Where are the irrational numbers?

cant_count

Thank you for suggesting Cantor - very interesting. Another quick question: there are integers, rational numbers, irrationals (which can be expressed using polynomials), transcendentals (which can't, but still can be defined, e.g. pi and e). What about numbers that cannot be defined in any way whatsoever? Does such a category have a name? And are there more of them than anything else? Are they ever used in mathematics?
(If they are still transcendentals, they'd need to be distinguished from the definable ones, I'd be thinking.) Perhaps they can't be discussed because they're undetectable or unprovable by definition? But there should be more of them than anything else, right? (Like dark matter ..!)
- Just wondering

cant_count

Sorry, meant dark energy.

micromass

Your intuition is correct! There are indeed things like undefinable and uncomputable numbers!! Check out http://en.wikipedia.org/wiki/Definable_real_number and http://en.wikipedia.org/wiki/Computable_number

Bacle

I don't know if this is what you're looking for, but there are efforts to "algebraize" parts of calculus and analysis; specifically with ideas like those of algebraic derivations on rings
( maps associated with a ring that satisfy the Leibniz property), which are(intended?) discrete substitutes of the derivative, which do not assume the continuum. These algebraic substitutes are also used to solve differential equations in differential Galois theory. There are also areas like discrete differential geometry which try to do the same.

gb7nash

No, I was talking about planck measurements in real life. I'll be the first to admit I know very little about it, but I think this is more of a physics problem than a calculus problem.

Bourbaki1123

I just want to point out that in fact the set of computable numbers has measure zero (which I think is mentioned in the article, but it's an important point so I want to emphasize it), so almost all numbers in R have no algorithm for arbitrary precision decimal approximation. This is something people seem to often fail to take into consideration, it is actually pretty counter intuitive before you have some grounding in the theory of computation.