Theories with models of prescribed cardinalities
Citations Over Time
Abstract
The Löwenheim–Skolem theorem states that if a theory has an infinite model it has models of all cardinalities greater than or equal to the cardinality of the language in which the theory is defined. A natural question is what happens if there is a model whose cardinality is less than that of the language. If κ is an infinite cardinal less than the first measurable cardinal and κ < κ ω , the Rabin–Keiler theorem [1, p. 139] gives an example of a theory which has a model of cardinality κ in which every element is the interpretation of a constant and all other models have cardinality μ ≥ κ ω . Keisler has also shown that if a theory has a model of cardinality κ it has models of all cardinalities μ ≥ κ ω . We will show that within the bounds of the above theorems anything can happen. The main result is as follows.
Related Papers
- → The Idea of an Exact Number: Children's Understanding of Cardinality and Equinumerosity(2013)110 cited
- → Factorials and the finite sequences of sets(2019)8 cited
- → Theories with models of prescribed cardinalities(1977)3 cited
- → Descendingly Incomplete Ultrafilters and the Cardinality of Ultrapowers(1972)3 cited
- → Children’s early understanding of the successor function(2023)