1,105 research outputs found
Representing Scott sets in algebraic settings
We prove that for every Scott set there are -saturated real closed
fields and models of Presburger arithmetic
Models of true arithmetic are integer parts of nice real closed fields
Exploring further the connection between exponentiation on real closed fields
and the existence of an integer part modelling strong fragments of arithmetic,
we demonstrate that each model of true arithmetic is an integer part of an
exponential real closed field that is elementary equivalent to the reals with
exponentiation
Some new results on decidability for elementary algebra and geometry
We carry out a systematic study of decidability for theories of (a) real
vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces,
Banach spaces and metric spaces, all formalised using a 2-sorted first-order
language. The theories for list (a) turn out to be decidable while the theories
for list (b) are not even arithmetical: the theory of 2-dimensional Banach
spaces, for example, has the same many-one degree as the set of truths of
second-order arithmetic.
We find that the purely universal and purely existential fragments of the
theory of normed spaces are decidable, as is the AE fragment of the theory of
metric spaces. These results are sharp of their type: reductions of Hilbert's
10th problem show that the EA fragments for metric and normed spaces and the AE
fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs
of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3
Axiomatics for the external numbers of nonstandard analysis
Neutrices are additive subgroups of a nonstandard model of the real numbers.
An external number is the algebraic sum of a nonstandard real number and a
neutrix. Due to the stability by some shifts, external numbers may be seen as
mathematical models for orders of magnitude. The algebraic properties of
external numbers gave rise to the so-called solids, which are extensions of
ordered fields, having a restricted distributivity law. However, necessary and
sufficient conditions can be given for distributivity to hold. In this article
we develop an axiomatics for the external numbers. The axioms are similar to,
but mostly somewhat weaker than the axioms for the real numbers and deal with
algebraic rules, Dedekind completeness and the Archimedean property. A
structure satisfying these axioms is called a complete arithmetical solid. We
show that the external numbers form a complete arithmetical solid, implying the
consistency of the axioms presented. We also show that the set of precise
elements (elements with minimal magnitude) has a built-in nonstandard model of
the rationals. Indeed the set of precise elements is situated between the
nonstandard rationals and the nonstandard reals whereas the set of non-precise
numbers is completely determined
Logical Dreams
We discuss the past and future of set theory, axiom systems and independence
results. We deal in particular with cardinal arithmetic
The non-unique Universe
The purpose of this paper is to elucidate, by means of concepts and theorems
drawn from mathematical logic, the conditions under which the existence of a
multiverse is a logical necessity in mathematical physics, and the implications
of Godel's incompleteness theorem for theories of everything.
Three conclusions are obtained in the final section: (i) the theory of the
structure of our universe might be an undecidable theory, and this constitutes
a potential epistemological limit for mathematical physics, but because such a
theory must be complete, there is no ontological barrier to the existence of a
final theory of everything; (ii) in terms of mathematical logic, there are two
different types of multiverse: classes of non-isomorphic but elementarily
equivalent models, and classes of model which are both non-isomorphic and
elementarily inequivalent; (iii) for a hypothetical theory of everything to
have only one possible model, and to thereby negate the possible existence of a
multiverse, that theory must be such that it admits only a finite model
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