6 research outputs found
Maximality Principles and Ressurection Axioms under a Laver-generic large cardinal
Set-theoretic axioms formulated in terms of existence of a Laver-generic
large cardinal were introduced in [16] and studied further in [17], [18], [20].
These axioms, let us call them Laver-genericity axioms, claim the existence of
a -Laver generic large cardinal for various classes
-of proper or semi-proper posets, and they still vary depending on
the notions of large cardinal involved, and a modification (tightness) of the
definition of Laver-genericity. Laver-genericity axioms we consider here are
divided into three groups depending on whether they imply that the Laver
generic large cardinal \k{appa} is , or it is
, or else it is very large and (see
the Trichotomy Theorem (Theorem 3.5)). Many set-theoretic axioms and principles
considered in the recent development of set theory follow from a
Laver-genericity axiom in one of these three groups, and by this, they are
placed uniformly in a global context (see Figure 3). In spite of this very
strong unifying feature of the Laver genericity axioms, we show that Maximality
Principle (MP) without parameters is independent over ZFC with any of the
Laver-genericity axioms we consider in our present context (Theorem 4.8,
Theorem 5.11). Similar independence is also shown for parameterized versions of
Maximality Principles (Theorem 6.1, Theorem 6.5). In contrast to these
independence results, we can show that local versions of Maximality Principle
as well as versions of Resurrection Axioms including the Unbounded Resurrection
Axioms of Tsaprounis follow from the existence of a tightly Laver-generic large
cardinal for a strong enough notion of large cardinal (Theorem 6.6, Theorem
7.1, Theorem 7.2)
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Set Theory
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Generic compactness reformulated
We point out a connection between reflection principles and generic large cardinals. One principle of pure reflection is introduced that is as strong as generic supercompactness of omega(2) by sigma-closed forcing. This new concept implies CH and extends the reflection principles for stationary sets in a canonical way