asaf's answers.

Ben Steffan

logic is something else

Thorgott

18:54

there's nothing deeper to know

well, I suppose knowing transfinie induction also might help at times

Ben Steffan

I almost used transfinite induction for a proof last week

now I know what the term means :)

Thorgott

Lurie does a bunch of transfinite inductions in HTT

Ben Steffan

yeah

Thorgott

I also had a brief set theory phase early on

1

Q: The Dimension Theorem via Transfinite Recursion

Dimension Theorem. Let $K$ be a field and $V$ be a vector space over $K$. Let $\mathscr{B}$ and $\mathscr{C}$ be bases of $V$. Then $|\mathscr{B}|=|\mathscr{C}|$. For finite-dimensional (a priori, finitely generated) vector spaces, this theorem admits a nice proof that roughly goes as follows: O...

sometimes I still think about this stupid question

Jakobian

@nickbros123 what do you mean by $S_\Omega$

either way, what are you going to do with your topology knowledge

because if you just want to use it for analysis or something, then bother with set theory too much

nickbros123

19:15

@Jakobian that's a philosophical question for which I don't have an answer right now

Ben Steffan

@Thorgott we all have our Jugendsünden :)

On an unrelated note, I hate Haynes Miller

giving proofs in papers is underrated I guess

Thorgott

lol

Ben Steffan

I also get to skim a classical paper of Husemoller, Moore & Stasheff where they call chain complexes "differential modules" lol

Xander Henderson

22:37

I have blueberries. My plan is to make either pancakes or waffles for dinner (because I don't generally eat breakfast, but I like breakfast foods). Pancakes are less work and fewer dirty dishes, but I really like waffles. It is a quandary...

Ben Steffan

mhhm, waffles

think_meaning_buildß

mhhm, pancakes 🥞

Ben Steffan

this town is not big enough for the two of us

think_meaning_buildß

bye

Xander Henderson

@BenSteffan Does that mean that you want me to kick you?!

Or are you going to show yourself the door?

:P

think_meaning_buildß

22:41

no, I'm leaving np

Ben Steffan

Not even a duel? :(

Sine of the Time

bro left

Ben Steffan

I just wanted to die Galois' death :(

Xander Henderson

PISTOLS AT DAWN!

Ben Steffan

pestles at dawn, and hope your opponent doesn't bring a mortar

Xander Henderson

22:43

It would make sense if they brought a mortar. Problems start if the bring a metate.

Joe

23:46

Hi, does anyone know how I can centre the text in the following example of set builder notation?

$$
\dim(X)=\sup\left\{
n\in\mathbb N \;\middle|\;
\begin{aligned}
\text{there is a strictly ascending chain}\\
\text{$Z_0\subset Z_1\subset\dots\subset Z_n$ of subsets of $X$ such that}\\
\text{each $Z_i$ is an irreducible closed subspace of $X$}
\end{aligned}
\right\} \, .
$$

(I'm trying to do this in a TeX document, and so it doesn't matter if the solution requires commands that aren't available in MathJax.)

Jakobian

@Joe try gather environment

$$
\dim(X)=\sup\left\{
n\in\mathbb N \;\middle|\;
\begin{gather}
\text{there is a strictly ascending chain}\\
\text{$Z_0\subset Z_1\subset\dots\subset Z_n$ of subsets of $X$ such that}\\
\text{each $Z_i$ is an irreducible closed subspace of $X$}
\end{gather}
\right\} \, .
$$

Ben Steffan

are you sure you want to have this kind of monster in whatever you're typing

Transcript for

$Mathematics$ Mathematics