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The modal logic of arithmetic potentialism and the universal algorithm
I investigate the modal commitments of various conceptions of the philosophy
of arithmetic potentialism. Specifically, I consider the natural potentialist
systems arising from the models of arithmetic under their natural extension
concepts, such as end-extensions, arbitrary extensions, conservative extensions
and more. In these potentialist systems, I show, the propositional modal
assertions that are valid with respect to all arithmetic assertions with
parameters are exactly the assertions of S4. With respect to sentences,
however, the validities of a model lie between S4 and S5, and these bounds are
sharp in that there are models realizing both endpoints. For a model of
arithmetic to validate S5 is precisely to fulfill the arithmetic maximality
principle, which asserts that every possibly necessary statement is already
true, and these models are equivalently characterized as those satisfying a
maximal theory. The main S4 analysis makes fundamental use of the
universal algorithm, of which this article provides a simplified,
self-contained account. The paper concludes with a discussion of how the
philosophical differences of several fundamentally different potentialist
attitudes---linear inevitability, convergent potentialism and radical branching
possibility---are expressed by their corresponding potentialist modal
validities.Comment: 38 pages. Inquiries and commentary can be made at
http://jdh.hamkins.org/arithmetic-potentialism-and-the-universal-algorithm.
Version v3 has further minor revisions, including additional reference
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
Maximin and maximal solutions for linear programming problems with possibilistic uncertainty
We consider linear programming problems with uncertain constraint coefficients described by intervals or, more generally, possi-bility distributions. The uncertainty is given a behavioral interpretation using coherent lower previsions from the theory of imprecise probabilities. We give a meaning to the linear programming problems by reformulating them as decision problems under such imprecise-probabilistic uncer-tainty. We provide expressions for and illustrations of the maximin and maximal solutions of these decision problems and present computational approaches for dealing with them
An algorithmic approach to the existence of ideal objects in commutative algebra
The existence of ideal objects, such as maximal ideals in nonzero rings,
plays a crucial role in commutative algebra. These are typically justified
using Zorn's lemma, and thus pose a challenge from a computational point of
view. Giving a constructive meaning to ideal objects is a problem which dates
back to Hilbert's program, and today is still a central theme in the area of
dynamical algebra, which focuses on the elimination of ideal objects via
syntactic methods. In this paper, we take an alternative approach based on
Kreisel's no counterexample interpretation and sequential algorithms. We first
give a computational interpretation to an abstract maximality principle in the
countable setting via an intuitive, state based algorithm. We then carry out a
concrete case study, in which we give an algorithmic account of the result that
in any commutative ring, the intersection of all prime ideals is contained in
its nilradical
Wrapped M2/M5 Duality
A microscopic accounting of the entropy of a generic 5D supersymmetric
rotating black hole, arising from wrapped M2-branes in Calabi-Yau compactified
M-theory, is an outstanding unsolved problem. In this paper we consider an
expansion around the zero-entropy, zero-temperature, maximally rotating ground
state for which the angular momentum J_L and graviphoton charge Q are related
by J_L^2=Q^3. At J_L=0 the near horizon geometry is AdS_2 x S^3. As J_L^2 goes
to Q^3 it becomes a singular quotient of AdS_3 x S^2: more precisely, a
quotient of the near horizon geometry of an M5 wrapped on a 4-cycle whose
self-intersection is the 2-cycle associated to the wrapped-M2 black hole. The
singularity of the AdS_3 quotient is identified as the usual one associated to
the zero-temperature limit, suggesting that the (0,4) wrapped-M5 CFT is dual
near maximality to the wrapped-M2 black hole. As evidence for this, the
microscopic (0,4) CFT entropy and the macroscopic rotating black hole entropy
are found to agree to leading order away from maximality.Comment: 10 pages, no figure
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