ginnungagap

Whenever you have a structure M of some language L you can take the so called complete theory of the structure, denoted Th(M), which is just the set of all L-sentences true in M. In particular if L is the language of PA and M are the standard natural number with the standard operations Th(M) is a theory called true arithmetic. This theory is complete (that's because Th(M) is always complete) and clearly enough to talk about the integers, but it escapes Gödel's theorem since it's axioms are not recursively axiomatizable.

You seem to be confusing "true" and "provable", as far as first order logic is concerned those are equivalent (by another theorem of Gödel, the completeness theorem), but the first in defined in terms of models and the second is purely syntactic

Iterestingly in the modern approach to set theory there still are both ordinal and cardinal numbers, but you can't say that the cardinal n is the set of all sets with cardinality n, because this is a proper class. Instead we choose a particular representative for this class, which also happens to be well ordered (this is in ZFC, you can still pick representatives for equipotence classes in ZF via Scott's trick but they are not necessarily well ordered or well orderable).

The community should really let this drift away quietly out of respect for a legend, instead of inviting him to conferences. After the S^6 business I'm disappointed by the organisers in Heidelberg.

Normality also includes the distribution of the substrings in the definition, it's not just "every finite string of numbers compares somewhere"

The abstract algebra definition is that p is a prime if whenever p divides a product ab then p divides at least one of a and b.

Having no proper factors is the definition of an irreducible element.

The two definitions agree for the integers and nice algebraic structure (UFDs) but there are algebraic structures in which they are not the same which explains why there are two differenr definitions and names in abstract algebra

Not directly related to the article but if you visit Italy make sure you go to Florence and see the dome. I was there less than a week ago, it wasn't my first time in Florence but it was the first time I went up to the lantern on the dome, it is that impressive

Now those advertisements for "wahrscheinlich das beste bier der welt" I used to see while I lived in Germany finally makes sense! At the time I thought they had confidence issues or just wanted to be very precise

There is a deep result stating that for a field F there are only three possibilities for what the dimension of its algebraic closure can be as an F-vector space: it can be 1, if F is algebraically closed, it can be 2, as happens for R and other real closed fields, or it can be infinite, as it happens for Q, but there is no other option!

My point is that is isn't just a function, but a measurable one (with weak restriction on the measure space it has as domain) so we can't call it just "function" as you were suggesting.

The actual name might be debatable, but a shortened name rather than the full definition definitely makes sense here. Random variable seems like a good choice of name to me, thinking about the intuition we're formalizing with it.

random variable as a shortening of "measurable function from a probability space" doesn't seem too bad to me

In Europe the norm is to do a Master after a Bachelor, some Master programmes are heavily research oriented and can give you a taste of what's to come if you choose to keep following that path. (I definitely agree with the point you're making, just pointing out that there are different possibilities)

I always find the example of PDEs interesting, depending on which mathematician you talk to PDEs could be anywhere from "as pure as it gets" to "extremely applied"

For me writing things down by myself is a huge part of the learning process (I take notes by hand in class even though I'm fluent enough with LaTeX to take them on my laptop directly) so I've never tried decks of flashcards made by other people, but you can find plenty of them for languages, divided by level or topic.

Personally I used Langenscheidt's Basic German Vocabulary¹, which contains the most common 4000 German words, further divided into the top 2000 and the 2001st-4000th and by topic, with example sentences for all of them. I made a card for each of the top 2000 words and included the example sentences in the cards as well.

[1] https://www.amazon.com/Basic-German-Vocabulary-Langenscheidt... (I found a used copy sold online for about 10€, it can be a bit of a pain to get one)

I can also recommend Anki, I used it to improve my German vocabulary and it was rather helpful!

I study maths and I never thought about using flashcards for it, but it's a cool idea, I might try it out!

Isomorphic to a free abelian group means that it behaves like Zⁿ for some n,that is the set of n-uples of integers with operations defined pointwise.

A simple example is Z², this is the set of all pairs of integers (a,b) where addition is defined as (a,b)+(c,d)=(a+c,b+d).

A more interesting example is the set of pairs (a,b√2) where a and b are integers and addition is defined as (a,b√2)+(c,d√2)=(a+c,(b+d)√2).

If you draw those in R² they look like repeating parallelograms (squares actually for the first), hence lattices

Is there a way to do something similar for a phone's lock screen background?

Or maybe even for a laptop

Plenty of problems have no solution, even in PDEs land where we're quite willing to redefine what's a solution (weak solutions, distributions etc.)

There is an interesting music video[1] by Nigel Stanford featuring Chladni plates among other cool phenomena. That's how I first learned about Chladni plates

[1] https://vimeo.com/111593305

I don't think we're talking about the same reddit, in my experience their search engine has consistently been awful! On different occasions I was unable to retrieve a post I almost remembered the exact title of via the reddit search engine, but found it quickly through google with the "inurl:reddit.com/r/subreddit" option

There are no finite algebraically closed fields, but I see your point, it depends on what you have to do really, if you want to do any kind of analysis completeness is great, but being more on the logic/model theory side I like all algebraically closed fields!

>How can I share that feeling inside my brain algebraically closed fields give me?

That's kinda like returning home after a long day, because algebraically closed fields are about the nicest structure you can hope to find yourself working in!

You still need to worry about the domain, in general there are plenty of real valued discontinuous linear functions (or functionals) from an infinite dimensional Banach space to R.

In finite dimension however linear functions are continuous

Also maybe it's just my phone but on mobile the sidebar covers the whole website making it impossible to read

I don't see the point of introducing sigma-algebras if you're not doing probability based on measure theory.

As others have said I wouldn't suggest this exposition to someone learning probability for the first time, but it's not as bad if you're familiar with the material and need a quick review.

Consider a graph on R^2 where two points are joined by an edge iff they have distance 1, in the 50s people wondered what the chromatic number of this graph is and established the trivial lower bound of 4 and upper bound of 7.

By a classic result of De Bruijn and Erdos this problem is equivalent to finding the maximum chromatic number of a finite unit distance graph in the plane, but no further progress was made in decades.

In a preprint published yesterday De Grey claims to have constructed a 1567 vertexes graph which can't be 4-coloured, as can be seen in the comment section of his polymath proposal this was raised to 1585, the non-4-colorability of this new graph seems to have been verified independently with SAT solvers by a couple of people, so that's promising.

As usual that's huge if true, the result looks promising, but time will tell

For the Langlands program he was awarded the Abel prize just a few weeks ago, following the link "Read more about Robert P. Langlands" at the end of the official page [1] for the Abel prize you can read a short and informal explanation of his work written by Alex Bellos

[1] http://www.abelprize.no/nyheter/vis.html?tid=73025

From the introduction to Shelah's Classification Theory and the number of nonisomorphic models (an extremely technical book with 700+ pages):

  So we shall now explain how to read the book. 
  The right way is to put it on your desk in the day, below your pillow in the night,
  devoting yourself to the reading,
  and solving the exercises till you know it by heart.

The simplified version is that if buy 1500 tickets one of those will win you 1000$ and 1500/54=27 tickets will win 50$ each for a total of 2350$

Group theory originated in a very visual and geometric ways studying the symmetries of various objects, before the modern definition of group was given. Many groups studied today are still geometrical in nature: the dihedral groups are the groups of symmetries of regular polygons, the symmetric groups are the groups of permutations of a set with N elements and so on and so forth.

After the notion of group was formalized (a set with an operation satisfying some axioms) it was natural to ask oneself what was the advantage of this apparently more general and abstract definition. Turns out (that's a theorem of Cayley) that every group is isomorphic to a subgroup of a symmetric group (that is a group of permutations on some set) so arguably we didn't get that much more generality while moving from the first geometric groups to the modern abstract ones!

Sure, but ZFC by itself also deals with every interesting set, I was just being nitpicky of my own assertion.

I'm not familiar with TG, what's the relation between it and ZFC+some large cardinal axiom?