• Corpus ID: 55769713

Large Cardinals and the Iterative Conception of Set

@inproceedings{Barton2018LargeCA,
  title={Large Cardinals and the Iterative Conception of Set},
  author={Neil Barton},
  year={2018}
}
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 for the iterative conception, and assert that the length of the iterative stages is as long as possible. In this paper, we argue that whether or not large cardinal principles count as maximality principles depends on prior commitments concerning the richness of the subset… 
1 Citations

Might All Infinities Be the Same Size?

  • A. Pruss
  • Philosophy
    Australasian Journal of Philosophy
  • 2019
ABSTRACT Cantor proved that no set has a bijection between itself and its power set. This is widely taken to have shown that there infinitely many sizes of infinite sets. The argument depends on the

References

SHOWING 1-10 OF 49 REFERENCES

A multiverse perspective on the axiom of constructiblity

I shall argue that the commonly held V 6 L via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist

Independence and Large Cardinals

The manner in which large cardinal axioms provide a canonical means for climbing the hierarchy of interpretability and serve as an intermediary in the comparison of systems from conceptually distinct domains is discussed.

How to be a minimalist about sets

According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical

Large Cardinals and Determinacy

The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC.

Maximality and ontology: how axiom content varies across philosophical frameworks

Critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception of set is provided, arguing that forms of multiversism and actualism face complementary problems.

Useful Axioms

It is argued that to a large extent this is possible for certain initial fragments of the universe of sets and the question of whether and to what extent forcing axioms can provide a "complete" semantics for set theory is addressed.

The Graph Conception of Set

It is argued that the axiom AFA is justified on the conception, which provides, contra Rieger (Mind 109:241–253, 2000), a rationale for restricting attention to the system based on this axiom.

Maximality Principles in Set Theory

In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality

Set theory and the continuum problem

Part I. General background. Some basics of class-set theory. The natural number. Superinduction, well ordering and choice. Ordinal numbers. Order isomorphism and transfinite recursion. Rank.

Bounded forcing axioms as principles of generic absoluteness

It is shown that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a $\Sigma^1_3$ sentence with parameters is forceable, then it is true.