Mathematics

The original Goodstein process proceeds by writing natural numbers in nested exponential k -normal form, then successively raising the base to k+1 and subtracting one from the end result. Such sequences always reach zero, but this fact is unprovable in Peano arithmetic. In this paper we instead consider notations for natural numbers based on the Ackermann function. We define two new Goodstein processes, obtaining new independence results for ACA ′ 0 and ACA + 0 , theories of second order arithmetic related to the existence of Turing jumps.

We advance the theory of strong colorings over partitions, studying both positive and negative Ramsey relations at the level of the uncountable. A correspondence between combinatorial properties of partitions and chain conditions of natural forcing notions for destroying strong colorings over them is uncovered and enables us to prove positive Ramsey relations for ??1 from weak forms of Martin's Axiom, thereby answering two questions from [CKS21]. Positive Ramsey relations for ??2 and higher cardinals are established as well and without making use of large cardinals. We also provide a group of pump-up theorems for strong colorings over partitions. Some of them solve more problems from [CKS21].

The definition of negation has to be referred to the totality of a theory and at last to what is defined as the organization of a scientific theory; in other words, the definition of negation is of a structural kind, rather than of an objective kind or a subjective kind. The paper starts by remarking that the ancient Greek word for truth was aletheia, which is a double negation, i.e. unveiling. Not before the 1968 the double negation law was re-evaluated, since it was recognized that its failure represents more appropriately than the failure of of excluded middle law the borderline between classical logic and almost all non-classical kinds of logic. Moreover, the failure of this law is easily recognized within a scientific text; this fact allows a new kind of logical analysis of a text. As an example, the analysis of Kolmogorov 1932 paper shows that he reasoned according to arguments of non-classical logic about the foundations of the intuitionist logic. The negation is defined as a unary operation which, under the problem of deciding whether a doubly negated proposition is equal to the corresponding affirmative proposition or not, leads to a subdivision into classical logic and intuitionist logic.

Complexity and decidability of logics is a major research area involving a huge range of different logical systems. This calls for a unified and systematic approach for the field. We introduce a research program based on an algebraic approach to complexity classifications of fragments of first-order logic (FO) and beyond. Our base system GRA, or general relation algebra, is equiexpressive with FO. It resembles cylindric algebra but employs a finite signature with only seven different operators. We provide a comprehensive classification of the decidability and complexity of the systems obtained by limiting the allowed sets of operators. We also give algebraic characterizations of the best known decidable fragments of FO. Furthermore, to move beyond FO, we introduce the notion of a generalized operator and briefly study related systems.

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form ∀∃!∧p=q . For a logic L algebraized by a quasivariety Q we show that the AE-subclasses of Q correspond to certain natural expansions of L , which we call {\em algebraic expansions}. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of Abelian ℓ -groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics.

This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions f: ω ω → ω ω . I prove that these cardinals can be organized into two diagrams analogous to the standard Cichoń diagram, prove several independence results and investigate the relation between these cardinals and the standard cardinal invariants on omega. In the second half of the thesis I look at forcing axioms compatible with CH. First I consider Jensen's subcomplete and subproper forcing. I generalize these notions to larger classes which are (apparently) much more nicely behaved structurally. I prove iteration and preservation theorems for both classes and use these to produce many new models of the subcomplete forcing axiom. Finally I deal with dee-complete forcing and its associated axiom DCFA. Extending a well-known result of Shelah, I show that if a tree of height ω 1 with no branch can be embedded into an ω 1 -tree, possibly with uncountable branches, then it can be specialized without adding reals. As a consequence I give a fleshed out proof that DCFA implies there are no Kurepa trees, even if CH fails.

We consider all 16 unary operations that, given a homogeneous binary relation R, define a new one by a boolean combination of xRy and yRx. Operations can be composed, and connected by pointwise-defined logical junctors. We consider the usual properties of relations, and allow them to be lifted by prepending an operation. We investigate extensional equality between lifted properties (e.g. a relation is connex iff its complement is asymmetric), and give a table to decide this equality. Supported by a counter-example generator and a resolution theorem prover, we investigate all 3-atom implications between lifted properties, and give a sound and complete axiom set for them (containing e.g. "if R's complement is left Euclidean and R is right serial, then R's symmetric kernel is left serial").

Vardanyan's Theorem states that quantified provability logic is ? 0 2 -complete, and in particular impossible to recursively axiomatize for consistent theories containing a minimum of arithmetic. However, the proof of this fact cannot be performed in a strictly positive signature. The logic QRC 1 was previously introduced as a candidate first-order provability logic. Here we show that QRC 1 is indeed complete with respect to arithmetical semantics. This is achieved via a Solovay-type construction applied to constant domain Kripke models. As a corollary we conclude that QRC 1 is also the quantified provability logic of HA.

We show that the forcing axiom for countably compact, ω 2 -Knaster, well-met posets is inconsistent. This is supplemental to an inconsistency result of Shelah and sets a new limit to the generalization of Martin's Axiom to the stage of ω 2 .

In this paper we continue the study of the variety MG of monadic Gödel algebras. These algebras are the equivalent algebraic semantics of the S5-modal expansion of Gödel logic, which is equivalent to the one-variable monadic fragment of first-order Gödel logic. We show three families of locally finite subvarieties of MG and give their equational bases. We also introduce a topological duality for monadic Gödel algebras and, as an application of this representation theorem, we characterize congruences and give characterizations of the locally finite subvarieties mentioned above by means of their dual spaces. Finally, we study some further properties of the subvariety generated by monadic Gödel chains: we present a characteristic chain for this variety, we prove that a Glivenko-type theorem holds for these algebras and we characterize free algebras over n generators.