/sci/ - Science & Math

We consider set of binary sequences [math]\gamma \in \{0,1\}^{\mathbb N}[/math].
Let [math]A_n \subset \{0,1\}^{\mathbb N}[/math] be the set of sequences which among their first 2n entries have the same number of 0's and 1's.

E.g. [math]A_3[/math] holds, among others, all sequences starting with
0,1,0,1,1,0
but not those starting with e.g.
1,1,0,1,1,0

1. What's [math]\dfrac{|A_n|}{2^{2n}}[/math] as a function of n, i.e. how many sequences have that type?
2. Likewise, taking the union below n, what's [math]{|\bigcup_{k\le n}A_k|}/{2^{2n}}[/math]
3. Taking the union of such set together, in the countable limit, what's the ratio [math]|\bigcup_{n\in{\mathbb n}} A_n|[/math] over [math]|\{0,1\}^{\mathbb N}|[/math]?

(My motivation is that I want to find a countable collection of decidable/detachable subsets of sequences, such that the countable union isn't decidable anymore, and one where the canonical measure on them isn't trivial (0 or 1). I assume the value in 3 is strictly between 0 and 1, although I'm not sure.)

>>15098263
The babby version is that, in arithmetic, no matter what axioms you add on top of the current arithmetic theory T you work with, you can always come up with a relatively simple statement G_T (as in it doesn't have too many quantifiers), such that neither G_T nor not(G_T) is provable.

By his other theorem, this means that arithmetic "doesn't know" that it's a theory about just the standard numbers. For the 19th century Peano theory of arithmetic, this implies that there's so called "non-standard models" with numbers that come after the ones we usually work with.

>Vaccine developed in 8 months
>"Many more people tested because of the emergency"
>"Not as much bureaucracy this time"

How does that mean potential side-effects years down the line are mitigated? Aren't vaccines usually tested for like five years or some shit?