Inner-Model Reflection Principles

Barton, Neil; Caicedo, Andrés Eduardo; Fuchs, G.; Hamkins, Joel David; Reitz, Jonas; Schindler, Ralf

doi:10.1007/s11225-019-09860-7

Cited by 4 publications

(2 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…That is, the general consensus is that it is impossible to develop a foundational theory for mathematics in standard first-order language that lacks both pathological features of ZF-all the more so because the downward Löwenheim-Skolem theorem pertains to standard first-order theories in general. It is therefore not surprising that this is not an active research field: those working in the foundations of mathematics are well aware of "our" problem, but ongoing research in set theory focuses on other topics such as, for example, large cardinals, forcing, and inner models-see, e.g., [8][9][10] for some recent works. That being said, the finitely axiomatized theory T that we present here is a nonstandard solution to "our" problem: it entails a rather drastic departure from the language, ontology, and logic of ZF.…”

Section: Related Workmentioning

confidence: 99%

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

Cabbolet¹

2021

Axioms

View full text Add to dashboard Cite

It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

show abstract

Section: Related Workmentioning

confidence: 99%

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

Cabbolet¹

2021

Axioms

View full text Add to dashboard Cite

show abstract

“…Clearly if an inner model thinks that it is not canonically definable, then it is a model of an inner model reflection principle (see Definition 4.4 below). Then [2] ask for upper bounds to the existence of a model of inner model reflection, thus essentially the same question.…”

Section: Then We Havementioning

confidence: 99%

Stably Measurable Cardinals

Welch¹

2020

J. symb. log.

View full text Add to dashboard Cite

We define a weak iterability notion that is sufficient for a number of arguments concerning Σ 1 -definability at uncountable regular cardinals. In particular we give its exact consistency strength first in terms of the second uniform indiscernible for bounded subsets of κ: u 2 (κ), and secondly to give the consistency strength of a property of L ücke's. Theorem The following are equiconsistent:(i) There exists κ which is stably measurable;(ii) for some cardinal κ, u 2 (κ) = (κ);(iii) The Σ 1 -club property holds at a cardinal κ.Here (κ) is the height of the smallest M ≺ Σ 1 H(κ + ) containing κ + 1 and all of H(κ). Let Φ(κ) be the assertion: And a form of converse:Theorem Suppose there is no sharp for an inner model with a strong cardinal. Then in the core model K we have: "∃κΦ(κ)" is (set)-generically absolute ←→ There are arbitrarily large stably measurable cardinals.When u 2 (κ) < (κ) we give some results on inner model reflection. §1. Introduction. There are a number of properties in the literature that fall in the region of being weaker than measurability, but stronger than 0 # , and thus inconsistent with the universe being that of the constructible sets.Actual cardinals of this nature have been well known and are usually of ancient pedigree: Ramsey cardinals, Rowbottom cardinals, Erd ős cardinals, and the like (cf. for example, [7]). Some concepts are naturally not going to prove the existence of such large cardinals, again for example, descriptive set theoretical properties which are about V +1 do not establish the existence of such large cardinals but rather may prove the consistency of large cardinal properties in an inner model. Weak generic absoluteness results, perhaps again only about R, may require some property such as closure of sets under #'s, or more, throughout the whole universe.An example of this is afforded by admissible measurability (defined below):Theorem ([15], Theorem 4, Lemma 1). Let Ψ be the statement:If K is the core model then Ψ K is (set)-generically absolute if and only if there are arbitrarily large admissibly measurable cardinals in K.

show abstract

Are Large Cardinal Axioms Restrictive?

Barton

2023

Philosophia Mathematica

Self Cite

View full text Add to dashboard Cite

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of their usual foundational roles.

show abstract

Inner-Model Reflection Principles

Cited by 4 publications

References 19 publications

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

Stably Measurable Cardinals

Are Large Cardinal Axioms Restrictive?

Contact Info

Product

Resources

About