On: The field of “useful reals” between rational and r...

[emacdona]

The author claims in the notes that "The useful reals are similar, but not quite equivalent to other ideas in mathematics, such as [...] computable numbers."

Is that correct? What is the complement of the Computable Numbers in the Useful Reals? What is the complement of the Useful Reals in the Computable Numbers?

I've always thought of Computable Numbers as all numbers able to be represented by a finite string, ie: a computer program that would generate the number to any desired precision. How does that differ from the set of numbers with a finite symbolic representation?

Hmmmm... maybe by asking that question I've led myself to the answer. Chaitin's Constant has symbolic representations, one of which being the Wikipedia page that describes it: https://en.wikipedia.org/wiki/Chaitin%27s_constant. Does that mean it's included in the complement of the Computable Numbers in the Useful Reals? Are the Computable numbers a subset of the Useful Reals?

[Sniffnoy → emacdona]

The standard term is "definable", not "useful": https://en.wikipedia.org/wiki/Definable_real_number

But yes, Chaitin's constant is an example of a number that is definable but not computable.

[bloomer → Sniffnoy]

Yeah I think this terminology is odd because I think that the computable numbers are much more “useful” than the definable numbers.

[emacdona → emacdona]

I say "it" when I mention Chaitin's Constant, but really I believe it's an entire set of constants. Is that set countable? So many questions... :-)

[emacdona → emacdona]

Looks like the wikipedia page says there's a Chaitin's constant for each Computable Function, so yeah, countable. That's if I'm reading it correctly. Even if the constant differs for every program that computes a given Computable Function... still countable, though (if I'm doing my math right).

[tromp → emacdona]

Yep, it's countable. I have defined a relatively simple one in https://tromp.github.io/cl/Binary_lambda_calculus.html#Halti...