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What are large cardinals and large cardinal axioms?

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Let us now return to our overview of large cardinal axioms. After strongly inaccessible cardinals there are Mahlo cardinals, indescribable cardinals, and ineffable cardinals. read more

A large cardinal axiom is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property. Most working set theorists believe that the large cardinal axioms that are currently being considered are consistent with ZFC. read more

Can all the large cardinal axioms not presently known to be inconsistent with ZFC be arranged in a strict linear order based on the following criterion. Let A(1) and A(2) be any distinct pair of these axioms and let both be adjoined to ZFC. read more

that some basic large cardinal axioms can be obtained by generalizing combinatorial properties of the natural numbers to uncountable sets. Here is a rough outline of what I’m going to say. read more

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large cardinal in nLab
Source: ncatlab.org

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