query: what is actually a "definable" real number?
positive_passages.docid: 17022
positive_passages.text: Definable real numbers
negative_passages.docid: 80019
negative_passages.text: $\varepsilon$-number countability without choice
negative_passages.docid: 118314
negative_passages.text: How to define $x^a$ for arbitrary real numbers $x$ and $a$
negative_passages.docid: 86866
negative_passages.text: Is there a number whose absolute value is negative?
negative_passages.docid: 116355
negative_passages.text: Do singularities have a "real" as opposed to mathematical or idealized existence?
negative_passages.docid: 34924
negative_passages.text: What is the domain of $x^x$ as a real valued function?
negative_passages.docid: 113344
negative_passages.text: How do we prove that something is unprovable?
negative_passages.docid: 143274
negative_passages.text: What is the smallest unknown natural number?
negative_passages.docid: 13273
negative_passages.text: Cardinality of the set of all real functions of real variable
negative_passages.docid: 65624
negative_passages.text: Real vs. Floating Point vs. Money
negative_passages.docid: 50555
negative_passages.text: Approximation of real number to the power of irrational
negative_passages.docid: 4946
negative_passages.text: Non-existence of irrational numbers?
negative_passages.docid: 11405
negative_passages.text: If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
negative_passages.docid: 134699
negative_passages.text: Can *any* real polynomial be factored linearly (plus a constant) over $\mathbb{R}$?
negative_passages.docid: 22391
negative_passages.text: Is the set of all pairs of real numbers uncountable?
negative_passages.docid: 975
negative_passages.text: Is there a known well ordering of the reals?
negative_passages.docid: 112910
negative_passages.text: Is the set of real numbers really uncountably infinite?