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.
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].
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).
The topic chosen for this purpose, "Comparing Infinite Sets
within the Cantorian Set Theory", is part of an advanced mathematics course.
Cantor was able to show that it is possible to generate, 'bigger and bigger infinite sets
from ones that we already have.
were identified as sets bijective to a proper subset of themselves.
describes the uses of infinity as he explains elementary set theory, functions (including inverse functions), counting infinite sets
(including Hilbert's Infinite Hotel), infinite Cardinals (including the two infinities), well-ordered sets (such as the arithmetic of ordinals and cardinals as ordinals), inductions and numbers, including number theory, prime numbers, including the Riemann Zeta function.
For one thing, there's no way to simply count the elements of two infinite sets
and determine which set has more.