Continuum Hypothesis identified as an open problem

Determine the truth or falsity of the Continuum Hypothesis as one of the open arithmetic and set-theoretic problems highlighted in the context of large cardinal investigations.

Background

Following Cohen’s independence result, the paper reports Gödel’s proposal to investigate large cardinals to illuminate open problems in arithmetic and set theory, explicitly including the Continuum Hypothesis (CH).

The identification of CH as an open problem underscores the need for additional axioms or semantic insights beyond ZFC to decide its status in the intended set-theoretic reality.

References

In the commentary on Cohen’s result from 1964, G¨ odel formulates the program of investigation of large cardinals, which could shed some light on the “open arithmetic and set-theoretic problems, including the Continuum Hypothesis” (see [12], pp. 260-261).

— Kurt Gödel and the Logic of Concepts (2406.05442 - Kostić et al., 8 Jun 2024) in Section 3.2.1