The h Bar

yuggib

17:01

it can be seen as a set $\{\varphi(x),x\in\omega\}$

if that is definable, then the reals are definable, for each real is a particular sequence of rationals (that satisfies certain properties)

Slereah

But those properties have to be written with a finite number of symbols

yuggib

oh indeed

Slereah

Each such sequence of symbols can define only one real

But there is $> \aleph_0$ reals

yuggib

i will do that: $(\forall m,n\in\omega)\lvert x_m -x_n\rvert\leq m^{-1}+n^{-1}$

here it is the property written with a finite number of symbols

a real is a such a sequence, so I have defined it

take another sequence, you have another real

and so on

Slereah

Yes, but that is one real

Not all reals

yuggib

17:07

each real is a different sequence

Slereah

You need one such formula for every real numbers to say that every real number is definable

Yes, but not all of them are finite

ACuriousMind

@yuggib I think @Slereah's point would be that you can't define every sequence of rationals with a finite number of symbols. Your "take another sequence" might be the step where you need moer than that to say which sequence you're taking.

Slereah

What he said

ACuriousMind

I'm not entirely sure why one cares about the distinction between definable and undefinable, though.

yuggib

sorry, but is not $\{\varphi(x),x\in\omega\}$ definable, as per @Slereah definition??

Slereah

17:12

Depends on the length of $\varphi$!

yuggib

ok, so it is a very silly definition

Slereah

Nah not really

It's not very commonly used, though

yuggib

it does not allow recursive definitions?

Slereah

It does

But if you have a random sequence of digits $0.a_1a_2a_3...$

No definition will be finite

And those constitute the bulk of real numbers

ACuriousMind

Wait, how do you express "random" here? ;)

Slereah

17:16

YOU KNOW WHAT I MEAN

It is a stronger version of computable numbers, basically

yuggib

well, you recursively define the sequence that approaches that number

Slereah

How

yuggib

thus defining the number itself

Slereah

How do you do it in a finite way

ACuriousMind

@Slereah Well, I would agree that this is a bit silly - you take a definition of "definable" that obviously only allows a countable set to be definable and then you act surprised that the reals are mostly not definable?

Slereah

17:17

I don't act surprised

ACuriousMind

It's not at all obvious why this idea of "definable" is relevant for anything

Slereah

It's just the usual definition?

ACuriousMind

But what is it used for?

Slereah

definable numbers are used usually in computation theory contexts and stuff

ACuriousMind

Oct 13 at 11:04, by yuggib

"A good definition should be the hypothesis of a theorem"

yuggib

17:18

it seems just bullshit to me

Slereah

Rude.

yuggib

no no, it is really just bullshit

Slereah

Why

yuggib

because it gives the false impression that you can't define a real number

while obviously you can

simply giving a sequence of rationals

Slereah

It just says that you cannot write a finite formula for every real numbers

Is all

0celo7

17:20

Wow this is some emeritus level math...

Slereah

The same way computable numbers are the set of all numbers that can be computed by an algorithm

0celo7

How do I get this smart

Slereah

Definable numbers are slightly larger since some uncomputable numbers are definable

Like Chaitin's constant and such

yuggib

bah

Slereah

While it's not an overly useful set it does exist

No need to get all up in arms about it

yuggib

17:21

but when you use it to say

ACuriousMind

Let's just say it is a bad choice of name for the definition :P

yuggib

50 mins ago, by Slereah

Most irrational numbers are undefinable, though :p

it is really misleading

Slereah

Well what other name would fit better

yuggib

unimportant class of numbers

ACuriousMind

horismos, maybe? :D

Slereah

17:23

"It can be encapsulated by a finite number of symbols" is p. good for "definable"

yuggib

the class is unimportant, not the numbers themselves

Slereah

I hope so because every number you've ever heard of is in it :p

yuggib

ah, so $\pi$ is in it

Slereah

Yes

yuggib

and $\sqrt{2}$

in the definable numbers?

Slereah

17:25

π = sup((ℝ+ ∩ (◡sin “ {0})), ℝ, ◡ < )

All computable numbers are part of the definable numbers

All algebraic numbers are part of computable numbers

yuggib

that shitty notation does not help

Slereah

Well welcome to math :p

yuggib

shut up

Slereah

It's the smallest positive number such that $\sin(x) = 0$

Err wait

No

No, that's correct, nvm

strictly positive tho

yuggib

how would that be definable?

you are telling that a function from the reals to themselves is definable?

from all the reals, even the undefinable ones

to all of them

you're not convincing me at all

Slereah

17:29

Ah whatever

yuggib

you're just playing smart

Slereah

I don't care enough to continue

yuggib

ah ok, so $\sin$ is definable, but most of the reals on which it is defined not

ACuriousMind

135

A: Is the analysis as taught in universities in fact the analysis of definable numbers?

The concept of definable real number, although seemingly easy to reason with at first, is actually laden with subtle metamathematical dangers to which both your question and the Wikipedia article to which you link fall prey. In particular, the Wikipedia article contains a number of fundamental er...

Slereah

Yes.

ACuriousMind

17:30

It seems that MO largely agrees with @yuggib on this one.

The concept of "definability" seems to fail to carry any kind of real meaning:

> If ZFC is consistent, then there is a model of ZFC in which every real number and indeed every set-theoretic object is definable.

yuggib

6 mins ago, by Slereah

Well welcome to math :p

Slereah

"The statement is valid externally, as a meta-statement about the model. Internally, inside the model, we cannot even express the statement."

ACuriousMind

Yes, so the point is that "definability" is a statement relative to the model.

Slereah

Sure

yuggib

model here is in mathematical sense

ACuriousMind

17:35

Well, that's entirely different from saying "most real numbers are not definable"

I would be equally justified in saying "all real numbers are definable" unless you specify the model in which you are working

And then you'd have to convince me that that is really a model I want to work in ;)

Slereah

Hm

Isn't the set of formulas always countable, tho

ACuriousMind

> Thus, just knowing that there are only countably many formulas does not actually provide us with the function that maps a definition φ to the object that it defines.

The MO post does address this

Slereah

But just the fact that it is countable at least provides the notion that not all real numbers can be defined

It just doesn't specify which is which

Depending on the theory you use

ACuriousMind

But ZFC is consistent with all real numbers being definable, apparently!

Slereah

I will have to investigate this

ACuriousMind

17:39

Read the part after the theorem about there being forcing extensions of ZFC models

Slereah

Seems a bit strange

yuggib

more I see it, more it seems the usual matter of vicious circle

Slereah

Well it's kind of a metamathematics issue

Those tend to be tricky

yuggib

well, I must have become too metamathematical, since I was not convinced from the beginning @_@

I don't know if I must be proud or ashamed of that

ACuriousMind

yesterday, by yuggib

I'm flawed

yuggib

17:41

exactly

that's the ultimate correct statement

gotta go now ;)

see ya later

DanielSank

@Danu ::shudders::

Slereah

Hm

Do you think that if two points can be linked by piecewise null geodesics, they can be linked by a null curve

I suspect that is the case but Penrose seems to not prove this directly

ACuriousMind

@DanielSank: How do the statistics fare?

Slereah

Might need to do a SE question on it

Well, and vice versa

He has a theorem saying that if there's a piecewise null geodesic curve, the endpoints can also be joined by a null geodesic

So obviously they can be joined by a null curve

But I dunno about the reciprocal

0celo7

I believe I proved that earlier.

Slereah

17:52

Did u

0celo7

But whatever.

ACuriousMind

@Slereah Uh, a null geodesic is a piecewise null geodesic, no?

Slereah

@ACuriousMind : Well same problem as before

Joints

They need to be smoothed a bit

ACuriousMind

Not following you. You said he proved that a piecewise null geodesic means there is a null geodesic. But the converse is trivial unless I'm misunderstanding something.

Slereah

Well if there are two points joined by a (non-geodesic) null curve

Is there a piecewise null geodesic curve joining the two

ACuriousMind

17:54

Ah

Slereah

I suspect that yes since I think you can apply the same method as for timelike curves while staying in the light cone

hm

@0celo7 Was that a proof that two null related points can always be linked by a geodesic?

0celo7

18:10

I'm back.

@Slereah A piecewise one.

i.e. with cusps

You use the null cone to get a null geodesic

then you go a bit

start a new null cone

etc.

and this always works because the convex normal neighborhoods form an open cover

Slereah

Sounds about right yeah

0celo7

@ACuriousMind convex normal neighborhoods form an open cover?

right?

Slereah

thx

0celo7

so there might be corners...

ACuriousMind

@0celo7 Remember that you explained to me what a convex normal nhbd is? I have no idea

0celo7

18:14

well they have to

because every point has one

so the union is the manifold

@ACuriousMind I don't remember this

Alzheimer's, remember?

Slereah

Proving equivalence with piecewise geodesics isn't fun but it sure makes theorems easier :p

"Just put the two curves together!"

Bam

"Thus, for any p, q ∈ U, there exists a unique geodesic γpq : [0, 1] → U which connects p with q."

That could be useful for such a proof, too!

ACuriousMind

@0celo7 No, I don't remember

Slereah

although

I fear I will have to use

HE D:

lightning strike and scared horses

Frau Blücher (from Young Frankenstein)

►

ACuriousMind

@Slereah HIM, you mean. Learn your grammar.

Slereah

Terrible joke

24 hours ago, by Slereah

Because "HE" might occur in "HE MAN AND THE MASTERS OF THE UNIVERSE", for instance

Also I made it first :p

ACuriousMind

18:24

I think it's even older, but the transcript is hard to search for that

user685252

English is wierd.

0celo7

@user685252 No, it isn't.

Slereah

*grommor

user685252

Spelling is wierder :P

Slereah

Hm

Perhaps I should

Read all HE first :(

0celo7

18:33

@user685252 No, it isn't.

@Slereah I thought you did that already

Slereah

Or at least the first 6 chapters or so

No

I'm not a NERD

reading is for nerds, you nerd

DanielSank

@ACuriousMind Well indeed. I am having an annoying sign issue on one particular problem.

Try computing the moments of a square distribution. Then compute the generating function and get the moments from that.

I wind up with a sign inconsistency

0celo7

Hmm

deleted post, and one @

user685252

Snooze you lose.

yuggib

@DanielSank eager to see the answer of JD to your question? ;-P

I am curious...since for once it is in some sense in my field (and therefore I can argue with him a little bit)

user685252

18:41

Don't bother

yuggib

sadly I cannot find the link to it anymore...

0celo7

Read 25 pages of GR and you're on his level there.

ACuriousMind

3

Q: Why does a damped quantum harmonic oscillator have the same decay rate as the equivalent classical system?

$\newcommand{ket}[1]{|#1\rangle} \newcommand{bbraket}[3]{\langle #1 | #2 | #3 \rangle}$ Why does the decay rate for a damped quantum harmonic oscillator exactly match the classical limit? Background Consider a localized quantum system $S$ connected to a 1D wave-supporting continuum. If $S$ is e...

user685252

TA da^

yuggib

thks

I avoid discussions with him anyways

user685252

18:43

Still don't waste your energy

yuggib

but that time I am curious :D

however, no sign of it yet...

ACuriousMind

@yuggib No, I'm curious. Get another adjective!

user685252

Time and energy is all you will waste.

yuggib

ahahah ok...

I am intrigued

ACuriousMind

Better.

yuggib

18:45

I could change my name to AnIntriguedYuggib

user685252

Since when did you become the grammar police @ACuriousMind :P

ACuriousMind

@yuggib Shouldn't that be YuggibDeugirtniNa? ;)

0celo7

wtf

yuggib

ahahah indeed

0celo7

what does that mean o.o

ACuriousMind

18:47

@0celo7 You'll figure it out.

Someday.

Probably.

user685252

Maybe

Hopefully

Doubtfully

note missing^ periods :P

ACuriousMind

@user685252 You pregnant?

...or why did you miss your periods?

0celo7

lele

user685252

Please no sexist attempts at a joke.

0celo7

@ACuriousMind Probably not

yuggib

18:51

stress or insufficient eating can be a cause

0celo7

@user685252 What

user685252

Just a friendly reminder that is an automatic banning offence in most rooms, if flagged. But I never have or will flag anything. @ACuriousMind

:-)

ACuriousMind

@user685252 I'm sorry if I offended you, but I fail to see the sexism there. It doesn't present one sex as inferior to the other and it doesn't use a stereotype.

user685252

There are women mods.

Jiminion

(wisely holding tongue....)

user685252

18:57

No apologies needed.

Mark Mitchison

@user685252 "Periods" only has a single interpretation, i.e. menstruation, in many English varieties (I am assuming you are only familiar with American English).

0celo7

@user685252 What??

user685252

What about punctuation? @MarkMitchison

Mark Mitchison

No. It's called a "full stop".

Of course it also means an "epoch"

0celo7

@MarkMitchison Not in real English lol

user685252

18:59

BEng

Mark Mitchison

@0celo7 What, like the English spoken by English people? ;)

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