The h Bar

yuggib

22:06

A completely random and unrelated question (since the majority of the live ones are mathematicians): do you think that the continuum hypothesis is true?

ACuriousMind

@yuggib Yes

It seems entirely absurd to me that there could be something between the integers and the reals. But then again, it also seems entirely absurd to me that we can't prove or disprove that.

Danu

It doesn't really matter... As far as I can tell

yuggib

@ACuriousMind You can take one sphere, rearrange the points, and obtain two spheres identical to the original one...absurdity is common where there is the axiom of choice

Danu

It's independent of ZFC so...

What does it even mean to ask if it's true?

Truth has no real meaning

yuggib

@Danu It surely matters; adding an axiom you get a richer theory, and could prove more theorems

Danu

22:10

As long as you refuse to specify your axioms

yuggib

now, which one is more reasonable to add? $CH$, $GCH$, $\overline{GCH}$?

ACuriousMind

@yuggib Yeah...but Banach-Tarski I can live with because it takes non-measurable subsets and moves them around - I understand why my intuition fails.

@Danu If you're a platonist, it has.

Danu

@ACuriousMind In mathematics, I mean.

Are you asking about what set of axioms is the one that applies to the physical world we live in?

Truth in that sense?

ACuriousMind

(I thought it was understood by the "do you think" that the question is not about "what does ZFC say")

yuggib

As you wish; the language and the set of axioms define the mathematical theory

Danu

22:14

@ACuriousMind I didn't take it as "what does ZFC say" because that'd be a stupid question :P

yuggib

and adding an axiom gives a richer theory

as long as the aforementioned theory does not becomes contradictory adding a true statement

Danu

*doesn't become

ACuriousMind

Is taking CH and not-GCH together possible? Has it weird consequences?

yuggib

CH and not-GCH should be possible (in the sense that holds only for $\aleph_1$, but not in general)

There are mathematical results that assume CH, or GCH, in order to be true

ACuriousMind

Then again, GCH implies choice, why not just take GCH and happily prove things, works for choice so far?

Danu

22:17

@ACuriousMind I still hope that these fundamentals of set theory etc. don't really matter for reality

ACuriousMind

@Danu For reality, they probably don't. For our description of reality, they do - since we describe reality with math.

yuggib

@ACuriousMind yes, you could... but I read that is the belief of logicians (I do not know why however) that GCH and also CH should be false

Danu

@ACuriousMind You think reality is independent? Then the statements that our theories make should also (tend to) be independent, in the long run.

Slereah

@yuggib It's neither true nor false

You can make a system with it, or its negation

yuggib

@Slereah yes, that is exactly the point

Slereah

22:21

One could be inconsistent, I suppose

Danu

@Slereah The question was about physical reality, not about mathematical universes

Slereah

The point is more "is it useful"

Danu

(at least that's what ACM said :P)

Slereah

Well the thing is, can it even describe a physical reality

Is there anything in the physical universe that requires cardinalities that big

yuggib

@Slereah the cardinality of the reals is not uncommon

Danu

22:23

Between rationals and reals isn't that big...

Slereah

It's not uncommon but is it necessary

that is the question

ACuriousMind

@Danu Our statements can't be independent, because we are forming them with math, which depends on it. I don't think reality is "independent" in the sense that you should be able to describe it with any axiom system, but I certainly think that "reality" does not do math.

yuggib

@Slereah If you think that a finite volume of space contains an infinite number of points

Danu

@ACuriousMind Our statements don't have to depend on CH (just as ZF doesn't)

ACuriousMind

If you now ask me what "reality" is, I have reached the fourth panel of this xkcd.

yuggib

22:25

and that there is e.g. a point at a distance $\pi$ from the origin

then you need $2^{\aleph_0}$ points

ACuriousMind

@Danu Yeah, well, but they do depend on ZF. Why would you hope they don't depend on CH but be fine with them depending on ZF?

Slereah

@yuggib : but for instance

While the reals are continuous

The definable numbers are not

Danu

@ACuriousMind I was just pointing out that your statement was wrong :P

yuggib

@Slereah What do you mean by definable? We can easily define real numbers

ACuriousMind

@Danu Oh, I was replying to your "these fundamental of set theory" and didn#t think we were still specifically talking about CH :P

Danu

22:27

@ACuriousMind Ah

Slereah

@yuggib Definable numbers are a subset of the real numbers

They are numbers that can be defined using a finite number of symbols

ACuriousMind

The Wiki article on definable reals has a nice number of warning flags on it

Slereah

@ACuriousMind : fortunately it's not only a wikipedia article :p

Danu

@ACuriousMind And definable numbers also include $\pi$, for instance according to the article (i.e. they're not a subset of the naturals)

yuggib

that does not mean it is a mathematical theory, but it is a model of it

@Danu since the statements obtained with a rigorous logic are in agreement with experience, reality should be a model of a logic theory

ACuriousMind

22:29

"reality should be a model of a logic theory"

Danu

@ACuriousMind Also not obvious to me at all

Slereah

the wiki article does seem pretty shit

yuggib

@Slereah It is possible to define real numbers by dedekind cuts

Construction of the real numbers

In mathematics, there are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. == Synthetic approach == The synthetic approach axiomatica...

ACuriousMind

Now that is what Physics chatroom discussions should be about :D

yuggib

and since we can write that construction (even if it is recursive)

Slereah

22:30

@yuggib : You define the whole set but most of those are not constructible!

Danu

I find constructionism kind of unreasonable.

yuggib

@Slereah why are not constructible?? the reals are constructible

Slereah

Well once you start going on about if a math axiom is "true", that is the kind of madness that lies ahead :p

ACuriousMind

@Danu Well, I guess everything in reality can be expressed as some statement, and then the collection of these statements is a model of some theory.

user54412

I think there's a miscommunication above between defining/constructing a set and defining its elements.

Slereah

22:31

I didn't even begin to talk about physical measurements and Craig axiomatization yet!

yuggib

@ACuriousMind Given a language, a model is roughly speaking a universe of objects such that it is possible to give meaning to each symbol in the universe, in a way that respects the rules of the language

and that seems a quite reasonable way of defining the link between a logic theory and what we observe

ACuriousMind

I...cautiously agree.

user54412

Is anyone else bothered by the fact that logicians use the word "model" in exactly the opposite way physicists do?

yuggib

@ChrisWhite :D

the model of logic may be called by a physicist a "toy model" (an exact one)

ACuriousMind

I'm already bothered enough by overloading of symbols and imprecisions. I guess if I started learning logic I would go completely mad.

Not sure if that's a bad thing.

Danu

22:43

Damn, the English Language & Usage users are curt

yuggib

gotta go to bed, cya

ACuriousMind

@yuggib good night

@Danu You've mentioned this before :P

Danu

@ACuriousMind I was reminded of it, once again

NeuroFuzzy

@Danu Do tell

HDE 226868

@ChrisWhite Okay, thanks.

Danu

22:56

@NeuroFuzzy You can just join the chat room

But I've had a bit of a chat now and it ain't so bad atm ;)

NeuroFuzzy

@Danu hehe "No."

user54412

@HDE226868 See also this, where it mentions the caveat of large scatter in doing exactly what you want to do.

user54412

As though large scatter ever stopped an astronomer from drawing a straight line on a log-log plot.

NeuroFuzzy

@ACuriousMind Do you blame @0celo7 or @Slereah more?

0celo7

@NeuroFuzzy "blame" is such an evil word

Slereah

23:09

0celo7

@ACuriousMind Jesus Christ

ACuriousMind

@NeuroFuzzy If I say that, they will know how to break my meter again.

0celo7

got my license

finally

I can drive like a normal human being now

NeuroFuzzy

Hehe so I ran into some people who suggested some GR notes to me

the first (or second?) section in the "differential geometry" portion is titled "how to defend yourself from mathematicians"

Slereah

@0celo7 prior to his license

0celo7

23:18

well I had a paper license

NeuroFuzzy

Wait. What?

"∃ Weird examples:
(1) (Friedman, Donaldson): {($\mathbb{R}^4$, identity)} is a differentiable manifold. But there is another differentiable structure on $\mathbb{R}^4$ which is NOT diffeomorphic to the usual one."

Slereah

Is this the thing about how in 4 dimensions you have non trivial bullshit manifolds

0celo7

4 dimensions is nontrivial bullshit in general

Stan Shunpike

@NeuroFuzzy 10hrs a day, that's a lot of reading lol. Tho i suspect i do that amount. I just dont usually think about it as a goal

Seems like less if I don't have a goal

0celo7

@ACuriousMind I promise I did not do that on purpose

I still don't understand what electrons have to do with Dirac belts

@JohnDuffield

NeuroFuzzy

23:36

@StanShunpike Oh, well including programming or other work.

i just usually get really annoyed/stressed if i spend all day playing video games or something, so I think it's better for my health if I cut that out.

0celo7

@ACuriousMind I did my civic duty

you get annoyed playing video games...

some people are jus strange

ACuriousMind

@0celo7 Very good.

NeuroFuzzy

@0celo7 :p no, I mean if I go, "I'm going to get through chapter 7 of XYZ" and then it's a week later and i haven't started it, i'm not a happy camper

0celo7

@ACuriousMind ::waits for cookie::

oh

ACuriousMind

I have never thought "I'm going to get through chapter 7 of XYZ". Too lazy.

0celo7

23:41

that's happened to me

@ACuriousMind how's BBS

ACuriousMind

@0celo7 ::gives cookie::

0celo7

@ACuriousMind ::slaps cookie:: bish I don't need yo charity

I'm a college student not a pauper

ACuriousMind

@0celo7 Well, I've not yet arrived at the parts I haven't seen before, but I can't say that I like it

0celo7

where are you

ACuriousMind

Not sure if I dislike it

NeuroFuzzy

23:42

germany, probably.

0celo7

smart ass

@ACuriousMind omg omg

Lang

drinks

Mineralwasser

he must import it or something, you can't get that shit over here

HDE 226868

@ChrisWhite Hm. It allows you to write $L/L_{\text{bulge}}=\text{constant}$. Interesting.

0celo7

I'm gonna ask him for the source, I haven't had real sparkling water in years

ACuriousMind

...there is no sparkling water in the US? oO

0celo7

not the real stuff

this was in a glass bottle and everything

HDE 226868

23:44

^^^^^ For some cases, at least.

0celo7

let's see...ODE homework and start on the English papel

I think Freire killed some people when he started talking about hyperbolic trig

ACuriousMind

@0celo7 I stopped at "Kac-Moody algebras" when I got too annoyed they didn't actually tell me anything of substance about them :D

0celo7

all the people who took AP calc just died

@ACuriousMind yes.

I hate that section

unless you know exactly what they are, that section is completely useless

one of the reasons I got BLT is that they have a whole chapter on Kac Moody CFTs

of course, I haven't made it through the 100 pages of partition functions to actually read that chapter

there's someone in my physics class named "Bentley Beard"

2

ACuriousMind

Sigh...perhaps I should try to read BLT, but I'm beginning to suspect there is no good book on string theory.

HDE 226868

@0celo7 Why?

0celo7

23:48

@HDE226868 AP calc does not cover hyperbolic trig

they had no clue what he was talking about

we were doing homogeneous ODEs and he had to do the integral $$\int\frac{\mathrm{d}v}{\sqrt{v^2-v}}$$

(I think) and that's $\cosh^{-1} v$ or something

@ACuriousMind I think I agree

GSW is still on my list

I've read some parts for clarification

BLT is just...so...dry

makes me yawn just thinking of it

ACuriousMind

Well, I think "dry" does not deter me :D

0celo7

you're a smarter/braver/less fashionable man than me

so maybe you can do it

and my roommate is playing LoL, time to relocate

ACuriousMind

@0celo7 I'll wait for motivation to come around. But currently I'm rather inclined to just relax. :P

obe

@0celo7 Would I be a cool roommate?

ACuriousMind

@0celo7 You coping with dorm life alright?

Slereah

23:55

Inverse functions of square roots of squares are always trigonometric or hyperbolic shit

it is the law

0celo7

@ACuriousMind I guess

two things really bug me

1) my roommate talks in his sleep and wakes me up

2) he plays LoL when I sit down to do homework

I've never seen him doing homework, I have hours every night

ACuriousMind

Both things that would also annoy me to hell if I had to share a room.

0celo7

@obe how on Earth would I know

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