TL;DR
This paper proves that strongly summable ultrafilters are rapid and that every rapid ultrafilter induces a closed left ideal of rapid ultrafilters, establishing foundational properties of ultrafilters on natural numbers.
Contribution
It provides new proofs of properties of ultrafilters, showing the connection between strongly summable and rapid ultrafilters and their algebraic structure.
Findings
Strongly summable ultrafilters are rapid.
Every rapid ultrafilter induces a closed left ideal of rapid ultrafilters.
Rapid minimal idempotents exist in all models with rapid ultrafilters.
Abstract
This short note contains the proofs of two small but somewhat surprising results about ultrafilters on : 1. strongly summable ultrafilters are rapid, 2. every rapid ultrafilter induces a closed left ideal of rapid ultrafilters. As a consequence, there will be rapid minimal idempotents in all models of set theory with rapid ultrafilters.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Limits and Structures in Graph Theory · Mathematical Dynamics and Fractals