model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...

, a branch of

mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

, Chang's conjecture, attributed to

Chen Chung Chang Chen Chung Chang (Chinese: 张晨钟) was a mathematician who worked in model theory. He obtained his PhD from Berkeley in 1955 on "Cardinal and Ordinal Factorization of Relation Types" under Alfred Tarski. He wrote the standard text on model th ...

by , states that every model of type (ω₂,ω₁) for a countable language has an elementary submodel of type (ω₁, ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinality β. The usual notation is

(\omega_2,\omega_1)\twoheadrightarrow(\omega_1,\omega)

. The axiom of constructibility implies that Chang's conjecture fails.

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proved the consistency of Chang's conjecture from the consistency of an ω₁- Erdős cardinal. Hans-Dieter Donder showed a weak version of the reverse implication: if CC is not only consistent but actually holds, then ω₂ is ω₁-Erdős in K. More generally, Chang's conjecture for two pairs (α,β), (γ,δ) of cardinals is the claim that every model of type (α,β) for a countable language has an elementary submodel of type (γ,δ). The consistency of

(\omega_3,\omega_2)\twoheadrightarrow(\omega_2,\omega_1)

was shown by

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from the consistency of a

huge cardinal In mathematics, a cardinal number κ is called huge if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point (set theory), critical point κ and :^M \subset M.\! Here, ''&al ...

References

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Model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...

Set theory Conjectures {{mathlogic-stub