581 research outputs found
Large transitive models in local {\rm ZFC}
This paper is a sequel to \cite{Tz10}, where a local version of ZFC, LZFC,
was introduced and examined and transitive models of ZFC with properties that
resemble large cardinal properties, namely Mahlo and -indescribable
models, were considered. By analogy we refer to such models as "large models",
and the properties in question as "large model properties". Continuing here in
the same spirit we consider further large model properties, that resemble
stronger large cardinals, namely, "elementarily embeddable", "extendible" and
"strongly extendible", "critical" and "strongly critical", "self-critical'' and
"strongly self-critical", the definitions of which involve elementary
embeddings. Each large model property gives rise to a localization axiom
saying that every set belongs to a transitive model of
ZFC satisfying . The theories + are local analogues of the theories ZFC+"there
is a proper class of large cardinals ", where is a large cardinal
property. If is the property of strong extendibility, it is shown
that proves Powerset and -Collection. In order to
refute over LZFC, we combine the existence of strongly critical models
with an axiom of different flavor, the Tall Model Axiom (). can also
be refuted by plus the axiom saying that "there is a greatest
cardinal", although it is not known if is consistent over LZFC.
Finally Vop\v{e}nka's Principle () and its impact on LZFC are examined. It
is shown that proves Powerset and Replacement, i.e., ZFC
is fully recovered. The same is true for some weaker variants of . Moreover the theories LZFC+ and ZFC+ are shown
to be identical.Comment: 32 page
Definable orthogonality classes in accessible categories are small
We lower substantially the strength of the assumptions needed for the
validity of certain results in category theory and homotopy theory which were
known to follow from Vopenka's principle. We prove that the necessary
large-cardinal hypotheses depend on the complexity of the formulas defining the
given classes, in the sense of the Levy hierarchy. For example, the statement
that, for a class S of morphisms in a locally presentable category C of
structures, the orthogonal class of objects is a small-orthogonality class
(hence reflective) is provable in ZFC if S is \Sigma_1, while it follows from
the existence of a proper class of supercompact cardinals if S is \Sigma_2, and
from the existence of a proper class of what we call C(n)-extendible cardinals
if S is \Sigma_{n+2} for n bigger than or equal to 1. These cardinals form a
new hierarchy, and we show that Vopenka's principle is equivalent to the
existence of C(n)-extendible cardinals for all n. As a consequence, we prove
that the existence of cohomological localizations of simplicial sets, a
long-standing open problem in algebraic topology, is implied by the existence
of arbitrarily large supercompact cardinals. This result follows from the fact
that cohomology equivalences are \Sigma_2. In contrast with this fact, homology
equivalences are \Sigma_1, from which it follows (as is well known) that the
existence of homological localizations is provable in ZFC.Comment: 38 pages; some results have been improved and former inaccuracies
have been correcte
The modal logic of set-theoretic potentialism and the potentialist maximality principles
We analyze the precise modal commitments of several natural varieties of
set-theoretic potentialism, using tools we develop for a general
model-theoretic account of potentialism, building on those of Hamkins, Leibman
and L\"owe, including the use of buttons, switches, dials and ratchets. Among
the potentialist conceptions we consider are: rank potentialism (true in all
larger ); Grothendieck-Zermelo potentialism (true in all larger
for inaccessible cardinals ); transitive-set potentialism
(true in all larger transitive sets); forcing potentialism (true in all forcing
extensions); countable-transitive-model potentialism (true in all larger
countable transitive models of ZFC); countable-model potentialism (true in all
larger countable models of ZFC); and others. In each case, we identify lower
bounds for the modal validities, which are generally either S4.2 or S4.3, and
an upper bound of S5, proving in each case that these bounds are optimal. The
validity of S5 in a world is a potentialist maximality principle, an
interesting set-theoretic principle of its own. The results can be viewed as
providing an analysis of the modal commitments of the various set-theoretic
multiverse conceptions corresponding to each potentialist account.Comment: 36 pages. Commentary can be made about this article at
http://jdh.hamkins.org/set-theoretic-potentialism. Minor revisions in v2;
further minor revisions in v
Set-Theoretic Geology
A ground of the universe V is a transitive proper class W subset V, such that
W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G]
for some W-generic filter G subset P in W . The model V satisfies the ground
axiom GA if there are no such W properly contained in V . The model W is a
bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle
of V is the intersection of all grounds of V . The generic mantle of V is the
intersection of all grounds of all set-forcing extensions of V . The generic
HOD, written gHOD, is the intersection of all HODs of all set-forcing
extensions. The generic HOD is always a model of ZFC, and the generic mantle is
always a model of ZF. Every model of ZFC is the mantle and generic mantle of
another model of ZFC. We prove this theorem while also controlling the HOD of
the final model, as well as the generic HOD. Iteratively taking the mantle
penetrates down through the inner mantles to what we call the outer core, what
remains when all outer layers of forcing have been stripped away. Many
fundamental questions remain open.Comment: 44 pages; commentary concerning this article can be made at
http://jdh.hamkins.org/set-theoreticgeology
Non-Absoluteness of Model Existence at
In [FHK13], the authors considered the question whether model-existence of
-sentences is absolute for transitive models of ZFC, in
the sense that if are transitive models of ZFC with the same
ordinals, and , then if and only if .
From [FHK13] we know that the answer is positive for and under
the negation of CH, the answer is negative for all . Under GCH, and
assuming the consistency of a supercompact cardinal, the answer remains
negative for each , except the case when which is an
open question in [FHK13].
We answer the open question by providing a negative answer under GCH even for
. Our examples are incomplete sentences. In fact, the same
sentences can be used to prove a negative answer under GCH for all
assuming the consistency of a Mahlo cardinal. Thus, the large cardinal
assumption is relaxed from a supercompact in [FHK13] to a Mahlo cardinal.
Finally, we consider the absoluteness question for the
-amalgamation property of -sentences (under
substructure). We prove that assuming GCH, -amalgamation is
non-absolute for . This answers a question from [SS]. The
cases and infinite remain open. As a corollary we get that
it is non-absolute that the amalgamation spectrum of an
-sentence is empty
- …