ginnungagap

This is irrelevant to your point, just change the numbers used as example, but we do not know whether pi+e is irrational! (Even though nobody believes it to be rational)

It's not clear to me what this approach offers over Dedekind cuts, you are specifying a real number by saying in which rational intervals it lies, a Dedekind cut specifies a real number by saying which rationals are below it and which are above, and translating back and forth between the two is immediate

A guy on reddit made a website that does exactly that, but I cannot find anymore at the moment

As far as I know there is no relationship between BrewDog and Flying Dog despite the names

Why would they not be?

That's not a name I was expecting to see on HN! I haven't had the luck of taking one of his courses, but I discussed many interesting puzzles with him at a conference last year, great guy!

Microsoft seems unable to do language settings as well. I bought a laptop in Germany that came preinstalled with windows, so the first thing I did was switch the OS's language to English. At some point during setup it also asked for my region or some similarly called info and I said Italy. The result is that now most of the OS is in English but random pieces of it revert to either German or Italian every now and then, especially error messages and similar things... Luckily they are languages I somewhat understand at least!

Ding was offered a translator but insisted on speaking English himself apparently!

"He said he didn’t need a translator since, “I feel very comfortable now,” " from https://chess24.com/en/read/news/ding-liren-passes-acid-test...

What are those trivially defined {r0,r1} for the "interleaving digits" map f:(0,1) -> (0,1)^2 defined by taking a real x in (0,1), writing its decimal expansion as

x=0.a1a2a3a4a5a6...

(always pick the nonrepeating decimal expansion in case x has two) and sending it to the pair (y,z) where

y=0.a1a3a5... z=0.a2a4a6...

(Using (0,1) is no issue since we can compose later with bijections (0,1) -> R)

Edit: I'm not sure why I cannot reply to your reply, but {π,π} is not even in (0,1)^2

We can also do a map f:R -> R^2 directly, it's just slightly more annoying to write. Take a real number x, written in decimal expansion as

x=bn...b2b1b0.a0a1a2a3a4...

where again we choose the representation which is not repeating in case x has two. We also choose our first index n so that bn is zero but b(n-1) is not. We now map x to the pair (y,z), where

y=...b5b3b1.a1a3a5...

z=...b4b2b0.a0a2a4...

So that, for example, (π,π) is the image of x=33.114411559922...

He's associating to each x in [0,1) an infinite sequence of natural numbers (just like decimal representations do), and there's uncountably many such sequences

There is no need to exhibit bijections, if you have an injection X -> Y and an injection Y -> X, then X and Y have the same cardinality. This is the statement of the Cantor-Schroeder-Bernstein theorem.

If you assume the axiom of choice it's enough to have surjections both way.

First of all the first link explicitely constructs (in a very strong sense) an explicit bijection R->R^2 so I don't know how why you seem to think otherwise.

It is true that there is no continuous bijection R -> R^2. It is also true that this has nothing to with cardinality, which is only about the existence of bijections (and indeed is defined for arbitrary sets with no topological structure for which continuity makes no sense)

Whether |R|=aleph_1 is independent of ZFC (but of course there are bijections both between R and R^2 and between aleph_1 and aleph_1^2, or between X and X^2 for any infinite set)

Oral exams are very common in Italian universities (many courses have both a written and an oral exam!), I was surprised when I moved to Germany for my masters and met students from various countries who had never taken one!

The concept looks extremely interesting, but (like the vast majority of language learning tools) Albanian is not supported. I might give it a try for German though

I think that the behaviour you're describing is due to the Leidenfrost effect!

https://en.m.wikipedia.org/wiki/Leidenfrost_effect

I do research mostly in general topology and topological dynamics, but I don't really see how AI would help, producing homeomorphic (topologically equivalent) spaces is not really the point.

In general even presenting a topological space to a computer is not easy. This can be done for very good spaces, for example those admitting (finite) triangulations, and there are some algorithms there, but my understanding is that you run into undecidability issues very quickly. But for the kind of spaces I work with (which are usually compact metrizable connected, so way better than arbitrary topological spaces) there seems to be no hope to get a computer to work with them.

This also requires universes/inaccessible cardinals to even be stated for categories that are not locally small. But as I mentioned in another comment assuming enough universes exist is not a big deal for set theorists

This is routinely dealt with through Grothendieck universes. Those are a fancy name for what is pretty much an inaccessible level of the cumulative hierarchy, indeed ZFC+"every set belongs to a Grothendieck universe" is equiconsistent with ZFC+"there is a proper class of inaccessible cardinals". This is not a strong assumption over pure ZFC compared to those set theorists interested in large cardinals work with

Personally I don't mind seeing only the LaTeX code and looking at the pdf once in a while after compiling. But a second screen to keep open and accessible background papers or notes would be very convenient

This is very common behaviour (and known as the "park wherever the f you want" button among my friends) in Italy as well. But italians are not exactly the best example of drivers to imitate and learn from

head is used twice in the program

Thanks for the detailed answer! So if I understand correctly you're planning to use a small computer case instead of an eGPU specific one? This is probably a stupid question but I don't quite understand it connects to the laptop

My gaming laptop's mainboard recently got fried from overheating, could you tell me a bit more about this setup/where to look to find out more? I've never even considered this possibility, I was tempted to just give up on laptops for gaming for good this time and build a computer, but your setup sounds very interesting

If I remember correctly there are also tricks like

  lea rax, [eiz+rbx*4]

instead of

  lea rax, [rbx*4]

because the latter will produce an heavier binary by being interpreted as

  lea rax, [00000000h+rbx*4]

The periodic tiling conjecture is true in dimension 2: if you have a single tile that tiles Z^2 aperiodically by translations, then it also tiles Z^2 by translations periodically, this is a result of Bhattacharya (and the reason why the expectation was for the conjecture to be true in general)

That's amazing! Thanks for the link

I am quite sad that "The secret life of videocassettes in Iran" is not a real article

I'm european and I've always been surprised at how far east Vienna is compared to where I imagine it to be. I mentioned this to many people and quite a few very surprised that Vienna is considerably further east than Prague (if you go straight down from Vienna you'll reach Zagreb)

I've been reading the dune series too and while the first book was amazing and the second was good, they get worse fast after that in my opinion