Ordinals and axioms

Concepts

• Partial order relation from Discrete mathematics 10 exercisesA relation preceq on a set S that is reflexive, antisymmetric and transitive, and therefore orders some pairs of elements of S.
• Well-ordering 10 exercisesA set A is well-ordered by < if < is a total order on A and every nonempty subset of A has a least element.
• Transfinite induction 10 exercisesPrinciple that proves a property holds for all ordinal numbers On by showing that if it holds for all ordinals below a given α, i…
• Axiom of choice 10 exercisesStatement that every family of nonempty sets admits a choice function f that picks one element from each set in the family.
• Zorn's lemma 10 exercisesZorn's lemma is a principle equivalent to the axiom of choice stating that every nonempty partially ordered set in which every ch…
• Zermelo–Fraenkel axioms 10 exercisesThe Zermelo-Fraenkel axioms (ZF) are the standard axiom system of set theory, formulated in first-order logic. They allow the usu…
• Construction of the naturals 10 exercisesThe natural numbers are a set ℕ built inside set theory as finite ordinals, where each natural is the set of all naturals smaller…
• Construction of the reals 10 exercisesThe real numbers ℝ form a complete ordered field extending ℚ , in which every bounded Cauchy sequence converges. They are built f…