Cyclic Cardinals: An Axiomatic Proposal to Preserve ω and Render ℵ₁, ℵ₂, … into Cyclic Structures

This work proposes an alternative set theory that preserves well-founded countable ordinals(particularly ω ) and reinterprets uncountable cardinals (ℵ1,ℵ2,…) as non-well-founded cyclic classes. The Anti-Foundation Axiom (AFA) was adopted to allow structures with cycles, and cyclic cardinals were definedthrough bisimulation on labeled graphs. Basic operations (successor, sum, and product of ordinals) remainstandard below ℵ0, while higher levels admit structural periodicity. Relative consistency with ZF−Foundation +AFA was sketched, pointing out relationships with hypersets and coinductive semantics, as well as providingformal examples.