/sci/ - Science & Math

>>11672806
There's some related yikes insights, such as the Specker sequence.

Of course it's relatively easy to define some uncomputable reals by case-wise definition on their decimal expansion.
For any predicate P, let [math]a_n^P[/math] be 1 is [math] P(n) [/math] is true and 0 otherwise.
Since
[math]\sum_{n=1}^\infty \dfrac{1}{2^n}=1[/math],

we have that any
[math]X^P:=\sum_{n=1}^\infty \dfrac{a_n^P}{2^n}[/math]

is a real with [math] 0 \le X^P \le 1 [/math]. Now enumerate all textfiles by dovetailing over the number of characters resp. ascii characters from 1 to 256. All C++ programs will be among them. It's uncomputable whether a program in a Turing complete machine will halt. Let P be whether the n'th text in your enumeration is a valid C++ program that will halt and you got yourself a number _defined_ between 0 and 1 that has digits you can't compute.

Now Specker's insight finds you can't even escape such shitty incidents by restricting yourself to monotonically increasing definitions of numbers.

Of course, you can take LEM as axiom, and then a corollary of your theory is that
[math] \forall (n\in {\mathbb N}). \ (a_n=0) \lor (a_n=1) [/math]
even though you know that there's infinitely many [math] k\in {\mathbb N} [/math] for which the value of a_k can really only be accessed by Plato in heaven.

The [math] n [/math]'s for which you can find the value of a_n is CE, since you can zic-zac-parallelize the C++ program execution in the same way you can run through all rational numbers and those programs that halt are in your set. But that set isn't all of [math] \mathbb N [/math] and you will never be able to know which n'th are wasted computation that will run forever.
That's what I mean with computer sciency dovetailing intution. Not all infinite subsets of [math] \mathbb N [/math] are in constructive bijection with [math]\mathbb N [/math], and there's your loophole where you can assets that all infinite uncoutable sets are actually of this type

>>11672806
There's some related yikes insights, such as the Specker sequence.

Of course it's relatively easy to define some uncomputable reals by case-wise definition on their decimal expansion. For any predicate P, let [math]a_n^P[math] be 1 is [math] P(n) [/math] is true and 0 otherwise. Since [math]\sum_{n=1}^\infty \dfrac{1}{2^n}=1[/math], we have that any [math]X^P:=\sum_{n=1}^\infty \dfrac{a_n^P}{2^n}[/math] is a real with [math] 0 \le X^P \le 1 [/math]. Now enumerate all textfiles by dovetailing over the number of characters resp. ascii characters from 1 to 256. All C++ programs will be among them. It's uncomputable whether a program in a Turing complete machine will halt. Let P be whether the n'th text in your enumeration is a valid C++ program that will halt and you got yourself a number _defined_ between 0 and 1 that has digits you can't compute.
Now Specker's insight finds you can't even escape such shitty incidents by restricting yourself to monotonically increasing definitions of numbers.

Of course, you can take LEM as axiom, and then a corollary of your theory is that
[math] \forall (n\in {\mathbb N}). \ (a_n=0) \lor (a_n=1) [math]
even though you know that there's infinitely many [math] k\in {\mathbb N} [/math] for which the value of a_k can really only be accessed by Plato in heaven.

The [math] n [/math]'s for which you can find the value of a_n is CE, since you can zic-zac-parallelize the C++ program execution in the same way you can run through all rational numbers and those programs that halt are in your set. But that set isn't all of [math] \mathbb N [/math] and you will never be able to know which n'th are wasted computation that will run forever.
That's what I mean with computer sciency dovetailing intution. Not all infinite subsets of [math] \mathbb N [/math] are in constructive bijection with [math]\mathbb N [/math], and there's your loophole where you can assets that all infinite uncoutable sets are actually of this type