Difference definable vs. computable - Mathematics Stack Exchange Yes, all computable numbers are definable but not all definable numbers are computable Informally, a computable number is one for which we can write a computer program which calculates it to any desired accuracy So, even though $\pi$ is irrational (and transcendental), it is computable since we know several formulae which can easily be
Constructive vs computable real numbers - Mathematics Stack Exchange So for all practical purposes you could just work with the set of computable real numbers - and that is what constructive analysts do Computable real numbers are algebraically closed - every reasonable operation you do with them will result in another computable numbers That is the essence of the statement
Are there any examples of non-computable real numbers? For example, $\pi$ is computable although it is irrational, i e endless decimal fraction It was just a luck, that there are some simple periodic formulas to calcualte $\pi$ If it wasn't than we were unable to calculate $\pi$ ans it was non-computable If so, that we can't provide any examples of non-computable numbers? Is that right?
How can we define the terms computable and partially computable for . . . $\begingroup$ In the first condition you wrote I think it should be enough to write "g is computable" In the second condition you write "g is partial computable and not computable" The idea seems about right but it is probably a little better to re-phrase it as "g is partial computable but not total" $\endgroup$ –
computational complexity - How can the computable numbers be countable . . . It is tricky, though, since the set of computable numbers (or at least the set of Godel numbers that map to TMs that define the computable numbers) is not computably enumerable, meaning that you can't generate a list of the relevant TMs and hence list out all the computable numbers On the other hand, given a computable number in some form it