In case your diet is not up to par(too much sugary and fried stuff, say), you could try hiring a nutritionist especially given that you have $ for a psychiatrist. Not that diet/nutrition science is all that rigorous, but at least they could optimize your diet to the extent that you start feeling a bit better.
Intermittent fasting is another tool. And the most important one of them all - good night's sleep.
This index needs real time updating. As a citizen of Kazakhstan I know that I don't need a visa to visit Andorra for example. Given the recent world events, I am sure the visa requirements for those affected have changed(and changing) considerably.
Not sure what's on the vid. Didn't watch it. But if I understand you correctly, the following is impossible as shown in Cantor's simple diagonalization argument:
> Follow each branch as deeply as possible counting all sub branches until there are no sub branches left.
Good point, the depth first search algorithm is a bit lack luster for this case:
1 -> .1
2 -> .01
3 -> .001
4 -> .0001
5 -> .00001
6 -> .000001
. -> .0∞1
I would definitely recommend not judging the quality of the video based upon the quality of my reasoning in the experiment.
That would only count the rationals which you can definitely count off a carefully arranged matrix of rationals. But reals = rationals and irrationals and the irrationals are uncountable.
> Going further R (points on a line) to R^2 (points on a plane) is also the same cardinality. The proofs are over my head but they're out there.
It's just a matter of finding a suitable set of functions. For example, you can try proving |R^2| = |(0, 1) x (0, 1)| = |(0, 1)| = |R| where R stands for reals.The middle equality can be proven using Schröder–Bernstein theorem.
I follow the line of the proof but finding those actual steps is where being out of the pure math game for 10 years catches up to me. I went into college on a pure math major but transferred to a CS degree. Only needed about 2 classes extra at the end but was tired of school and a simple BS in mathematics wouldn't get me too much in a generic CS career so I just stopped.
> Take the infinities of all numbers > 0 and then all even numbers > 0.
Define your number first, then we'll work it out from there. From the looks of it, you are considering the integers.
> So you have
> 1,2,3,4,5,6,.... 2,4,6,8,........
> Why can't we just consider both infinities to be the same size (they go on forever), but the item in a given position simply differs.
These sets have the same size. Two sets have the same size if you can find a function from one to the other which is bijective meaning if two sets match exactly element for element(elements don't have to be the same ones), they have the same size. For example, the sets {1, 2, 3} and {a, b, c} have the same size because we can match 1 to a, 2 to b and 3 to c. Or we can match 1 to c, 2 to a and 3 to b. So these two "matching" examples constitute two bijective functions. Going back to your example, the function f from {1,2,3,4,5,6,....} to {2,4,6,8,........} given by f(x) = 2x for x in {1,2,3,4,5,6,....} is bijective and therefore lines up the two sets in one to one correspondence. To prove f is bijective, we can find the inverse for f, multiply f and its inverse and get an identity or show f is surjective and injective. These are slightly technical, but not too bad. Any intro to discrete math textbook contains this material.
Intermittent fasting is another tool. And the most important one of them all - good night's sleep.