9,545 research outputs found
Incomparable -like models of set theory
We show that the analogues of the Hamkins embedding theorems, proved for the
countable models of set theory, do not hold when extended to the uncountable
realm of -like models of set theory. Specifically, under the
hypothesis and suitable consistency assumptions, we show that
there is a family of many -like models of ZFC, all
with the same ordinals, that are pairwise incomparable under embeddability;
there can be a transitive -like model of ZFC that does not embed into
its own constructible universe; and there can be an -like model of PA
whose structure of hereditarily finite sets is not universal for the
-like models of set theory.Comment: 15 pages. Commentary concerning this article can be made at
http://jdh.hamkins.org/incomparable-omega-one-like-models-of-set-theor
Cardinal arithmetic for skeptics
We present a survey of some results of the pcf-theory and their applications
to cardinal arithmetic. We review basics notions (in section 1), briefly look
at history in section 2 (and some personal history in section 3). We present
main results on pcf in section 5 and describe applications to cardinal
arithmetic in section 6. The limitations on independence proofs are discussed
in section 7, and in section 8 we discuss the status of two axioms that arise
in the new setting. Applications to other areas are found in section 9.Comment: 14 page
Satisfaction is not absolute
We prove that the satisfaction relation
of first-order logic is not absolute between models of set theory having the
structure and the formulas all in common. Two models of
set theory can have the same natural numbers, for example, and the same
standard model of arithmetic ,
yet disagree on their theories of arithmetic truth; two models of set theory
can have the same natural numbers and the same arithmetic truths, yet disagree
on their truths-about-truth, at any desired level of the iterated
truth-predicate hierarchy; two models of set theory can have the same natural
numbers and the same reals, yet disagree on projective truth; two models of set
theory can have the same or the same
rank-initial segment , yet disagree on which
assertions are true in these structures.
On the basis of these mathematical results, we argue that a philosophical
commitment to the determinateness of the theory of truth for a structure cannot
be seen as a consequence solely of the determinateness of the structure in
which that truth resides. The determinate nature of arithmetic truth, for
example, is not a consequence of the determinate nature of the arithmetic
structure itself, but rather, we argue, is an
additional higher-order commitment requiring its own analysis and
justification.Comment: 34 pages. Commentary concerning this article can be made at
http://jdh.hamkins.org/satisfaction-is-not-absolut
The power set function
We survey old and recent results on the problem of finding a complete set of
rules describing the behavior of the power function, i.e. the function which
takes a cardinal to the cardinality of its power
Classical realizability and arithmetical formul{\ae}
In this paper we treat the specification problem in classical realizability
(as defined in [20]) in the case of arithmetical formul{\ae}. In the continuity
of [10] and [11], we characterize the universal realizers of a formula as being
the winning strategies for a game (defined according to the formula). In the
first section we recall the definition of classical realizability, as well as a
few technical results. In Section 5, we introduce in more details the
specification problem and the intuition of the game-theoretic point of view we
adopt later. We first present a game , that we prove to be adequate and
complete if the language contains no instructions "quote" [18], using
interaction constants to do substitution over execution threads. Then we show
that as soon as the language contain "quote", the game is no more complete, and
present a second game that is both adequate and complete in the general
case. In the last Section, we draw attention to a model-theoretic point of
view, and use our specification result to show that arithmetical formul{\ae}
are absolute for realizability models.Comment: arXiv admin note: text overlap with arXiv:1101.4364 by other author
Models of expansions of N with no end extensions
We deal with models of Peano arithmetic (specifically with a question of Ali
Enayat). The methods are from creature forcing. We find an expansion of N such
that its theory has models with no (elementary) end extensions. In fact there
is a Borel uncountable set of subsets of N such that expanding N by any
uncountably many of them suffice. Also we find arithmetically closed A with no
definably closed ultrafilter on it
Dependent choice, properness, and generic absoluteness
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory
Reduced Perplexity: A simplified perspective on assessing probabilistic forecasts
A simple, intuitive approach to the assessment of probabilistic inferences is
introduced. The Shannon information metrics are translated to the probability
domain. The translation shows that the negative logarithmic score and the
geometric mean are equivalent measures of the accuracy of a probabilistic
inference. Thus there is both a quantitative reduction in perplexity, which is
the inverse of the geometric mean of the probabilities, as good inference
algorithms reduce the uncertainty and a qualitative reduction due to the
increased clarity between the original set of probabilistic forecasts and their
central tendency, the geometric mean. Further insight is provided by showing
that the R\'enyi and Tsallis entropy functions translated to the probability
domain are both the weighted generalized mean of the distribution. The
generalized mean of probabilistic forecasts forms a spectrum of performance
metrics referred to as a Risk Profile. The arithmetic mean is used to measure
the decisiveness, while the -2/3 mean is used to measure the robustness.Comment: 21 pages, 5 figures, conference paper presented at Recent Advances in
Info-Metrics, Washington, DC, 2014. Accepted for a book chapter in "Recent
innovations in info-metrics: a cross-disciplinary perspective on information
and information processing" by Oxford University Pres
The universal finite set
We define a certain finite set in set theory and prove
that it exhibits a universal extension property: it can be any desired
particular finite set in the right set-theoretic universe and it can become
successively any desired larger finite set in top-extensions of that universe.
Specifically, ZFC proves the set is finite; the definition has
complexity , so that any affirmative instance of it is
verified in any sufficiently large rank-initial segment of the universe
; the set is empty in any transitive model and others; and if
defines the set in some countable model of ZFC and y\of z
for some finite set in , then there is a top-extension of to a model
in which defines the new set . Thus, the set shows that no
model of set theory can realize a maximal theory with its natural
number parameters, although this is possible without parameters. Using the
universal finite set, we prove that the validities of top-extensional
set-theoretic potentialism, the modal principles valid in the Kripke model of
all countable models of set theory, each accessing its top-extensions, are
precisely the assertions of S4. Furthermore, if ZFC is consistent, then there
are models of ZFC realizing the top-extensional maximality principle.Comment: 16 pages. Commentary can be made at
http://jdh.hamkins.org/the-universal-finite-set. Version 2 makes minor
changes, including a footnote concerning the history of the universal
algorithm and additional reference
Chains of saturated models in AECs
We study when a union of saturated models is saturated in the framework of
tame abstract elementary classes (AECs) with amalgamation. We prove:
If is a tame AEC with amalgamation satisfying a natural definition of
superstability (which follows from categoricity in a high-enough cardinal),
then for all high-enough :
* The union of an increasing chain of -saturated models is
-saturated.
* There exists a type-full good -frame with underlying class the
saturated models of size .
* There exists a unique limit model of size .
Our proofs use independence calculus and a generalization of averages to this
non first-order context.Comment: 27 page
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