Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets
Writing in an admirably succinct and clear style, he covers logic, sets, functions, counting infinite sets
, infinite cardinals, well-ordered sets, inductions and numbers, prime numbers, and logic and meta-mathematics.
In particular, Cantor's theory of multiple infinite sets
, which is at the very foundation of contemporary set theory in all of its versions, yields a well-defined mathematical calculus which allows the "size" or cardinality of various infinite sets
to be considered and compared.
plates and napkins on a table) to establish they're the same size to using the operation to establish when infinite sets
have the same size.
As students of mathematics we invariably confront infinite sets
1] can be expressed as a disjoint union of a class C of countably infinite sets
, the cardinality of C being [[omega].
First, we identified constructivist perspectives that have been, or could be used to describe thinking about infinite sets
, specifically, the set of natural numbers N.
This includes a problem list compile by the speakers that reflects some of the most important questions in various areas of logic, and the presenters cover such topics as a stationary-tower-free proof of the derives model theorem, a proof of an absoluteness theorem, a simple inductive measure analysis of cardinals under the axiom of determinacy, the complexity of index set and Ehrenfeucht theories, computable structures in familiar classes, the classes of separating sets, voting rules for infinite sets
and Boolean algebras, "very mad families," Borel boundedness and the lattice surrounding property, and Steinhous sets and Jackson sets.
Of course, the counting numbers can be paired with even "smaller" infinite sets
such as the prime numbers.
Infinite wisdom A mathematician proposed a new approach to resolving a longstanding question about infinite sets
of numbers (164: 139).
We discovered that there were three different ways our students thought about Z[subscript n], namely as infinite sets
, element-set combinations, and representative elements.
The topic chosen for this purpose, "Comparing Infinite Sets
within the Cantorian Set Theory", is part of an advanced mathematics course.