Periodicity in the cumulative hierarchy
Gabriel Goldberg, Farmer Schlutzenberg

TL;DR
This paper explores the structure of elementary embeddings within the cumulative hierarchy of set theory without the Axiom of Choice, revealing conditions for definability and periodicity related to rank-to-rank embeddings.
Contribution
It provides a detailed analysis of elementary embeddings in the cumulative hierarchy, characterizing when such embeddings are definable and establishing the eventual periodicity under certain large cardinal assumptions.
Findings
Embeddings are definable iff the ordinal is odd.
Embeddings are not definable from elements of V_α.
Under large cardinal assumptions, the hierarchy exhibits periodicity.
Abstract
We investigate the structure of rank-to-rank elementary embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels of the cumulative hierarchy are defined via iterated application of the power set operation, starting from , and taking unions at limit stages. Assuming that is a (non-trivial) elementary embedding, we show that the structure of is fundamentally different to that of . We show that is definable from parameters over iff is an odd ordinal. Moreover, if is odd then is definable over from the parameter , and uniformly so. This parameter is optimal in that is not definable from any parameter which is an element of . In the case that ,…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms
