query: The Archimedean property of the real-number system: completeness axiom
positive_passages.docid: 143067
positive_passages.text: Show given any $x\in\mathbb{R}$ show there exists a unique $n\in\mathbb{Z}$ such that $n-1\leq x <n$.
negative_passages.docid: 106011
negative_passages.text: Proving it has all real roots
negative_passages.docid: 109964
negative_passages.text: the converse of Pythagoras Theorem
negative_passages.docid: 139992
negative_passages.text: Is power-associativity finitely axiomatizable?
negative_passages.docid: 156660
negative_passages.text: The set of all strictly increasing sequences of natural numbers
negative_passages.docid: 116009
negative_passages.text: Proving that if a set is both open and closed then it is equal to the real numbers
negative_passages.docid: 30519
negative_passages.text: "Real" cardinality, say, $\aleph_\pi$?
negative_passages.docid: 121732
negative_passages.text: Functional completeness of $\{\text{or},\text{ xor}, \text{ xnor}\}$
negative_passages.docid: 9146
negative_passages.text: Proving that a list of perfect square numbers is complete
negative_passages.docid: 17022
negative_passages.text: Definable real numbers
negative_passages.docid: 153969
negative_passages.text: The "elements" of a real number
negative_passages.docid: 135025
negative_passages.text: Axiomatization of a Topology using the Boundary
negative_passages.docid: 138897
negative_passages.text: help with some Hilbert style proofs in a propositional logic axiom system.
negative_passages.docid: 118189
negative_passages.text: Proving the pairing axiom from the rest of ZF
negative_passages.docid: 150385
negative_passages.text: The use of the Axiom of Choice in this proof of the sequence lemma
negative_passages.docid: 118617
negative_passages.text: Completeness and Incompleteness
negative_passages.docid: 100759
negative_passages.text: Extending exponentiation to reals