So far, we have discussed Actualist and Potentialist approaches to set theory. In this chapter I will discuss a third major family of approaches to set theory: anti-Objectivist views on which some questions in the language of set theory lack determinate right answers. Such views are fairly popular. For example, many philosophers and mathematicians find it plausible that there’s no fact of the matter about the Continuum Hypothesis (the claim that there is no set whose cardinality is strictly between that of the integers and the real numbers), which has famously been shown not to be provable or refutable from the ZFC axioms of set theory.

In this chapter, I will discuss some major examples of anti-Objectivist philosophies of mathematics, and some concerns that arise for them.