78 research outputs found
On colimits and elementary embeddings
We give a sharper version of a theorem of Rosicky, Trnkova and Adamek, and a
new proof of a theorem of Rosicky, both about colimit preservation between
categories of structures. Unlike the original proofs, which use
category-theoretic methods, we use set-theoretic arguments involving elementary
embeddings given by large cardinals such as alpha-strongly compact and
C^(n)-extendible cardinals.Comment: 17 page
A Short Guide to Gödel's Second Incompleteness Theorem\ud
The usual proof of Godel's second incompleteness theorem for weak theories like I Sigma 'subscript 1' is long and technically cumbersome. The details are rarely given in full and in most cases they are skipped altogether with dismissing vague sentences alluding to the reader's ability to fill them in. In the first part of this note we provide a guide through the main technical points of the usual proof of Godel's theorem for weak theories. In the second part we present a different and simpler proof of the theorem for Zermelo-Fraenkel set theory, due to T. Jech, and we observe that it can be stretched to encompass weak theories, while avoiding many of the technicalities that are required in the usual proofs
Superstrong and other large cardinals are never Laver indestructible
Superstrong cardinals are never Laver indestructible. Similarly, almost huge
cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals,
extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly
superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals,
superstrongly unfoldable cardinals, \Sigma_n-reflecting cardinals,
\Sigma_n-correct cardinals and \Sigma_n-extendible cardinals (all for n>2) are
never Laver indestructible. In fact, all these large cardinal properties are
superdestructible: if \kappa\ exhibits any of them, with corresponding target
\theta, then in any forcing extension arising from nontrivial strategically
<\kappa-closed forcing Q in V_\theta, the cardinal \kappa\ will exhibit none of
the large cardinal properties with target \theta\ or larger.Comment: 19 pages. Commentary concerning this article can be made at
http://jdh.hamkins.org/superstrong-never-indestructible. Minor changes in v
Intrinsic Justification for Large Cardinals and Structural Reflection
We deal with the complex issue of whether large cardinals are intrinsically
justified principles of set theory (we call this the Intrinsicness Issue). In
order to do this, we review, in a systematic fashion, (1.) the abstract
principles that have been formulated to motivate them, as well as (2.) their
mathematical expressions, and assess the justifiability of both on the grounds
of the (iterative) concept of set. A parallel, but closely linked, issue is
whether there exist mathematical principles able to yield all known large
cardinals (we call this the Universality Issue), and we also test principles
for their responses to this issue. Finally, we discuss the first author's
Structural Reflection Principles (SRPs), and their response to Intrinsicness
and Universality. We conclude the paper with some considerations on the global
justifiability of SRPs, and on alternative construals of the concept of set
also potentially able to intrinsically justify large cardinals
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
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