Related papers: Generalized ordinal analysis and reflection princi…
There is no infinite sequence of $\Pi^1_1$-sound extensions of $\mathsf{ACA}_0$ each of which proves $\Pi^1_1$-reflection of the next. This engenders a well-founded ``reflection ranking'' of $\Pi^1_1$-sound extensions of $\mathsf{ACA}_0$.…
We study transfinite extensions of Japaridze's provability logic GLP and the well-founded relations that naturally occur within them. Every ordinal induces a partial order over the class of "words," which are iterated consistency statements…
In this paper, we argue that formal systems of first order Arithmetic that admit Goedelian undecidable propositions validly are abnormally non-constructive. We argue that, in such systems, the strong representation of primitive recursive…
We introduce layer systems for proving generalizations of the modularity of confluence for first-order rewrite systems. Layer systems specify how terms can be divided into layers. We establish structural conditions on those systems that…
There are many examples of optimization problems whose associated polyhedra can be described much nicer, and with way less inequalities, by projections of higher dimensional polyhedra than this would be possible in the original space.…
We introduce and axiomatize the notion of a reflective cardinal, use it to give semantics to higher order set theory, and explore connections between the notion of reflective cardinals and large cardinal axioms.
The classical Gelfand--Naimark theorems provide important insight into the structure of general and of commutative C*-algebras. It is shown that these can be generalized to certain ordered *-algebras. More precisely, for $\sigma$-bounded…
A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which…
Contemporary complexity theory has been instrumental in providing novel rigorous definitions for some classic philosophical concepts, including emergence. In an attempt to provide an account of emergence that is consistent with complexity…
We show that first-order logic can be translated into a very simple and weak logic, and thus set theory can be formalized in this weak logic. This weak logical system is equivalent to the equational theory of Boolean algebras with three…
For the generalized oscillator, we prove a Rellich type theorem, or characterize the order of growth of eigenfunctions. The proofs are given by an extensive use of commutator arguments invented recently by Ito and Skibsted. These arguments…
Continuing [Fuchino, Ottenbreit and Sakai[9, 10]] and [Fuchino and Ottenbreit[11]], we further study reflection principles in connection with the L\"owenheim-Skolem Theorems of stationary logics. In this paper, we mainly analyze the…
We introduce a new covering property, defined in terms of order types of sequences of open sets, rather than in terms of cardinalities of families. The most general form of this compactness notion depends on two ordinal parameters. In the…
Let V be the universe of sets and V_{\alpha} the sets of rank \leq\alpha. We develop some axiom schemata for set theory based on the following three assumptions: 1. V \models ZFC 2. V is large with respect to the class of ordinals 3. V is…
We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the…
We present a unified approach to (bi-)orthogonal basis sets for gravitating systems. Central to our discussion is the notion of mutual gravitational energy, which gives rise to the self-energy inner product on mass densities. We consider a…
Classically, any structure for a signature $\Sigma$ may be completed to a model of a desired regular theory $T$ by means of the chase construction or small object argument. Moreover, this exhibits $\mathrm{Mod}(T)$ as weakly reflective in…
Given a reflection group $G$ acting on a complex vector space $V$, a reflection map is the composition of an embedding $X \hookrightarrow V$ with the orbit map $V\to\mathbb C^p$ that maps a $G$-orbit to a point. Reflection maps can be very…
We present a higher well-ordering principle which is equivalent (over Simpson's set theoretic version of $\text{ATR}_0$) to the existence of transitive models of Kripke-Platek set theory, and thus to $\Pi^1_1$-comprehension. This is a…
We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class…