Extending labels to all finite sets for levels of standardness

Ascertain whether SPOTS can be reformulated to allow labels indexing standardness levels to range over all finite subsets of N, thereby providing levels of standardness indexed by all natural numbers rather than only standard finite subsets.

Background

In SPOTS, labels a, b range over standard finite subsets of N, which implies that levels of standardness are indexed by standard natural numbers. This design is tied to the goal of avoiding strong forms of Choice while enabling multi-level nonstandard methods.

The authors note that related theories with broader indexing have been developed under ZFC, suggesting feasibility in stronger set-theoretic contexts. Whether a similar generalization is possible within the axiomatic constraints of SPOTS remains an explicit open question.