The h Bar

^^

( ͡~ ͜ʖ ͡°)

(ง ͠° ͟ل͜ ͡°)ง,

Don't you just love when the dude sitting in front of you smells like skunk?

@0celo7 That's a weird fetish.

2:20 PM

@ACuriousMind Don't worry, I still like mismatched socks ;)

<3

our star wall is a mess

It's been a while since a good (mat-phys) question was asked on the website... ;-(

no wonder we have a bad rep

@yuggib "has the invention of ZFC increased GDP"

@0celo7 Neither a physics nor a good question :P

2:22 PM

I think it's good.

ZFC might have been used for the development of stochastic theory which is used in economics

Interesting point

Whose predictions might be good for increasing the GDP

(Maybe you'd have to ask an economist)

@Slereah and mathematical finance ruined the world and was responsible for a part of the financial crisis we are in...

"If you are a good economist, a virtuous economist, you are reborn as a physicist. But if you are an evil, wicked economist, you are reborn as a sociologist."

2:23 PM

so ZFC is evil

@yuggib ::refrains from discussing politics::

@yuggib it is, but not for that reason

@ACuriousMind has TQFT increased GDP

Well, science publishing companies are businesses

So maybe

no, because the TQFT people could have done something better with their time

but that is true of all activities

Except the Best Activity

engineering?

2:32 PM

@0celo7 Let $\kappa$ be a strongly inaccessible cardinal; $V_\kappa$ the corresponding von Neumann universe. Then $V_\kappa\vDash ZFC$, i.e. $V_\kappa$ is a model of ZFC. Therefore ZFC is consistent by Gödel's completeness theorem, but not maximally consistent by Gödel's incompleteness theorem.

gold mining, obviously

because gold

@yuggib what the hell

All this math nonsense does not rise up to the level of Einstein and the Evidence

I hope I don't have to do any set theory

it's too complicated

set theory is my maximum math

@0celo7 Just to quote your ZFC being evil for other reasons...

2:35 PM

even more maximum than analysis

You don't use a lot of set theory in physics

@yuggib ZFC is evil, no theorem needed

@Slereah I'm considering adding a math double major

Don't you fool

I know math 307 touches on set theory, but Denzler said it doesn't talk about ZFC

Do you want to learn about measure theory

2:36 PM

@Slereah actually, yes

@0celo7 You don't use a lot fo set theory in math either... :-D

let's go back to the powerset

so we have $2^{\{a,b\}}$

what then

you want to know why it is the powerset or what?

I want to know why $\{0,1\}=2$

and that isomorphic thing

with the maps

2:38 PM

it's just the definition of integers in ZFC

ok, motivation?

you define $0=\emptyset$

It's simple enough?

And you can do it with only sets

$1=\{\emptyset,\{\emptyset\}\}$

WHAT DOES THAT MEAN

2:39 PM

$2=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$

That is like asking "What does the decimal system means"

and so on

wtf

then you see that $2=\{0,1\}$ by definition

It's just a representation

2:40 PM

what does $\{\emptyset ,\{\emptyset\}\}$ mean!!!!!

and that these objects satisfy all the properties of the natural numbers

@0celo7 It's a set. What do you expect it to "mean"?

@yuggib no!!!!

FenderLesPaul

@0celo7 lol read the abstract: arxiv.org/pdf/1509.05174.pdf

The set that contains the empty set and a set containing the empty set

FenderLesPaul

2:40 PM

it has a fucking poem in it

@FenderLesPaul haha

BBS has a translated chinese poem in it

@Slereah wtf

that makes no sense

@0celo7 Hint: substitute the definitions of $0$ and $1$ inside $\{0,1\}$

1 = $\{\emptyset, \{\emptyset\}\}$
Define 0 = $\emptyset$
1 = $\{0, \{0\}\}$ ?

@Secret yes

The idea behind it is that when people tried building math from set theory

they wanted it to be the simplest system

With the fewest building blocks

hence sets do not contain anything but sets

2:43 PM

the "basic" set being the emptyset

wtf $2=\{0,\{0\},1\}$

this is too hard

ZFC is pretty simple really

2 = $\{0, 1\}$ = $\{0, \{0,\{0\}\}\}$ = $\{\emptyset, \{\emptyset,\{\emptyset\}\}\}$ ?

At its core, it only has like

5 different operations

You can build set theory from like

@Secret that's not what he had above

I'm very confused

2:44 PM

$\vert$, $\in$, $\forall$

Hm, do you need anything else

outside of variables

I am missing a $\{0\}$ where does that came from?

Good thing that @0celo7 hasn't found out about Dedekind cuts yet

:,(

I'm so bad at math

@Secret Yeah you're right...d'oh!! so $2=\{1,\{1\}\}$

ofc he ignores me not being able to get it

probably just thought I was an idiot

2:46 PM

@0celo7 You were right too

never believe to memory, or other people ;)

what does $\{1,\{1\}\}$ even mean!?!?!?

@0celo7 now it works

Calm down @0celo7

Breathe

it is the set containing one, and the set that contains one

Too tiring to type the $\{$ I am gonna use my mouse instead

2:48 PM

@Slereah I'm sick!

easier said than done

You are way overthinking this

@yuggib that makes no sense

@Secret what?

Typing many "$\{$" is too tedious for me atm

{}{}{}{}{}{}{}{}{}{}

Those are

Sets

A collection of objects

2:51 PM

D'oh!!!!!! Actually, this is not the definition I wanted ;-(

what

@Slereah let's talk about GR

Sure

Let's talk about topology of GR

A topology is a set

$0=\emptyset$, $1=\{\emptyset\}$, $2=\{\emptyset,\{\emptyset\}\}$...ok, so now $2=\{0,1\}$ indeed. That's what I wanted from the beginning

::@0celo7 would kill me if I was in the same room::

@Slereah let's talk about not GR

let's talk about crystals

FenderLesPaul

AdS/CMT

get the best of both worlds

2:55 PM

let's see

crystal symmetries are governed by finite groups

they are

:-D

a group is a set...

Spoiler

In modern math

Everything is a set

(Well except logic)

@Slereah A category might not.

2:56 PM

You can build category theory from sets

@ACuriousMind but a small category is

can I have the category of all categories

what about the set of all sets

@yuggib I said might

@0celo7 Haven't we been over that?

@ACuriousMind ;-) I know...but I wanted to annoy @0celo7

@ACuriousMind yes

er, no

not that I recall

2:57 PM

@yuggib
http://i.stack.imgur.com/ryFgn.png

Do I get the rules right?
(This is NOT a visualisation as per ACM definition)

Might not have been paying attention ~~~

@Secret Yes, exactly

the examples are good

box theory

But what's the justification of putting an increasingly number of nested sets within each other, It seems to get very messy when you get to n?

Well in practice, you don't have to deal with the sets themselves

You just deal with the property

3:00 PM

@Secret Because you want a way to define various types of sets by the number of objects they have

This is by the way similar to one implementation of integers in LISP

Where 0 is NULL and n+1 is (cons n NULL)

and actually define numbers that way

So why can't I have the set of all sets

Although as said, there are other ways to define integers from set theory

And the von Neumann one is pretty straightforward

@Slereah That's how my 1st year maths class taught us sets. They say everything becomes easy to see once you imagine the empty set as an empty box

What's the problem of the above way to define integers?

3:02 PM

@Secret That wouldn't work

Because {0,0} = 0

@Secret that $0$ is never defined

also that, yes

@0celo7 If you had the set of all sets, you'd run into Russell's paradox, which is exactly what ZFC wants to avoid.

the one on the right hand side of the first equality

a set cannot contain twice the same element

3:03 PM

Oops I mixed that up with multiset, sorry

Because you could use the axiom of comprehension to write $\{S\in\mathsf{Sets} \mid S \notin S\}$ if you had the set of all sets $\mathsf{Sets}$.

if $x \in S$ then $S \cup {x} = S$

That is why in set theory, an ordered pair <a,b> is defined as {a, {a,b}}

It does, yes

That's the von Neumann integer scheme

I think I prefer von neumann, as all I have to deal with is one stack of nested boxes

3:06 PM

Again, the right hand side $0$ is another thing, $\emptyset$

Yeah just define 0 as the empty set

the advantage of the other definition, is that $n\subset m$ whenever $n\leq m$ (i.e. the set theoretical inclusion is a (well-)order)

like this?

yes

Then the Peano axioms can be defined as $0 = \{\}$ and $s(n) = \{n\}$

and the theory of cardinal/ordinal numbers use that property a lot

(the one that $n\subset m$ whenever $n\leq m$)

3:08 PM

ok, this is a lot easier than numerical analysis!
In fact, I found most abstract algebra pretty striaght forward once you know the rules, since in some way the rules can be treated one step at a time

What is the axiom of comprehension

I need some of that

I don't comprehend this

Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets free of paradoxes such as Russell's paradox. Zermelo–Fraenkel set theory with the historically controversial axiom of choice included is commonly abbreviated ZFC, where C stands for choice (Ciesielski 1997). Many authors use ZF to refer to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Today ZFC is the standard form of axiomatic set theory and as...

Zermelo fried chicken

@0celo7 The axiom of comprehension says that you may define a new set if you have a set $X$ and some formula $\phi$ by $\{x\in X\mid \phi(x)\}$.

The crucial thing is that you have to have some $X$ to which you apply the formula, it is not enough to only have the formula to define a set.

so you cannot have $\phi$ acting on $\emptyset$?

3:13 PM

Well, you can, but that obviously just gives the empty set, no?

@JohnDuffield ... I'm not even going to bother untangling this, you're doing the same strawmans with the new references

@Secret $\{x\in\emptyset\mid \phi(x)\} = \emptyset$ regardless of what $\phi$ is.

Again. Time travel may be science fiction. CTCs are mainstream research, and are a valuable part of GR research. Derailing a discussion of mainstream CTCs with "time travel is science fiction" is not okay

oh yes, $\emptyset$ has no elements, thus there is nothing for $\phi$ to act on, so the reuslt is still empty

@ACuriousMind : What is the equivalent to the axiom of comprehension in that list

Axiom of Extensionality ax-ext |- (A.z(z e. x <-> z e. y) -> x = y)
Axiom of Replacement ax-rep |- (A.wE.yA.z(A.yph -> z = y) -> E.yA.z(z e. y <-> E.w(w e. x /\ A.yph)))
Axiom of Power Sets ax-pow |- E.yA.z(A.w(w e. z -> w e. x) -> z e. y)
Axiom of Union ax-un |- E.yA.z(E.w(z e. w /\ w e. x) -> z e. y)
Axiom of Regularity ax-reg |- (E.y y e. x -> E.y(y e. x /\ A.z(z e. y -> -. z e. x)))
Axiom of Infinity ax-inf |- E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))
Axiom of Choice ax-ac |- E.yA.zA.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <->…

3:15 PM

And ... really? Citing a discovery.com article? Pop articles are never known for scientific accuracy, they love to make broad sensationalistic claims based off of humbler claims in journals.

@Slereah It is an axiom schema, the one also called of specification

@Slereah I think you can contort yourself and derive it from the replacement one, if you really want to

@ManishEarth What is your full background on CTCs?

I am a time travel science freak and I would like to follow more closely on time travel researches in mainstream physics. which journal most commonly report abotu theoretical and experimental researches on time travel topics?

Recently I heard about quanutm postselection on resolving the grandfather paradox. What's your comment on that?

@ACuriousMind No, I think it is separated

@Secret I don't have a CTC background per se. Though some of my classmates are doing research in GR.

3:17 PM

"It tells us that the image of any set under a function is also a set (see the variant funimaex 4811)."

I am no expert on CTCs but I read quite a bunch of papers on them

I don't think much CTC research is going on right now. The various vacuum fluctuation proofs got rid of most of them

@yuggib Wiki says replacement + existence of empty set gives separation

"Many developments of set theory distinguish the uses of Replacement from uses the weaker axioms of Separation axsep 3686, Null Set axnul 3695, and Pairing axpr 3755, all of which we derive from Replacement"

Ah yes, apparently

But, well, I don't really know about this.

@ACuriousMind Yes, and replacement+separation gives the existence of the empty set

3:19 PM

@Secret No such thing as "experimental research" in this case.

Q: Chronology protection for non-geodesic CTCs and imprisoned curves

I still have a niggling doubt about the vacuum energy thing

1

As far as I can make out, the quantum part of the Chronology Protection Conjecture hinges on the fact that in curved space, in the semiclassical approximation, the stress energy tensor contains a term \begin{equation} \lim_{x\rightarrow y}\sum_{\gamma \neq \gamma_0} \frac{\sqrt{\Delta_\gamma (x...

^this one

@yuggib Ah

Then Slereah's list is incomplete though, isn't it?

@ACuriousMind or maybe his replacement is a stronger version

@Slereah you beat me, I was searching for that too, lol

I did not read them, only the name because of the awful notation :P

3:21 PM

Not quite sure how I could check that

@ACuriousMind I see.

I guess I could find a compact CTC and check by hand the stress energy tensor of the vacuum

Can't think of any compact CTC spacetime with only non-geodesic CTCs, though

I will have to give it a looksie

(This is not a question)

I am almost done!

Although IIRC CTCs that are geodesics are pretty rare

So probably not overly hard to find

But I suspect this would not help because I think the theorem is basically linked to just theee

Blueshifting effect in classical field theory with CTCs

You just have causal loops all over the place of particles going nowhere

@ACuriousMind Correction: it is the axiom of infinity plus axiom schema of separation that gives the existence of the empty set. Still it is not clear if that list is complete or not...

3:27 PM

It's not quite ZFC

It's ZFC + classes

?

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as Von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class...

Bernays-Gödel?

I know what a class is

yes

it is a different beast...it is not even the same language of logic

ZF/ZFC set theory: $(\in)$ where $\in$ is a binary relation; Bernays-Gödel: $(\in,V)$, where $\in$ is a binary relation and $V$ a unary relation.

3:35 PM

Is V the universal class here?

@Slereah No, it is a symbol that means (I think) that the object is a proper class

or something like that

Oh

I think that is something like $\vDash V(a)$ means that $a$ is not a set

(poor @0celo7 he just wanted physics)

Phys chem, but close enough to physics for me atm

This is what I need to fully understand in my honours project

hence that 3 weeks of numerical analysis!

because this article is otherwise not very comprehendable to me

3:51 PM

@Slereah dropping math major if this is what I have to look forward to

nah probably not

Unless you pick like MATH FOUNDATION or something

John Duffield

4:11 PM

@ManishEarth : answering a time travel question with a reasoned explanation of why it's not possible, backed by robust references including Einstein and Gödel is OK. Trying to censor such a contribution on specious that's not mainstream grounds is not OK. I gave you your journal reference which said ”outside science fiction, time travel is impossible”. PRL is mainstream. Get used to it.

@skillpatrol Got antibiotics, sedative cough meds for sleeping and a pack of cough drops.

John Duffield

@Slereah : sigh : curved space. And there's somebody else who doesn't understand the first thing about general relativity. That curved spacetime is not curved space. Or that spacetime is not the same as space. Or that there's no motion in spacetime. So you can't travel round a CTC.

4:31 PM

@JohnDuffield Can you have curved space for which spacetime is not curved?

@0celo7 try to sleep as much as possible pal

You could try to sweat it out.

I sleep under two covers every night.

I've tried that.

First thing you need to remember is to finish the entire antibiotics prescription

Yes.

I will start it when I go to lunch.

Currently hiding from the hundreds upon hundreds of people storming the dining hall.

Take meds right after you eat food. Never on an empty stomach, ok?

4:42 PM

Yes, mom

Eating is cheating.

4 mins ago, by 0celo7

I will start it when I go to lunch.

I don't mean just sleep under two covers. What I do is get a portable heater and put it on high and literally sweat it out for the night.

I always feel better in the morning

@skillpatrol That's merely an inductive argument. There's an obvious survival bias here.

I could do that at home.

But I have a roommate.

4:48 PM

Aren't we all time travelling? Or is it just me?

2