1,240 research outputs found
Initial segments and end-extensions of models of arithmetic
This thesis is organized into two independent parts. In the first part, we extend the recent work on generic cuts by Kaye and the author. The focus here is the properties of the pairs (M, I) where I is a generic cut of a model M. Amongst other results, we characterize the theory of such pairs, and prove that they are existentially closed in a natural category. In the second part, we construct end-extensions of models of arithmetic that are at least as strong as ATR. Two new constructions are presented. The first one uses a variant of Fodor’s Lemma in ATR to build an internally rather classless model. The second one uses some weak versions of the Galvin–Prikry Theorem in adjoining an ideal set to a model of second-order arithmetic
Expansions, omitting types, and standard systems
Recursive saturation and resplendence are two important notions in models of
arithmetic. Kaye, Kossak, and Kotlarski introduced the notion of arithmetic
saturation and argued that recursive saturation might not be as rigid as first
assumed.
In this thesis we give further examples of variations of recursive
saturation, all of which are connected with expandability properties similar to
resplendence. However, the expandability properties are stronger than
resplendence and implies, in one way or another, that the expansion not only
satisfies a theory, but also omits a type. We conjecture that a special version
of this expandability is in fact equivalent to arithmetic saturation. We prove
that another of these properties is equivalent to \beta-saturation. We also
introduce a variant on recursive saturation which makes sense in the context of
a standard predicate, and which is equivalent to a certain amount of ordinary
saturation.
The theory of all models which omit a certain type p(x) is also investigated.
We define a proof system, which proves a sentence if and only if it is true in
all models omitting the type p(x). The complexity of such proof systems are
discussed and some explicit examples of theories and types with high
complexity, in a special sense, are given.
We end the thesis by a small comment on Scott's problem. We prove that, under
the assumption of Martin's axiom, every Scott set of cardinality <2^{\aleph_0}
closed under arithmetic comprehension which has the countable chain condition
is the standard system of some model of PA. However, we do not know if there
exists any such uncountable Scott sets.Comment: Doctoral thesi
Interpolation in local theory extensions
In this paper we study interpolation in local extensions of a base theory. We
identify situations in which it is possible to obtain interpolants in a
hierarchical manner, by using a prover and a procedure for generating
interpolants in the base theory as black-boxes. We present several examples of
theory extensions in which interpolants can be computed this way, and discuss
applications in verification, knowledge representation, and modular reasoning
in combinations of local theories.Comment: 31 pages, 1 figur
Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
We use the conformal bootstrap to perform a precision study of the operator
spectrum of the critical 3d Ising model. We conjecture that the 3d Ising
spectrum minimizes the central charge c in the space of unitary solutions to
crossing symmetry. Because extremal solutions to crossing symmetry are uniquely
determined, we are able to precisely reconstruct the first several Z2-even
operator dimensions and their OPE coefficients. We observe that a sharp
transition in the operator spectrum occurs at the 3d Ising dimension
Delta_sigma=0.518154(15), and find strong numerical evidence that operators
decouple from the spectrum as one approaches the 3d Ising point. We compare
this behavior to the analogous situation in 2d, where the disappearance of
operators can be understood in terms of degenerate Virasoro representations.Comment: 55 pages, many figures; v2 - refs and comments added, to appear in a
special issue of J.Stat.Phys. in memory of Kenneth Wilso
On groups and initial segments in nonstandard models of Peano Arithmetic
This thesis concerns M-finite groups and a notion of discrete measure in models of Peano Arithmetic. First we look at a measure construction for arbitrary non-M-finite sets via suprema and infima of appropriate M-finite sets. The basic properties of the measures are covered, along with non-measurable sets and the use of end-extensions. Next we look at nonstandard finite permutations, introducing nonstandard symmetric and alternating groups. We show that the standard cut being strong is necessary and sufficient for coding of the cycle shape in the standard system to be equivalent to the cycle being contained within the external normal closure of the nonstandard symmetric group. Subsequently the normal subgroup structure of nonstandard symmetric and alternating groups is given as a result analogous to the result of Baer, Schreier and Ulam for infinite symmetric groups. The external structure of nonstandard cyclic groups of prime order is identified as that of infinite dimensional rational vector spaces and the normal subgroup structure of nonstandard projective special linear groups is given for models elementarily extending the standard model. Finally we discuss some applications of our measure to nonstandard finite groups
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