I am sorry if I sound a bit confused. I don’t speak English well and I am far more familiar with the mathematical terms in German. And I am confused by Solomoff induction.
My main point is the vast mathematical difference between “only” infinite (abzählbar unendlich) sets like the rational numbers or the computable numbers and the sets that are uncountable (überabzählbar) like the real numbers or the power set of the natural numbers.
If I have the set of real numbers between zero and one and “take” one number “randomly” I should “get” a non—computable number—in other words something that’s existence is more than doubtful. I certainly can not give a sample for that.
Thesenumbers were not named “irrational” for nothing.
Perhaps this is only about the word particular in “particular infinite input string”. Something like “you can take it only if it exists”. But should it not be formulated like that? It feels to me like smuggling the whole concept of real numbers in by the backdoor.
What if you use instead the following sentence:
“More precisely, suppose that a particular input string x0 of unknown length is about to be fed into U.
However, you know nothing about x0 other than that each term of the string is either 0 or 1.”
Would that form into be a sort of weak Solomoff induction? Or would the thing than collapse into complete nonsense?
I am sorry if I sound a bit confused. I don’t speak English well and I am far more familiar with the mathematical terms in German. And I am confused by Solomoff induction.
My main point is the vast mathematical difference between “only” infinite (abzählbar unendlich) sets like the rational numbers or the computable numbers and the sets that are uncountable (überabzählbar) like the real numbers or the power set of the natural numbers.
If I have the set of real numbers between zero and one and “take” one number “randomly” I should “get” a non—computable number—in other words something that’s existence is more than doubtful. I certainly can not give a sample for that.
These numbers were not named “irrational” for nothing.
Perhaps this is only about the word particular in “particular infinite input string”. Something like “you can take it only if it exists”. But should it not be formulated like that? It feels to me like smuggling the whole concept of real numbers in by the backdoor.
What if you use instead the following sentence:
“More precisely, suppose that a particular input string x0 of unknown length is about to be fed into U.
However, you know nothing about x0 other than that each term of the string is either 0 or 1.”
Would that form into be a sort of weak Solomoff induction? Or would the thing than collapse into complete nonsense?