yoneda-emma - Tumblr blog

[yoneda-emma]

i was watching my girlfriend screenshare nightcore and we did a beautiful dance. this is what true love looks like

[yoneda-emma]

momiji helping carry energy drinks

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everyone knows that space is very very cold, and the sun is very very hot. so i assume there's a bit of space kind of near the sun which is just right. balmy space

[yoneda-emma]

i love that the satellites in 17776 r like actual satellites or whatever. satellite rpf. i see news abt the actual ones and i go hey thats my friend from football

[yoneda-emma]

The way I got taught Gödel numberings (Which is just the fancy term for "one to one mapping between a certain class of objects and the natural numbers") is with the following bijection between the set of natural numbers and the set of arbitrary sequences of natural numbers: Say you have a (potentially infinite) sequence (a,b,c,d,[...]). Then you can map this sequence to the natural number 2^a * 3^b * 5^c * 7^d * [...] going through all prime numbers until we've gone the length of the sequence. This already gives us a mapping from our sequences onto the natural numbers.

This mapping is reversible because the prime factorization of any natural number is unique - say we take the number 280. The prime factorization of 280 is 2^3 * 5 * 7 = 2^3 * 3^0 * 5^1 * 7^1 , so our corresponding sequence is (3,0,1,1).

This way we can map any natural number to a sequence of natural numbers, with the exception of 0. If we want to cover 0 to, however, then we can just subtract one at the start. We end up with a function f that maps a given sequence a = (a_1, a_2, a_3, (...), a_n) of natural numbers to individual natural numbers: f(a) = -1 * 2^a_1 * 3^a_2 * 5^a_3 (...) Or, in closed form: f(a) = -1 + ∏p_n^a_n , Where p_n is the nth prime number. I'm pretty sure you could modify this idea slightly to get a mapping between turing machines and natural numbers, but I havent actually double checked it since the calculations get very tedious. The formal definition of a turing machine is as a tuple containing a bunch of sets(the set of states, the set of state transitions, etc). If you assign a number to each of the states, represent each state transition as a new tuple, order each set where needed to get even more tuples, then encode all of these subtuples using this function we just made, and then encode the individual resulting numbers again, you should be able to get an injective mapping from the set of all turing machines into the natural numbers, which would be enough to prove our statement that the set of computable numbers is countable.

edit: bunch of typos

I think one fun side effect of studying uni level math is that simultaneously the complex numbers start seeming simpler than you thought but at the same time the real numbers are *way* more fucked up than you thought

[yoneda-emma]

Yeah the formal proof is pretty much just that, it just goes: - A number is computable if there's a turing machine that can write that number on an empty output band (this is just one of the main definitions of computability) - There's a countable number of turing machines, which you'd prove formally by coming up with some map from turing machines onto natural numbers. I cant be bothered to formally define one of these right now since it tends to be very dry and fiddly, but one classical way to do this usually comes down to taking a product of primes and encoding the turing machine in the exponents. You could also just write down the turing machine in ascii characters (in some rigorous standardized way that you come up with) and take all of the bits together as one long number. - Since theres a countable number of turing machines, and there cant be more computable numbers than turing machines, there's a countable number of computable numbers.

I think one fun side effect of studying uni level math is that simultaneously the complex numbers start seeming simpler than you thought but at the same time the real numbers are *way* more fucked up than you thought

[yoneda-emma]

You often see people bailing water out of a sinking boat make the rookie mistake of throwing the water back into the sea. That's no good, that water's just going to come right back in again through the hole in your boat and all your hard work will be undone. You need to find somewhere else to put the water.

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my dumb ass really just thought "damn pizza is expensive, maybe i should pirate it"

[yoneda-emma]

you try to get a touhou character into bunny girl outfit and she just eats you

two bites

[yoneda-emma]

malls are dying because they don't have blacksmith, apothecary, alehouse or peddler's

[yoneda-emma]

if you befriend enough weirdos you can get lectures for free

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[yoneda-emma]

The bite of a cat is incredibly venomous but their love cures it instantly

[yoneda-emma]

yo they named a city after the xbox emulator