@LeakyNun The reason I said that my version is close to your version, is that my chains $f$ are defined in nearly the same way as your $P$. But by using "ordered-pair" functions instead of "statement" functions, I'm able to sidestep self-reference issues (since we're allowed to quantify over sets but not statements).

I think one consequence of being so stubborn on doing the above will be introducing sets with very weird order types and I probably hit an insurmountable roadblock as these sets might violate one or more of the ZFC axioms

@Secret The definition of $\epsilon_1$ is "the smallest ordinal $\alpha$ above $\epsilon_0$ that satisfies $\omega^\alpha=\alpha$"."

Indeed, the definition of $\epsilon_\kappa$ is "the $\kappa$-eth ordinal that satisfies $\omega^\alpha=\alpha$."

(Zero-indexed, I guess, so that $\epsilon_0$ is really the first, $\epsilon_1$ is really the second, etc. But $\epsilon_\omega$ is the $\omega$-eth.)

a simple question just to be sure, if $E=C([0,1],\mathbb{R})$ and $\psi$ a continuous function from $E$ to $\mathbb{R}$ such that $\psi(f)=f(0)$ then $\ker\psi=\{f\in E, \psi(f)=\{0\}\}$ is not always $\{0\}$ right ?

It would be the set of functions that satisfies $f(0)=0$, no?

If the above fails then we learn a very important lesson:

@AkivaWeinberger yes

> Ordinal tetration actually grows with each step, instead of staying constant, unlike the finite case

this not means that $f\equiv0$ right ? @AkivaWeinberger

Right

Just any function that has a value of $0$ at zero

So, like $f(x)=x$, for example

all we can say is that $\ker\psi=\psi^{-1}(\{0\})$

@AkivaWeinberger so I mentioned prolog because it's very like logic

however, I've not been able to debug my program until now

so here's my program

I used 1 instead of a.

press the play button to run the program

Cool!

Right, it lets you use recursion like that

basically what I wrote

It seems to be more relaxed about quantifiers

Of course, it's possible to make an infinite loop in that, right?

@AkivaWeinberger certainly.

this is an infinite loop

a :- a. to check whether a is true, check whether a is true.

If we were allowed to use self-reference that in logical statements, then we'd be able to write things that have infinite loops and would never evaluate to "true" or "false"

Since we want our logical statements to always either be "true" or "false," this would be very bad

I understand

well I just wanted to bring up prolog :p

I suppose it's always possible to write "such and such Prolog program halts and returns True" in logic :P

or "such and such Prolog program halts and returns False," or "such and such Prolog program never halts"

(I think you need something at least as strong as Peano Arithmetic to do this.

Peano Arithmetic can model Turing machines, famously, and Turing machines are equivalent to any programming language.)

The unsolvability of the Halting Problem, then, tells us that there's no general algorithm to decide if a given statement in PA is provable.

In fact, actually, I think that last fact was the main theorem of Turing's famous paper.

That there's no algorithm that decides if a given statement in PA is provable.

Did you try Google (or Google Scholar or Google Books)?

I actually have no idea if Google Scholar would be of any help

No luck

Google books has no peek-inside feature for this one

it's not in libgen? huh

@AkivaWeinberger I just wanted to bring up prolog :p

Hm, it's technically available in a library 2 miles from me (according to the "find in a library" feature)

What city are you in, again?

Munich.

There exists an injection $f:\Bbb R \to \Bbb R$ such that $f(\Bbb R)$ contains no nontrivial interval right ?

hey @Astyx have you had some basic optics, i.e. are you familiar with the lens maker's formula?

@ShaVuklia isn't that the formula for focal length of a lens?

I have done basic optics, I don't know this formula as such though

yea @SBM

@Danu There's a bunch of libraries there that have it, apparently, like the Bayerische Staatsbibliothek

Yeah I know that formula

@ShaVuklia $\frac1f = \frac1u + \frac1v$

?

bye

good night

like, I get conflicting information on the sign convections regarding the radii of curvature

bye @SBM

bye

@Astyx hi

Hi

in my book, they change the sign of the radius of the curvature when the light approaches the lens from the opposite direction

@Danu Tell me if this means anything to you

while I would think that the curvature being "convex" (for the lens) makes in invariantly positive

some sources say: just look at whether your lens is convex or not. and I'd agree. but it's not what my book is doing

@AkivaWeinberger I can probably find where that library is supposed to be...

Mmm radius is not really the appropriate term here

that's how to call it

I guess it's more the coordinate of the center of the circle

@Danu Just go here, click on "find in a library", and type in your ZIP code

(typically a radius is positive)

(or postal code or whatever you call it in Germany)

Cool, thanks @Akiva!

yea that's why there are conventions

@Danu That specific library is in 80539, for example.

@Semi are you there?

So if you draw the light coming from the left, you'd have the axis pointing to the right

Which explains why the sign changes when the ray goes the other way

uhm, which axis?

The x axis

@AkivaWeinberger I can pick the book up on Friday. Great success!

(Your options seem to be Bayerische Staatsbibliothek; Deutsches Museum, Bibliothek; and Technische Universität München, Universitätsbibliothek @Danu)

The one going through the centers of each circle (or sphere rather)

@Danu Why can't you go there now? Is it in storage or something?

okay, and we take the center of the lens as zero? @Astyx

@jakebird451 everyone knows vector math.

@AkivaWeinberger German beaurocracy

@Sha No, you'd take the intersection of both circles as 0

De facto we draw lenses really disproportionnally : the radius is much greater than the diameter of the intersection of both spheres (mathematically speaking)

yea apparently it the "Cartesian sign convention"

That means coordinates

hi

@TheGreatDuck I have the equations $t = \frac{\vec{p_{i_P}} - \vec{p_{i_T}}}{\vec{v_{i_T}} - \vec{v_{i_p}}}$, $|\vec{v_{i_p}}| = k$. Is there a way I can use the second into the first without converting the formulas into component form?

@Sha A terrible drawing

@jakebird451 ...

All I said was that everyone knows vector math. You should too.

I appreciate the drawing, but how does that help?

Understanding the sign of $R_2$ and $R_1$

oh yea

What's on the left has negative coordinate, what's on the right positive

right, so a convex lens will always have one negative coordinate

And conventionnally the lights flows from the left

and a concave lens too

okayy

I think this is a good trick :P

@Astyx wrong drawing. That one had a broken link sketchtoy.com/68150535

thanks

@TheGreatDuck Link's not broken for me

for me neither

@Sha Glad I could help

also @Astyx I saw you are accepted to your school of preference?

well I found the correct link

Not accepted yet, elligible

at least, it looks right

oh right

well congrats anyways

Which means I'll be taking oral exams soon

ah right

Thanks. I am really happy about it

haha I can imagine

@TheGreatDuck Ha :p I hadn't taken the time to watch it all

???

oooh

@TheGreatDuck Do you know any vector tricks to help me on my grand adventure?

yeah

@jakebird451 there are no "vector tricks"...

Just making a joke

they're just arrays...

I'm also a bit proud that my name is one of less than 150 on this elligibility list :p

And stressed about it too

I'm really happy for you. I hope I get in the university I want too, next year...

I hope you will too

You deserve it

To phrase my question a bit better, is there a website that lists the mathematical formulas & principles about vectors?

In particular, I am looking for something that covers how to handle $|\vec{v}|$ type formulas.

idk

I don't even really know much with vectors

(I was trying to tell you to go ask someone else)

(everyone doesn't include me)

@jakebird451 Could you give an example?

@Secret maybe some day I should teach you the ZFC axioms one by one :p

@AkivaWeinberger The example of what I need to do is listed above: chat.stackexchange.com/transcript/message/38035957#38035957

... that is, if I'm even qualified to teach you. @Secret

You seemed to done fine

@jakebird451 Hm. I don't know how I'd do that, then

Using the Von Neumann encoding?

(I think that's what it's called)

not that one

that one uses power set

Oh, I don't mean the $V_n$ sets

I mean the $n+1=n\cup\{n\}$ thingy

yes, that one

Meh

I prefer to write $n^+ = n \cup \{ n \}$

Right, sure

Finite ordinals.

@AkivaWeinberger what?

$<\Leftrightarrow\in$ for those, which is cool

$\omega^+ = \omega \cup \{\omega\}$

What I am trying to do is find the 3D vector to launch a projectile that intercepts a target, given that the starting speed of the projectile is a known constant and acceleration is 0.

@AkivaWeinberger that's also true for infinite ordinals

@Secret I expected $4$ and $3$ but alright

@jakebird451 fyi, if you're messaging me because of this comment keep in mind that vector manipulation of the sort you refer and geometry aren't as interconnected as you believe. A lot of the geometric systems rely more on relating them back to euclidean geometry imo. That's how I prefer to think of things. chat.stackexchange.com/transcript/message/38024535#38024535

Too much ordinal in my brain at the moment

@Secret prove the existence of $\omega^\omega$

would that be too difficult?

cc @AkivaWeinberger

$\omega^{\omega}=\sup(\omega^n|n\in \Bbb{N})$ ?

but um.... good luck on your whatever that you're doing. Vectors don't really have a lot of formulae except in analysis if I understand and those are just parametric equations.

@Secret So $|\{f:a\to b\}|={?}$

@Secret that's the definition

Guys... i need help with this problem. sketchtoy.com/68150565

@AkivaWeinberger $|b^a|$

I cannot rind the radius of a circle.

@Secret why the modulus sign?

@Secret $|\{f:0\to0\}|={?}$

cardinality

@AkivaWeinberger $0^0=1$ by empty product and there exists only one function from emptyset to empty set, the empty function

@Secret We don't know that the set $\{\omega^n:n\in\Bbb N\}$ (which you take the supremum (aka union) of) exists

@Secret Yup.

complementary note that $0^0 \ne 1$...

@LeakyNun @Secret Prove that $\omega+\omega$ exists instead.

@LeakyNun Well... a matter of debate :P

@AkivaWeinberger that's a better exercise

@TheGreatDuck which circle?

@LeakyNun That's true (because calculus and continuity issues), but it is commonly defined to be 1 as it makes a lot of maths smoother

Note: There's something called Z set theory (or ZC if you add choice)

@LeakyNun either one. They're the same.

@AkivaWeinberger what axioms do they throw away?

@TheGreatDuck you basically wrote $R_1$. I don't know what we're supposed to find.

It's the same as ZF(C) but without replacement.

hi

Replacement and regularity, actually. TIL. I thought it was just replacement.

Ah, no functions and exists self referential sets, I see

@LeakyNun Nevermind. It was a joke. Watch the rest of it. I cannot locate the radius.

In any case: Z (and ZC) set theory cannot prove the existence of $\omega+\omega$.

XD

@Secret we need to go over replacement to tell you how it has nothing to do with functions.

(Even if you add regularity.)

So you need replacement to show that $\omega+\omega$ exists.

it's actually a diagram someone posted earlier. I've been playing with the sketchpad.

@AkivaWeinberger I just need to add an axiom that $\omega + \omega$ exists.

can't I use C to compute $\omega \sqcup \omega$ though?

Hi @meow

@LeakyNun I don't think so (re: C)

@AkivaWeinberger why not?

@LeakyNun Isn't replacement requires the map f to be a function?