This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard (but still first-order) cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

The Maximality Principle $\mathrm{M}{\mathrm{P}}_{\mathrm{C}\mathrm{C}\mathrm{C}}$ is a scheme which states that if a sentence of the language of ZFC is true in some CCC forcing extension $V^{\mathbb{P}}$, and remains true in any further CCC-forcing extension of $V^{\mathbb{P}}$, then it is true in all CCC-forcing extensions of V, including V itself. A parameterized form of this principle, $\mathrm{M}{\mathrm{P}}_{\mathrm{C}\mathrm{C}\mathrm{C}}(\mathbb{R})$, makes this assertion for formulas taking real parameters. In this paper, we show that $\mathrm{M}{\mathrm{P}}_{\mathrm{C}\mathrm{C}\mathrm{C}}(\mathbb{R})$ has the same consistency strength as ZFC, solving an open problem of Hamkins. We extend this result further to parameter sets larger than $\mathbb{R}$.

A set $B\subseteq \mathbb{N}$ is called low for Martin-Löf random if every Martin-Löf random set is also Martin-Löf random relative to B. We show that a ${\mathrm{\Delta }}_2^0$ set B is low for Martin-Löf random if and only if the class of oracles which compress less efficiently than B, namely, the class ${𝒞}^B=\{A|\mathrm{\forall }n{K}^B(n){\le }^{+}{K}^A(n)\}$ is countable (where K denotes the prefix-free complexity and ${\le }^{+}$ denotes inequality modulo a constant. It follows that ${\mathrm{\Delta }}_2^0$ is the largest arithmetical class with this property and if ${𝒞}^B$ is uncountable, it contains a perfect ${\mathrm{\Pi }}_1^0$ set of reals. The proof introduces a new method for constructing nontrivial reals below a ${\mathrm{\Delta }}_2^0$ set which is not low for Martin-Löf random.

The probability that a fair coin tossed yesterday landed heads is either 0 or 1, but the probability that it would land heads was 0.5. In order to account for the latter type of probabilities, past probabilities, a temporal restriction operator is introduced and axiomatically characterized. It is used to construct a representation of conditional past probabilities. The logic of past probabilities turns out to be strictly weaker than the logic of standard probabilities.

We define notions of homomorphism, submodel, and sandwich of Kripke models, and we define two syntactic operators analogous to universal and existential closure. Then we prove an intuitionistic analogue of the generalized (dual of the) Lyndon-Łoś-Tarski Theorem, which characterizes the sentences preserved under inverse images of homomorphisms of Kripke models, an intuitionistic analogue of the generalized Łoś-Tarski Theorem, which characterizes the sentences preserved under submodels of Kripke models, and an intuitionistic analogue of the generalized Keisler Sandwich Theorem, which characterizes the sentences preserved under sandwiches of Kripke models. We also define several intuitionistic formula hierarchies analogous to the classical formula hierarchies ${\mathrm{\forall }}_n(={\mathrm{\Pi }}_n^0)$ and ${\mathrm{\exists }}_n(={\mathrm{\Sigma }}_n^0)$, and we show how our generalized syntactic preservation theorems specialize to these hierarchies. Each of these theorems implies the corresponding classical theorem in the case where the Kripke models force classical logic.

We introduce CE-cell decomposition, a modified version of the usual o-minimal cell decomposition. We show that if an o-minimal structure $ℛ$ admits CE-cell decomposition then any definable open set in $ℛ$ may be expressed as a finite union of definable open cells. The dense linear ordering and linear o-minimal expansions of ordered abelian groups are examples of such structures.

In order to accommodate his view that quantifiers are predicates of predicates within a type theory, Frege introduces a rule which allows a function name to be formed by removing a saturated name from another saturated name which contains it. This rule requires that each name has a rather rich syntactic structure, since one must be able to recognize the occurrences of a name in a larger name. However, I argue that Frege is unable to account for this syntactic structure. I argue that this problem undermines the inductive portion of Frege's proof that all of the names of his system denote in §§29–32 of The Basic Laws.

We study the Turing degrees which contain a real of effective packing dimension one. Downey and Greenberg showed that a c.e. degree has effective packing dimension one if and only if it is not c.e. traceable. In this paper, we show that this characterization fails in general. We construct a real $A{\le }_T{\mathrm{\varnothing }}^{\prime \prime }$ which is hyperimmune-free and not c.e. traceable such that every real $\alpha {\le }_{T}A$ has effective packing dimension 0. We construct a real $B{\le }_T{\mathrm{\varnothing }}^{\prime }$ which is not c.e. traceable such that every real $\alpha {\le }_{T}B$ has effective packing dimension 0.