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
- MathematicsAnn. Pure Appl. Logic
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
- PhilosophyJournal of Symbolic Logic
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
- MathematicsBulletin of Symbolic Logic
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.
AXIOM I0 AND HIGHER DEGREE THEORY
- MathematicsThe Journal of Symbolic Logic
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
- EconomicsJournal of Symbolic Logic
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.…
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…
SHOWING 1-10 OF 68 REFERENCES
Elementary Embeddings and Infinitary Combinatorics
- MathematicsJ. Symb. Log.
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
- MathematicsJ. Symb. Log.
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
- Mathematics, Philosophy
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…