Strong axioms of infinity and elementary embeddings

  title={Strong axioms of infinity and elementary embeddings},
  author={Robert Solovay and William N. Reinhardt and Akihiro Kanamori},
  journal={Annals of Mathematical Logic},

Elementary Embeddings and Algebra

This chapter describes algebraic consequences of the existence of a non-trivial elementary embedding of V λ into itself (Axiom I3). In addition to composition, the family of all elementary embeddings

Critical Points in an Algebra of Elementary Embeddings

Extending the Non-extendible: Shades of Infinity in Large Cardinals and Forcing Theories

This is an article whose intended scope is to deal with the question of infinity in formal mathematics, mainly in the context of the theory of large cardinals as it has developed over time since

The κ-closed unbounded filter and supercpmpact cardinals

The results indicate that AD is not an axiom which the authors can justify as intuitively true, a priori or by reason of its consequences, and they thus cannot add it to set theory (as an accepted axiom, evidently true in the cumulative hierarchy of sets).

Recent Advances in Ordinal Analysis: Π1 2 — CA and Related Systems

  • M. Rathjen
  • Mathematics
    Bulletin of Symbolic Logic
  • 1995
Recent success is reported in obtaining an ordinal analysis for the system of Π2 analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to Π1-formulae, giving hope for an ordinals analysis of Z2 in the foreseeable future.


A generic absoluteness theorem is obtained in the theory of I0, from which an analogue of Perfect Set Theorem for “projective” subsets of Vλ+1, and the Posner–Robinson follows as a corollary is obtained.

AD and the supercompactness of ℵ1

  • H. Becker
  • Economics
    Journal of Symbolic Logic
  • 1981
Since the discovery of forcing in the early sixties, it has been clear that many natural and interesting mathematical questions are not decidable from the classical axioms of set theory, ZFC.

The wholeness axiom and Laver sequences

Open Problems in the Theory of Ultrafilters

The purpose of this paper is to present a list of open questions in the theory of ultrafilters. Most of them seem almost impenetrable by the usual methods of set-theory. Needless to say, the list of

Very large set axioms over constructive set theories

. We investigate large set axioms defined in terms of elementary embeddings over constructive set theories, focusing on IKP and CZF . Most previously studied large set axioms, notably the constructive



Elementary Embeddings and Infinitary Combinatorics

One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings, j, from the universe, V into some transitive submodel, M, and it is shown that whenever κ is measurable, there is such j and M .

An undecidable arithmetical statement

Publisher Summary This chapter provides an alternative proof of the existence of formally undecidable sentences. Instead of the arithmetization of syntax and the diagonal process which were used by

Sets Constructible from Sequences of Ultrafilters

Kunen's methods are extended to arbitrary sequences U of ultrafilters and it is shown that in L[U] the generalized continuum hypothesis is true, there is a Souslin tree, andthere is a well-ordering of the reals.

Weakly normal filters and irregular ultrafilters

For a filter over a regular cardinal, least functions and the consequent notion of weak normality are described. The following two results, which make a basic connection between the existence of

Combinatorial characterization of supercompact cardinals

It is proved that supercompact cardinals can be characterized by combinatorial properties which are generalizations of ineffability. 0. Introduction. A A B is the symmetric difference of A and B.

On the role of supercompact and extendible cardinals in logic

It is proved that the existence of supercompact cardinal is equivalent to a certain Skolem-Löwenheim Theorem for second order logic, whereas the existence of extendible cardinal is equivalent to a

On a problem of Erdös, Hajnal and Rado