List of statements undecidable in ZFC

The mathematical statements discussed below are provably independent of ZFC (the Zermelo–Fraenkel axioms plus the axiom of choice, the canonical axiomatic set theory of contemporary mathematics), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC.

In 1931, Kurt Gödel proved the first ZFC independence result, namely that the consistency of ZFC itself was independent of ZFC (Gödel's second incompleteness theorem).

Moreover, the following statements are independent of ZFC:

We have the following chains of implication:

Another statement that is independent of ZFC is:

Several statements related to the existence of large cardinals cannot be proven in ZFC (assuming ZFC is consistent). These are independent of ZFC provided that they are consistent with ZFC, which most working set theorists believe to be the case. These statements are strong enough to imply the consistency of ZFC. This has the consequence (via Gödel's second incompleteness theorem) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent). The following statements belong to this class:

The following statements can be proven to be independent of ZFC assuming the consistency of a suitable large cardinal:

There are many cardinal invariants of the real line, connected with measure theory and statements related to the Baire category theorem, whose exact values are independent of ZFC. While nontrivial relations can be proved between them, most cardinal invariants can be any regular cardinal between ℵ₁ and 2^ℵ₀. This is a major area of study in the set theory of the real line (see Cichon diagram). MA has a tendency to set most interesting cardinal invariants equal to 2^ℵ₀.

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