This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically ...we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a
Π
2
-property formalized in an appropriate language for second order number theory is forcible from some
T
⊇
ZFC
+
large cardinals
if and only if it is consistent with the universal fragment of
T
if and only if it is realized in the model companion of
T
. In particular we show that the first order theory of
H
ω
1
is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for
Δ
0
-properties and for all universally Baire sets of reals. We will extend these results also to the theory of
H
ℵ
2
in a follow up of this paper.
Speech acts in mathematics Ruffino, Marco; San Mauro, Luca; Venturi, Giorgio
Synthese (Dordrecht),
10/2021, Letnik:
198, Številka:
10
Journal Article
Recenzirano
We offer a novel picture of mathematical language from the perspective of speech act theory. There are distinct speech acts within mathematics (not just assertions), and, as we intend to show, ...distinct illocutionary force indicators as well. Even mathematics in its most formalized version cannot do without some such indicators. This goes against a certain orthodoxy both in contemporary philosophy of mathematics (which tends to see mathematics as a realm in which no pragmatic features of ordinary language are present) and in speech act theory (which tends to pay attention solely to communication in ordinary language but not to formal languages). As we will comment, the recognition of distinct illocutionary acts within logic and mathematics and the incorporation of illocutionary force indicators in the formal language for both goes back to Frege’s conception of these topics. We are, therefore, going back to a Fregean perspective. This paper is part of a larger project of applying contemporary speech act theory to the scientific language of mathematics in order to uncover the varieties and regular combinations of illocutionary acts (silently) present in it. For reasons of space, we here concentrate only on assertive and declarative acts within mathematics, leaving the investigation of other kinds of acts for a future occasion.
ABSTRACT We present and discuss a change in the introduction of Hilbert’s Grundlagen der Geometrie between the first and the subsequent editions: the disappearance of the reference to the ...independence of the axioms. We briefly outline the theoretical relevance of the notion of independence in Hilbert’s work and we suggest that a possible reason for this disappearance is the discovery that Hilbert’s axioms were not, in fact, independent. In the end we show how this change gives textual evidence for the connection between the notions of independence and simplicity.
In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry ...and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution.
In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of ...models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.
Modelling Afthairetic Modality Venturi, Giorgio; Yago, Pedro
Journal of philosophical logic,
08/2024, Letnik:
53, Številka:
4
Journal Article
Recenzirano
Odprti dostop
Despite their controversial ontological status, the discussion on arbitrary objects has been reignited in recent years. According to the supporting views, they present interesting and unique ...qualities. Among those, two define their nature: their assuming of values, and the way in which they present properties. Leon Horsten has advanced a particular view on arbitrary objects which thoroughly describes the earlier, arguing they assume values according to a
sui generis
modality, which he calls
afthairetic
. In this paper, we offer a general method for defining the minimal system of this modality for any given first-order theory, and possible extensions of it that incorporate further aspects of Horsten’s account. The minimal system presents an unconventional inference rule, which deals with two different notions of derivability. For this reason and the failure of the
Necessitation
rule, in its full generality, the resulting system is non-normal. Then, we provide conditional soundness and completeness results for the minimal system and its extensions.
We present a generalization of the algebra-valued models of
ZF
where the axioms of set theory are not necessarily mapped to the top element of an algebra, but may get intermediate values, in a set of ...designated values. Under this generalization there are many algebras which are neither Boolean, nor Heyting, but that still validate
ZF
.
We propose a novel, ontological approach to studying mathematical propositions and proofs. By “ontological approach” we refer to the study of the categories of beings or concepts that, in their ...practice, mathematicians isolate as fruitful for the advancement of their scientific activity (like discovering and proving theorems, formulating conjectures, and providing explanations). We do so by developing what we call a “formal ontology” of proofs using semantic modeling tools (like RDF and OWL) developed by the computer science community. In this article, (i) we describe this new approach and, (ii) to provide an example, we apply it to the problem of the identity of proofs. We also describe open issues and further applications of this approach (for example, the study of purity of methods). We lay some foundations to investigate rigorously and at large scale intellectual moves and attitudes that underpin the advancement of mathematics through cognitive means (carving out investigationally valuable concepts and techniques) and social means (like communication, collaboration, revision, and criticism of specific categories, inferential patterns, and levels of analysis). Our approach complements other types of analysis of proofs such as reconstruction in a deductive system and examination through a proof-assistant.
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