@Simply getting ready for those large numbers :D

@Zophikel I can't even... not right now...

My brain is mush

And I did something that someone really good at large numbers is trying to grasp

think he's been trying to grasp it for about 20 minutes now

Before I told him what I was doing, he gave me a guess

Ψ(Ω_ω)

I was like WTF?!

And he explained to me what he thought

which was nowhere close to what I was actually doing

I think I found mindfuck level 2

I think I shall be taking a break this evening and shall write this weeks large number thing tomorrow morning

@amWhy @Zophikel @everyone.else Sleep sanely

@Simply you think I should supplement the diffclut stuff with easy stuff from time to time

Of course!

personally, the greatest supplement is rewriting the stuff altogether

For example, I gained much insight into large numbers by creating my own large function

(its the thing that's gonna give me nightmares tonight)

Huh, he says I'm at Ψ(Ω_(ω^4))

@user21820 I think I broke my brain

I made PAIN, right?

PAIN ≈ e_0

MIA-PAIN ≈ ⱷ(ω,0)

And now RMIA-PAIN ≈ Ψ(Ω_(ω^ω))

You need to join Discord @user21820

Don't even know what happened to getting to the n-argument Veblen function or small/large Veblen ordinals, so yeah

@Fawad Hey man

Hi

If you came for the large numbers, we start in the morning because my brain is melty

Uh,I have school homeworks

ah, okay

much better XD

@Simply it's finished :)

@SimplyBeautifulArt Please reread the comment of mine you responded to: "For example, in this chat room you needn't worry about informal...". So you are simply repeating my point, otherwise, why were you reponding to me?

@amWhy I think he was responding to me

also @amWhy i'll tell you the details of my test later

@Simply see ya guys

@amWhy sorry for pinging you so lately. Just wanted to point out on the room description

@Zophikel I will look at that in the morning. Good night!

@amWhy Enjoy your weekend! =) You reminded me of this:

phdcomics.com/comics/archive.php?comicid=1937

=P

@Zophikel Sometimes intuitive definitions are useful for assisting understanding. But one must always be able to provide precise definitions when requested. That is the peculiar but precious hallmark of mathematics. If we are unable to precisely define something, there could be many reasons. Perhaps we do not understand the fundamentals enough. Or perhaps we are looking for something vague. See this meta post for more explanation.

That doesn't mean we should stop asking. Rather it means we should just try to learn what is it that we are lacking in precision, and especially try to look from a stranger's point of view. The reason I say this is that sometimes being precise is actually the key to solving the problem; a lot of students cannot solve problems simply because they do not even know what precisely is being asked, and what precisely are the given assumptions and known facts.

Sometimes this means that instead of asking our original question we might want to ask about the fundamentals for that topic instead, because it might very well enable us to solve not just our original question but many others as well.

Makes sense? =)

@SimplyBeautifulArt What's Discord?

@user21820 discord is a voice chat/ messaging software which is very lightweight

runs on browsers too

kind of like skype but no bloating

@shredalert: Ah. But why would I need/want Discord? I currently use the old Gmail chat, but not online.

I almost never mix my online and real-life stuff, because who knows what people might do? =P

@user21820 was just giving a description because you asked earlier. :p

@shredalert: Yeap I know. I'm just wondering why SBA suggested it. =)

going to do a bit of analysis now

lots of balls I see

lol

I love the little bits on topology, and then I tend suffer from "boredom by limit". xD

Haha!

Okay see you!

see you later

@FabianGerhardt: Hi there!

Haha..

yeah hi :)

I think it's best we continue here

I'm pretty sure kolmogorov touched almost every field of mathematics in his time haha

Well about Kolmogorov complexity, it's not that useless. It's true that it's uncomputable and all that, but did you know that you can use it to prove things like the incompleteness theorems?

yeah its useful for very foundational things

thats why I find it interesting

he was a good physicist too

the ZFC thing, the only people disliking it also did computer aided proofs for the existence of good. so my preconceptions regarding disliking ZFC are not on your side

god*

not good

@FabianGerhardt: Unfortunately, that's a very narrow-minded and wrong notion.

ZFC is just one of many formal systems.

yeah and it seems ok to me. the others seem weird

never understood why one would toss the law of excluded middle

I'm saying actually that your preconception of people who do not like ZFC is narrow minded.

Distrusting ZFC has nothing to do with LEM.

And nothing to do with other systems.

And nothing to do with God.

yeah I know, still thats my only emotional connection to it

Proving things inside a system doesn't mean it is universally true. :P

yeah sure

just saying. ZFC seems ok, works so far. I'm fine

@FabianGerhardt But then it makes no sense. There are tons of systems that retain LEM and are not ZFC.

I wasn't talking about anything in particular btw haha

And naive set theory worked okay until the crisis.

lets wait for the next

Most mathematicians these days work with naive set theory anyway

haha

We don't know what's in store for ZFC until it comes, because the incompleteness theorem guarantees we can't know...

again I know that

maybe life's just hard

Okay so why bring up God? We both know it's silly to try to prove things about God within a formal system.

we're in the realm of philosophy now :p

goes back to playing with balls

because these are the only people, I've met, wo opposed ZFC. these people brought god into it. it does not have anything to do with ZFC itself. just said that, because I wanted to make my emotional standing clear.

@shredalert Yes we are. But there are practical issues here. We don't want to build a bridge whose stability is based on some theorem proven in a formal system that also proves 1=2...

@FabianGerhardt That's fine. But that's not the only people disliking ZFC. =)

I know

And I guess it's strange that those are the only people you've met.

Because Godel liked ZFC and he also attempted to prove existence of God.

Which is .......

just didn't meet many people with interests in logic

Yeah, I think that was partly why he ended up in an asylum

wikipedia says he did that for fun, to show how silly it is to do that

@shredalert Supposedly, it was CH, not God that pushed him crazy.

oh

CH drove Cantor nuts, that's for sure

what is CH?

Continuum Hypothesis

ah.

@FabianGerhardt I'm going to look that up! Didn't hear of that aspect before.

@user21820 not 100% sure, better check it!

The wikipedia article actually says the opposite to what you said lol.

It says Godel actually believed in God, or so most people believe.

thats not the opposite

Um okay it's not the opposite, but I interpreted your statement to imply that Godel didn't believe in God and was trying to show how silly it is.

Just to make clear, I myself think it's very silly.

=)

I'm trying to look it up now too.

yeah I was meaning to say that he believed that a formal proof of the existence of god is silly. but I'm probably wrong

my mind makes up things

the situations seems really confusing en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof

@FabianGerhardt I'd be interested to see if you have some reference for that Godel believed formally proving existence of God is silly. The linked wikipedia article seems to suggest that he actually took it seriously.

yeah exactly

Since Anselm's 'proof' is a well-known argument that many believers 'believe'.

Probably/Possibly including Godel.

yes makes more sense that way

It's surprising what people believe, including ourselves.

I'm surprised every day

Don't know whether you two have heard of the easily reproducible experiment that people always attempt to confirm what they believe instead of testing it, and hence get more and more convinced with zero evidence.

is that a recursive trick

Like this: I'm thinking of a pattern for triples of numbers, and one of them is (2,4,6). You can ask any triple and I'll tell you whether it follows the pattern or not. When you think you got the pattern, you can say so and we'll see whether you're right.

Of course, I don't expect this to work on Math people. =)

But you may be surprised how many non-Math people fall for it. Try on your parents and see! =D

I don't need experiments to tell me how much stupid stuff people believe. its painfully visible

Lol.

xD

The experiments are however one way to help people see this particular flaw in their reasoning.

Otherwise few will ever learn to be properly skeptical of claims.

hmm. I never had much hope, some people ask, others dont. always thought it would be a thing of upbringing

Oh I'm much more hopeful. In my experience, we can in fact teach people to learn this kind of things.

The reason we don't observe more people with critical thinking is simply because not enough teachers emphasize such things.

It takes time and effort though.

oh, no hope in school either

haha

Hahahah..

but thats probably stupid. there have been people making things better always

Strangely, the university I'm at recently introduced a compulsory course on critical thinking for all first-years regardless of major!

And not just one but a few, each on a different aspect of that.

Quite surprising, pleasantly.

haha that sounds so weird. wouldn't want to be in such a course

I don't know. I think I wouldn't need it at all, but then again, that's precisely the kind of mistake that I believe people who should take that course make when they don't want to take it.

haha, no I don't think feynman would have needed such a course and he would have known

Hahaha there are always exceptions.

I'm gonna go, continue writing my thesis

Ok see you around next time!

see you later @FabianGerhardt

@user21820 it's an app for chat

And if the people I talked to aren't crazy, then ,I just passed BHO

@user21820 Found a group of googologists

@SimplyBeautifulArt define bho

@Fawad Bachmann-Howard Ordinal, psi(e_(Omega+1))

the supremum of Madore's ordinal collapsing function

@SimplyBeautifulArt Okay!

That's my Discord

Ah I see. Well I use nearly no social apps or websites whatsoever.

Probably this chat-room is the most common place anyone can find me recently. =D

XD

Aw man, you should give it a try :P Its not like we talk about our life and stuff

XD They even play games related to large numbers, which makes me feel weird, but you know, whatever

Lol!

I know. But I have too many things to do and can't afford to spend too much time on other things.

One of the games is "so the first player start at zero..." and we go on trying to slowly build our way up to larger numbers

Haha!

also, healer classes in video games should not be so offensive

but when they are, they op af XD

in guild wars 2 every class could do everything

it was great until they broke pvp

lol

there was a skill for elementalists where you literally turned into a ball of lightning and zipped around

it was called "ride the lightning" as well xD

woo! This healer op

After an hour, we've finally reached 10^^(10^^100) @user21820

Gonna be a while XD

@SimplyBeautifulArt Why aren't you going straight up to BB(that)? =P

By the way, do you know that you can play Nim with ordinals instead of natural numbers?

And it's not that hard to find and prove the optimal strategy.

@user21820 We are supposed to go slowly

How slowly is slowly? Can't just plus 1?

Not that slowly

but like if we are on exponential level, don't bring in tetration

we at f[w2 + 1] level now

make that w3

w^2

:D

Ok take your time!

w^(w2) level now

:P My p-notation is fun to play with

w^w^w level now lol

and everyone uses their own notations

is good way to waste my day

and we taking a break at w^w^w^w^w

last time I played I went up to zeta_1

the game takes forever XD

Indeed.

@user21820 I have trouble giving precise definitions

Anyway to improve

@Zophikel: I don't doubt that you will improve! Learn to ask the right questions and it will be easier too. =)

@user21820 I was unable to state lipchitz community rigsorsuly

When I try to state things in detail

@Zophikel: Can you first state normal continuity rigorously? And check your spelling before you hit ENTER.

lol

@user21820 And we jumped to e0 level now

Lol.

In PAIN notation

P n # @[][] # k ≈ f[e0 * k](n)

You should learn PAIN and the other weird notation other people have

Aarex and Deedlit are on the Discord server too

Haha.

well, @Deedlit has been away for quite the while.

Pn#@[]#k ≈ f[ω^ω *k](n)

Pn#@[]#0#1 ≈ f[ω^(ω+1)](n)

Pn#@[]#0#k ≈ f[ω^(ω+1) *k](n)

Pn#@[1]#1 ≈ f[ω^(ω2)](n)

Hm...

Pn#@[1]#k ≈ f[ω^(ω2) *k](n)

Pn#@[]#1#@[1]#1 ≈ f[ω^(ω2) + ω^ω](n)

ah, so that's how my notation adds up ordinals

Pn#@[a]#1 ≈ f[ω^ω^a](n)

Pn#@[0#1]#1 ≈ f[ω^ω^ω](n)

Hm, no, wait, not quite right

Ugh, what slow time it takes to reach e0

Ah, I see

@[a#b#c#...] only represents every combination of ordinals up to ω^ω^ω, which is the limit

Then @[@[]#1] = ω^ω^ω

Er, hm, something like that

And then Pn#@[][k]#1 ≈ f[ek](n)

and then Pn#@[][][][]...#1 ≈ f[phi(a,0)](n)

so the limit of that version of PAIN is f[phi(ω,0)](n)

Heh, the game just moved to e2

@user21820 for normal continuity basically we have our function defined on an interval (a) we say that the function exists if: $\lim_{x \rightarrow a}(f(x)) \, = f(a)$

If this condition isn't satisfied then we say our function is discontinuous.

But @user21820 I feel like what I said wasn't detailed or precise :(

Well you could make it more precise by saying "for all a in A, then $\lim\limits_{x\to a}f(x)=f(a)$"

@Simply all right

Personally, if you wrote all that in the right format, I'd be satisfied. I suppose sometimes, what we mean is we want you to use less words (which are vague/ill-defined) and... uh, more math

For all a in A, then $\lim\limits_{x\to a}f(x)=f(a)$, if \limits_{x\to a}f(x) \neq f(a) the our function is discontinuous

@Simply is that better

Hm, we are doing single variable functions, right?

@Simply yeah but this can be generalized for multivariable functions

Hm, okay

@Simply which that definition was used in my proof

How do you rigorously write the definition of a limit in multiple variables? @Zophikel

@Simply it would look like this

For all f(a) in A, then $\frac{\partial }{\partial a_i}= $\lim\limits_{h\to 0} \frac{a_1,....,a_{I-1} -f(a_i, ..., a_i ...., a_n)}{h}$ then our function is continuous at f(a).

^ @Simply I had written this one before-hand when revisting my question

@Zophikel You wrote "the function exists" but it's supposed to be "the function is continuous at ___". Fill in the blanks.

@user21820 ahh sorry

Mhm, okay. Though LaTeX please lol

@user21820 how did I do, is It bad to look up things to be rigorous ?

@Zophikel: But that's the point. It is not that you are not able to be rigorous, but you have never put much effort into it before.

@user21820 what do you mean ?

You do need to be able to state the definitions without looking them up.

@user21820 all right ahh

But as I said, it's not that you can't do it. I'm sure you can.

You just need practice.

@user21820 in fact I can for some definitions

That's good! That's what you need to aim for, for all definitions.

@Zophikel: Let me give an example of a rigorous definition of continuity.

@user21820 what's a good way to practice this: i'm thinking everytime I look a definition within a proof I write down what I had to look up and then practice the definition

all right @user21820

A function f from R to R is said to be continuous at the point c in R iff f(x) -> f(c) as x -> c. The last part could be phrased as lim_{x->c} f(x) = f(c). In turn, you should know how to expand the definition of the limit.

@Zophikel I'm not sure what's a good way; you just need to attempt to write down the definition of every term you come across that you are not already absolutely certain of being able to write down.

@user21820 the way to expand this is to add the notion if our function is discontinuous at a certain poin or interval

That's not what I mean.

I mean to further expand the "lim_{x->c} f(x) = f(c)".

Do you know the formal ε-δ definition of the limit?

@user21820 yes

Off the tip of your fingers?

@user21820 I can try:

Write it out for this particular case and let me see.

ok

@user21820 I don't know it off the tip of my fingers :(

So that's where you know you need to focus your learning.

It's because you do not fully grasp the notion of limits.

ahh all right @user21820

$lim_{\delta->c} f(x) = L$

?

Informally, we can say that for any positive error margin ε, there is a positive window of size δ around c such that if x is in that window then f(x) is close to f(c) within the error margin.

@user21820 all right

Which you need to translate to rigorous quantifiers.

@user21820 so it would look like this:

$| lim_{x \rightarrow \delta}f(x) | < \ epislion$

?

No.

You really need to revise the meaning and definition of limits.

Do you have a proper teacher and a textbook?

I recommend Spivak's Calculus.

@user21820 I have a book i'm going through

Do you have a teacher?

@user21820 no not yet

Hmm.

Is your book available online?

@user21820 yeah

Where?

(Legally)

archive.org/stream/pdfy-_KqLYE14P4NqcS2j/…

@user21820 there are some extreme gaps in my rigor that I'm trying to address

@Zophikel: Page 35 has the definition of limit of a sequence, right?

But @user21820 for being more rigorous what do you recommend how do I learn how to state definitions without looking things up

@user21820 think so

Are you able to write that definition in logical form?

And do you actually understand it?

@user21820 I think I can

Write it.

$N > \, n $ implies $|S_n - s | < \eplision$

That's not even what's written there, and no that's wrong.

The logical form includes all quantifiers.

@user21820 oh I misread sorry

I'm not sure how to state it in quantifiers

Do you know the "forall" and "exists" quantifiers?

@user21820 yes

epsilon*

forall real ε>0 ( exists natural N ( ... ) ). Can you fill in the blanks?

@user21820 Yes

Fill it.

Wrong.

Write exactly what I wrote and fill in the blank.

Don't just write a partial answer.

sorry @user21820 and thanks for being patient with me

And be careful. I already said earlier that you weren't even copying what was in the textbook correctly.

For all real $\eplision\, > 0$ there exists a natural N such that: $N > \, n$ implies $ |S_n -s} < \eplision$

epsilon*

thanks @SimplyBeautifulArt

lol, no problem

Wrong. Copy exactly what I wrote, and fill in the blank.

> forall real ε>0 ( exists natural N ( ... ) ).

I do not want English. I want logic.

The brackets are important.

And you're still making the same mistake as before with your "N" and "n".

$\forall\varepsilon>0(\exists N\in\mathbb N(\forall n>N(|S_n-s|<\varepsilon)))$

?

@SimplyBeautifulArt: You're not supposed to help.... =S

:(

Oh! Sorry

Did I do it right though?

@user21820 can you give me another one

It's near perfect though, except to be clear you should say "forall natural n>N".

hmph, I need practice sometimes too you know

I know, that's why I just commented on your attempt.

Ah, okay

I'm being pedantic, but it's just a good practice at least at the start.

After you can be pedantic you can stop being pedantic. =D

@user21820 can you give me another one

i'm learning a lot here but I feel dumb at the same time

forall real ε>0 ( exists natural N ( forall natural m>N ( |s(m)-s| < ε ) ) ).

This is the definition of lim_{m->inf} s(m) = s.

@user21820 what page are you on

@Zophikel: Page 35 of your textbook. Is it the correct one?

@user21820 yeah just checking

But @user21820 can you give me another one

I will.

First, I want to ask questions.

@user

@Zophikel: First do what I originally asked for, namely expand the statement "lim_{x->c} f(x) = f(c)". It should be similar.

> A function f from R to R is said to be continuous at the point c in R iff f(x) -> f(c) as x -> c. The last part could be phrased as lim_{x->c} f(x) = f(c). In turn, you should know how to expand the definition of the limit.