Tag
Set theory
- How to Prove Two Nondegenerate Closed Intervals Have Equal CardinalityA linear bijection proves bounded nondegenerate closed intervals have equal cardinality; unbounded intervals with more than one point do too.
- What Is the Definition of a Weakly Compact Cardinal?A weakly compact cardinal is an uncountable inaccessible cardinal with strong partition and tree properties; here are the standard equivalent definitions.
- Is Consistency an Axiom of Mathematics?Consistency is usually a property of a formal mathematical theory, not a universal axiom; a theory can nevertheless include a formal consistency statement.
- Can We Prove the Irrationals Are Uncountable Without Cantor’s Diagonal Argument?Yes. A nested-interval argument proves the real numbers, and therefore the irrationals, are uncountable without decimal diagonalization.
- Does the Set of All Self-Membered Sets Contain Itself?In standard ZF set theory, no set contains itself, so the collection is empty; see why this differs from Russell’s paradox and what changes without Foundation.
- What Is the Difference Between an Arbitrary and a Finite Collection?An arbitrary collection may have any number of members, while a finite collection has finitely many; see how this affects unions and intersections.
- What Is the Difference Between Supremum and Maximum?A supremum is the least upper bound and need not belong to a set; a maximum is the greatest element, so every maximum is a supremum but not vice versa.
- How Do You Find the Cardinality of a Set Containing Sets?To find the cardinality of a set containing sets, count its distinct outer elements; do not add the numbers of elements inside those sets.
- What Is the Cartesian Product with the Empty Set?The Cartesian product of any set with the empty set is empty: neither A × ∅ nor ∅ × A contains an ordered pair.
- Can a Set Be an Element of Itself?In ZF and ZFC, the Axiom of Foundation rules out x ∈ x. See the short proof, why the axiom matters, and how non-well-founded theories differ.