Pcf Theory
نویسنده
چکیده
The abbreviation “PCF” stands for “Possible Cofinalities”. PCF theory was invented by Saharon Shelah to prove upper bounds on exponents of singular cardinals. The starting point of PCF theory is in the realization that the usual exponent function is too coarse for measuring the power set of singular cardinals. Consider the cardinal אω, which is the smallest singular cardinal and has countable cofinality. The usual exponent (אω) א0 measures the total number of countable subsets of אω and is larger than אω itself by straightforward diagonalization. The exponent (אω) א0 clearly satisfies (אω) א0 ≥ 20 and therefore has no upper bound in the list {אα : α ∈ On} of cardinal numbers (because 20 itself has none). The main conceptual change that PCF theory has generated lies in the fact that the number of countable subsets of אω which is required to cover all countable subsets of אω is always bounded by אω4.
منابع مشابه
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