two-cardinal problem

logic and mathematics

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July 14, 2026

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The "two-cardinal problem" arises in model theory, a branch of formal logic. It concerns how the cardinality (size) of a model's domain and the cardinality of a specific property within that model can vary.

In essence, it explores whether changing the size of a model also changes the size of certain subsets or properties within it. Researchers have found various answers, including proofs of independence from standard set theory axioms and conditional theorems based on common set theory hypotheses. This problem is distinct from Cantor's work on transfinite cardinal numbers, which deals with the sizes of infinite sets themselves.12

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metalogic
Satisfaction of a theory by a structure: finite and infinite models
…another direction, there are “two-cardinal” problems that arise from the possibilities of changing, from one model to another, not only the cardinality of the domain of the first model but also the cardinality of some chosen property (such as being a prime number). Various answers to these questions have…
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…another direction, there are “two-cardinal” problems that arise from the possibilities of changing, from one model to another, not only the cardinality of the domain of the first model but also the cardinality of some chosen property (such as being a prime number). Various answers to these questions have…
Read More