Compactness and Löwenheim-Skolem theorems in extensions of first-order logic

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Enrique Casanovas Ruiz-Fornells[en] Lindström’s theorem characterizes first-order logic as the most expressive among those that satisfy the countable Compactness and downward Löwenheim-Skolem theorems. Given the importance of this results in model theory, Lindström’s theorem justifies, to some extent, the privileged position of first-order logic in contemporary mathematics. Even though Lindström’s theorem gives a negative answer to the problem of finding a proper extension of first-order logic satisfying the same model-theoretical properties, the study of these extensions has been of great importance during the second half of the XX. century: logicians were trying to find systems that kept a balance between expressive power and rich model-theoretical properties. The goal of this essay is to prove Lindström’s theorem, along with its prerequisites, and to give weaker versions of the Compactness and Löwenheim-Skolem theorems for the logic L ( Q 1 ) (first-order logic with the quantifier "there exist uncountably many"), which we present as an example of extended logic with good model-theoretical properties

Decidability vs. undecidability. Logico-philosophico-historical remarks

The aim of the paper is to present the decidability problems from a philosophical and historical perspective as well as to indicate basic mathematical and logical results concerning (un)decidability of particular theories and problems

Categoricity, Open-Ended Schemas and Peano Arithmetic

One of the philosophical uses of Dedekind’s categoricity theorem for Peano Arithmetic is to provide support for semantic realism. To this end, the logical framework in which the proof of the theorem is conducted becomes highly significant. I examine different proposals regarding these logical frameworks and focus on the philosophical benefits of adopting open-ended schemas in contrast to second order logic as the logical medium of the proof. I investigate Pederson and Rossberg’s critique of the ontological advantages of open-ended arithmetic when it comes to establishing the categoricity of Peano Arithmetic and show that the critique is highly problematic. I argue that Pederson and Rossberg’s ontological criterion deliver the bizarre result that certain first order subsystems of Peano Arithmetic have a second order ontology. As a consequence, the application of the ontological criterion proposed by Pederson and Rossberg assigns a certain type of ontology to a theory, and a different, richer, ontology to one of its subtheories

Is everything a set? Quine and (Hyper)Pythagoreanism

Every student of Quine knows that his ontology consists in physical objects plus the entities required by mathematics, i.e., sets or classes. He came to this view after trying out nominalism, the view that there are only physical objects, and concluding that it cannot be made to work. Less well known is his having tried out, and ultimately rejected, the seemingly outlandish view that everything is a set or class, called “Pythagoreanism” or “(Hyper)Pythagoreanism.” I think he should not have rejected this view but embraced it, and I try to defend it by appealing only to premises that Quine accepted. Of course students of Quine also know that he came to think of ontology as having less scientific importance than he did in “On What There Is” and Word and Object. The scientific cash value of our whole theory of nature, he came to think, is measured not so much in what it says exists but rather in its structure. I think that Quine’s view on this point is immaterial to the fortunes of Pythagoreanism

On the Axiom of Canonicity

The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox (without reference to the L\"{o}wenheim-Skolem theorem) and a precise representation of the axiom of restriction in set theory proposed much earlier by Abraham Fraenkel. We discuss the main features of Suszko's contribution and hint at its possible further applications

Ernst Zermelo's Project of Infinitary Logic

This paper is a summary of a more comprehensive work Infinitarna Logika Ernsta Zermela (The Infinitary Logic of Ernst Zermelo) being currently under preparation for the research grant KBN 2H01A 00725 Metody nieskończonościowe w teorii definicji (Infinitary methods in the theory of definitions) headed by Professor JANUSZ CZELAKOWSKI at the Institute of Mathematics and Information Science of the University of Opole, Poland. The presentation of Zermelo's ideas is accompanied with some remarks concerning the development of infinitary logic.This paper is a summary of a more comprehensive work Infinitarna Logika Ernsta Zermela (The Infinitary Logic of Ernst Zermelo) being currently under preparation for the research grant KBN 2H01A 00725 Metody nieskończonościowe w teorii definicji (Infinitary methods in the theory of definitions) headed by Professor JANUSZ CZELAKOWSKI at the Institute of Mathematics and Information Science of the University of Opole, Poland. The presentation of Zermelo's ideas is accompanied with some remarks concerning the development of infinitary logic.

Why There is no General Solution to the Problem of Software Verification

How can we be certain that software is reliable? Is there any method that can verify the correctness of software for all cases of interest? Computer scientists and software engineers have informally assumed that there is no fully general solution to the verification problem. In this paper, we survey approaches to the problem of software verification and offer a new proof for why there can be no general solution.The National Security Agency through the Science of Security initiative contract #H98230-18-D-0009