The aim of this paper is to examine some formulations of fallibilism about knowledge and show that the precise formulation of fallibilism that can safely be regarded as true or plausible is not easily available. We start by discussing Jason Stanley's formulation of fallibilism and will sketch some difficulties that it will face. Although we mainly focus on his version of fallibilism, the same points can equally apply to variants of it. We will then examine Stanley's fallibilism in relation to his interest-relative invariantism, and reveal the difficulties Stanley's fallibilism faces in accommodating the interest-relativity. In fact, fallibilism should be an unnecessary additional concession to skepticism for interest-relative invariantists in general, and we argue that they even have a good reason to be infallibilists.

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Set theorists often take their subject as constituting a foundation for the rest of mathematics, in the sense that other abstract mathematical objects can be construed fundamentally as sets. In this way, they regard the set-theoretic universe as the universe of all mathematics. And although many set-theorists affirm the Platonic view that there is just one universe of all sets, nevertheless the most powerful set-theoretic tools developed over the past half century are actually methods of constructing alternative universes. With forcing and other methods, we can now produce diverse models of ZFC set theory having precise, exacting features. The fundamental object of study in set theory has thus become the model of set theory, and the subject consequently begins to exhibit a category-theoretic second-order nature. We have a multiverse of set-theoretic worlds, connected by forcing and large cardinal embeddings like constellations in a dark sky. In this article, I will discuss a few emerging developments illustrating this second-order nature. The work engages pleasantly with various philosophical views on the nature of mathematical existence.

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