Projectum

Projectum is a term used in set theory, particularly in the area of inner model theory and fine-structure analysis of premice (small canonical models used to approximate large cardinal theories). The projectum of a model records the least amount of the model that is needed to recover the rest of the model by definable operations; in formal work this is encoded as ρ^M_n, the n-th projectum of a premouse M. Intuitively, ρ^M_n is the smallest ordinal α such that M is generated by the definable Skolem hull over M|α using formulas of complexity up to level n. If the projectum is small, much of M is determined by a small initial segment; if it is large, more of M is needed to reconstruct it.

Projecta are used to define soundness (or solidity) of premice, and to guide comparisons and iteration arguments that build larger models from smaller ones. They interact with extenders, condensation lemmas, and the fine-structure recursion that underlies the construction of canonical inner models with large cardinal properties. The notion has precise technical definitions in the work of Jensen and later refinements by Steel and others; it is a standard tool in proving consistency results relative to large cardinals by comparing different mice.

Because projectum is a specialized concept, it appears mainly in advanced texts on inner model theory and is not part of standard set theory outside that area.