The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, namely, $\mathsf{ZFC}$ set theory, all mathematical objects are represented by sets, while ordinary, namely, non–set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass changes significantly upon the replacement of “second-order representations” by “third-order functions.” We discuss the implications and connections to the reverse mathematics program and its foundational claims regarding predicativist mathematics and Hilbert’s program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations.

We produce a connected real Lie group that, as a first-order structure in the group language, interprets the real field expanded with a predicate for the integers. Moreover, the domain of our interpretation is definable in the group.

We give a purely syntactical proof of the fixed point theorem for Sacchetti’s modal logics $\mathbf{K}+\square {(\square }^{n}p\to p)\to \square p$ ($n\ge 2$) of provability. From our proof, an effective procedure for constructing fixed points in these logics is obtained. We also show the existence of simple fixed points for particular modal formulas.

A visceral structure on $\mathfrak{M}$ is given by a definable base for a uniform topology on its universe M in which all basic open sets are infinite and any infinite definable subset $X\subseteq M$ has nonempty interior.

Assuming only viscerality, we show that the definable sets in $\mathfrak{M}$ satisfy some desirable topological tameness conditions. For example, any definable function $f:M\to M$ has a finite set of discontinuities; any definable function $f:M^n\to M^m$ is continuous on a nonnempty open set; and assuming definable finite choice, we obtain a cell decomposition result for definable sets. Under an additional topological assumption (“no space-filling functions”), we prove that the natural notion of topological dimension is invariant under definable bijections. These results generalize some of the theorems proved by Simon and Walsberg, who assumed dp-minimality in addition to viscerality. In the final section, we construct new examples of visceral structures.

We expand on the fixed point semantic approach of Kripke via the addition of two unary intensional operators: a paradoxicality operator Π where $\Pi (\mathrm{\Phi })$ is true at a fixed point if and only if Φ is paradoxical (i.e., if and only if Φ receives the third, non-classical value on all fixed points that extend the current fixed point), and an unbounded truth operator ${\mathrm{{\rm Y}}}_{\top }$ where ${\mathrm{{\rm Y}}}_{\top }(\mathrm{\Phi })$ is true at a fixed point if and only if any fixed point extending the current fixed point can be extended to one on which Φ receives the value true. We prove a generalized version of Kripke’s fixed point theorem guaranteeing the existence of models of this new language, as well as an expressive completeness result. We conclude with an exploration of the significant improvements in expressive power that result from the addition of these new operators, and we precisely identify what still cannot be said on this intensional extension of the Kripkean framework.

Let λ be a regular cardinal satisfying ${\mathit{\lambda }}^{<\mathit{\lambda }}=\mathit{\lambda }$ and $◇(S_{\mathit{\lambda }}^{{\mathit{\lambda }}^{+}})$. Then there exists a family of $2^{{\mathit{\lambda }}^{+}}$ many completely club rigid special ${\mathit{\lambda }}^{+}$-Aronszajn trees which are pairwise far.

Infinite-dimensional quasi-polyadic algebras and cylindric algebras are the best-known algebraizations of first-order logic. Quasi-polyadic equality algebras (QPEAs), due to Paul R. Halmos, are special polyadic algebras; nevertheless, for a long time, these algebras have been considered only as a version of cylindric algebras (CAs), because their behavior seemed to be very similar to that of CAs. However, in researching QPEAs, over time it turned out that many properties of QPEA are different from those of cylindric algebras. The reason is the presence of the polyadic operator, the transposition $p_{ij}$ (missing in CAs). The author states and documents that QPEAs have a dual position in algebraic logic. In many respects, this class is like CAs, following the cylindric paradigm, but, in other respects, as an effect of the polyadic paradigm (of the transposition), QPEAs are different from CAs. The paper is a survey on quasi-polyadic algebras with particular regard to the foregoing documentation, and it fills an existing gap in the literature.