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Every uncountable cardinal is singular in Gitik's model
Statement
The model satisfies that every uncountable cardinal has and is therefore singular.
Facts & Assumptions
Given: The completed Gitik symmetric model.
Gitik's symmetric submodel satisfies ZF: satisfies ZF, without assuming Choice.
Every natural number and are cardinals, every infinite cardinal is a limit ordinal, and on the natural numbers the cardinal operations are the published finite counting operations, with in the finite sense equal to in the cardinal sense: In ZF every infinite cardinal, viewed as an initial ordinal, is a limit ordinal.
Every limit ordinal has cofinality omega in Gitik's model: Every nonzero limit ordinal of has cofinality .
Cardinal (initial ordinal) and cardinality and Cofinality , and regular and singular cardinals: An infinite cardinal is singular exactly when its cofinality differs from the cardinal.
Proof
Work inside , which satisfies ZF by F1, and let be an uncountable cardinal. Then is infinite, so F2 makes it a limit ordinal; it is nonzero because is finite. F3 therefore gives .
Uncountability says , so the equality from step 1.1 gives . By F4, is singular. The only excluded endpoint is itself: it is countable and regular, so the corollary neither includes nor misclassifies it. No Choice principle is used.
Depends on
- Gitik's symmetric submodel satisfies ZF
- Every limit ordinal has cofinality omega in Gitik's model
- Every natural number and $\omega$ are cardinals, every infinite cardinal is a limit ordinal, and on the natural numbers the cardinal operations are the published finite counting operations, with $\lvert A \rvert$ in the finite sense equal to $\lvert A \rvert$ in the cardinal sense
- Cardinal (initial ordinal) and cardinality
- Cofinality $\operatorname{cf}(\alpha)$, and regular and singular cardinals
Used by
Dependency tree · two levels
40 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Schürz, Gitik's model, abstract and final theorem (standard reference, not scraped)