The CH, or Continuum Hypothesis, is a conjecture in set theory that states that there is no set whose cardinality is strictly between that of the integers and the real numbers. It is important in the study of large cardinals because it is closely related to the existence of large cardinal axioms, which are statements about the existence of sets with extremely large cardinalities.
The status of CH is a significant issue in the study of large cardinals in New Foundations, as it is a foundational theory that allows for the existence of large cardinals. If CH were proven to be true, it would have major implications for the existence of large cardinals in New Foundations and other foundational theories.
The current understanding of the status of CH in New Foundations is that it remains an open problem. While there have been some attempts to prove or disprove CH in New Foundations, no definitive answer has been reached. As such, it remains a topic of ongoing research and debate.
Yes, there are connections between the status of CH and large cardinal axioms in New Foundations. Some large cardinal axioms, such as the existence of a strongly compact cardinal, imply the negation of CH, while others, such as the existence of a measurable cardinal, are consistent with both the truth and falsity of CH.
If the status of CH were to be definitively resolved in New Foundations, it would have significant implications for our understanding of large cardinals and the foundations of mathematics. It could also impact other areas of mathematics, such as the study of infinity and the continuum, and have practical applications in fields such as computer science and cryptography.