Stable Theory - Superstable Theories

Superstable Theories

T is called superstable if it is stable for all sufficiently large cardinals, so all superstable theories are stable. For countable T superstability is equivalent to stability for all κ≥2ω. The following conditions on a theory T are equivalent:

• T is superstable.
• All types of T are ranked by at least one notion of rank.
• T is κ-stable for all sufficiently large cardinals κ
• T is κ-stable for all cardinals κ that are at least 2|T|.

If a theory is superstable but not totally transcendental it is called strictly superstable.

The number of countable models of a countable superstable theory must be 1, ℵ₀, ℵ₁, or 2ω. If the number of models is 1 the theory is totally transcendental. There are examples with 1, ℵ₀ or 2ω models, and it is not known if there are examples with ℵ₁ models if the continuum hypothesis does not hold. If a theory T is not superstable then the number of models of cardinality κ>|T| is 2κ.

Examples:

• The additive group of integers is superstable, but not totally transcendental. It has 2ω countable models.
• The theory with a countable number of unary relations P_i with model the positive integers where P_i(n) is interpreted as saying n is divisible by the nth prime is superstable but not totally transcendental.
• An abelian group A is superstable if and only if there are only finitely many pairs (p,n) with p prime, n a natural number, with pnA/pn+1A infinite.