# Why can the set of real numbers $\mathbb{R}$ be defined in words of finite length? [duplicate]

It's well-known that there exists some real numbers that cannot be defined in a string of finite length (Berry's paradox). However, why can the set of all real numbers be defined?

My gut feelings are

$$1$$. the definition has some nonconstructive descriptions, like the supremum and infimum principle;

$$2$$. or we are just talking about computable numbers?

I'm not familiar with real analysis & mathematical logic so there might be something I overlooked.

• That's not what Berry's paradox is about. Berry's paradox is about being very careful with what you choose as a valid way of defining things, not about how there are more real numbers than possible definitions. Dec 3 '19 at 8:37
• I'm not saying your question is wrong. I'm saying Berry's paradox is the wrong thing to reference here, as it has nothing to do with how many things you can define. Dec 3 '19 at 8:57
• Which of "the set of all real numbers" have an infinite length? Dec 3 '19 at 8:58
• @AsafKaragila The way I read the question, I think the issue here is how you can even say something like "all real numbers" if it is provably impossible to define each of them. Dec 3 '19 at 9:02
• @ekd123: Short answer is that this is a very subtle argument that is usually overlooked. And unfortunately it's hard to wrap your head around this without knowing some logic and set theory. Dec 3 '19 at 9:10