The goal of this paper is to explore the significance of Montague’s paradox—that is, any arithmetical theory $T\supseteq Q$ over a language containing a predicate $P\left(x\right)$ satisfying (T) $P\left(⌜\phi ⌝\right)\to \phi$ and (Nec) $T⊢\phi \hspace{0.17em}\therefore \hspace{0.17em}T⊢P\left(⌜\phi ⌝\right)$ is inconsistent—as a limitative result pertaining to the notions of formal, informal, and constructive provability, in their respective historical contexts. To this end, the paradox is reconstructed in a quantified extension $\mathcal{QLP}$ (the quantified logic of proofs) of Artemov’s logic of proofs ($\mathcal{LP}$). $\mathcal{QLP}$ contains both explicit modalities $t:\phi$ (“$t$ is a proof of $\phi$”) and also proof quantifiers $\left(\exists x\right)x:\phi$ (“there exists a proof of $\phi$”). In this system, the basis for the rule NEC is decomposed into a number of distinct principles governing how various modes of reasoning about proofs and provability can be internalized within the system itself. A conceptually motivated resolution to the paradox is proposed in the form of an argument for rejecting the unrestricted rule NEC on the basis of its subsumption of an intuitively invalid principle pertaining to the interaction of proof quantifiers and the proof-theorem relation expressed by explicit modalities.

We exhibit a simple inference rule, which is admissible but not derivable in BCK, proving that BCK is not structurally complete. The argument is proof-theoretical.

We continue the analysis of foundations of positive model theory as introduced by Ben Yaacov and Poizat. The objects of this analysis are $h$-inductive theories and their models, especially the “positively” existentially closed ones. We analyze topological properties of spaces of types, introduce forms of quantifier elimination, and characterize minimal completions of arbitrary $h$-inductive theories. The main technical tools consist of various forms of amalgamations in special classes of structures.

For a function $f$ with domain $[X{]}^n$, where $X\subseteq \mathbb{N}$, we say that $H\subseteq X$ is canonical for $f$ if there is a $\upsilon \subseteq n$ such that for any $x_0,\dots ,x_{n-1}$ and $y_0,\dots ,y_{n-1}$ in $H$, $f(x_0,\dots ,x_{n-1})=f(y_0,\dots ,y_{n-1})$ iff $x_i=y_i$ for all $i\in \upsilon$. The canonical Ramsey theorem is the statement that for any $n\in \mathbb{N}$, if $f:[\mathbb{N}{]}^n\to \mathbb{N}$, then there is an infinite $H\subseteq \mathbb{N}$ canonical for $f$. This paper is concerned with a model-theoretic study of a finite version of the canonical Ramsey theorem with a largeness condition and also a version of the Kanamori–McAloon principle. As a consequence, we produce new indicators for cuts satisfying $PA$.

We define the bounded jump of $A$ by $A^b=\{x\in \omega \mid \exists i\le x[{\phi }_i\left(x\right)\downarrow \wedge {\Phi }_x^{A\upharpoonright \upharpoonright {\phi }_i\left(x\right)}\left(x\right)\downarrow \left]\right\}$ and let $A^{nb}$ denote the $n$th bounded jump. We demonstrate several properties of the bounded jump, including the fact that it is strictly increasing and order-preserving on the bounded Turing ($bT$) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for $n\ge 2$ we have $X{\le }_{bT}{\varnothing }^{nb}\iff X$ is ${\omega }^n$-c.e. $\iff X{\le }_1{\varnothing }^{nb}$, extending the classical result that $X{\le }_{bT}\varnothing \text{'}\iff X$ is $\omega$-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every $X$ such that ${\varnothing }^b{\le }_{bT}X{\le }_{bT}{\varnothing }^{2b}$, there is a $Y{\le }_{bT}{\varnothing }^b$ such that $Y^b{\equiv }_{bT}X$.

We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than ${\aleph }_1$, with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. The distinction is important since a proof using finite supports is more amenable to generalizations to cardinals greater than ${\aleph }_1$.