Set Theory and Logic

Recursive Inaccessibility and the Inductive Hypothesis as to Countable Sets in ZFC

Authors: Amel Mara
Comments: 32 Pages.

The significance of the Inductive Hypothesis is examined with respect to the Principle of Mathematical Induction. A few relevant theorems that involve functions in set theory are specified with respect to the Inductive Hypothesis. The countability of rational numbers is reviewed, as to Cantor’s "intuition" (i.e., the "zig-zag" method of enumerating rational numbers) and constructive formulas that would map the set of natural numbers to a subset of the rational numbers (e.g., a multiplicative inverse function, a divisive function). Von Neumann and Zermelo ordinals are introduced to support the definition of a non-dense, well-ordered set of numbers. It is determined that for a specific infinite set of ordinals with a maximal element that is a limit ordinal, the set must contain at least one successor ordinal that cannot be recursively accessed in a finite number of steps from a specified base ordinal.
Category: Set Theory and Logic