97 research outputs found
On the inevitability of the consistency operator
We examine recursive monotonic functions on the Lindenbaum algebra of
. We prove that no such function sends every consistent
to a sentence with deductive strength strictly between and
. We generalize this result to iterates
of consistency into the effective transfinite. We then prove that for any
recursive monotonic function , if there is an iterate of that
bounds everywhere, then must be somewhere equal to an iterate of
The Complexity of Orbits of Computably Enumerable Sets
The goal of this paper is to announce there is a single orbit of the c.e.
sets with inclusion, \E, such that the question of membership in this orbit
is -complete. This result and proof have a number of nice
corollaries: the Scott rank of \E is \wock +1; not all orbits are
elementarily definable; there is no arithmetic description of all orbits of
\E; for all finite , there is a properly
orbit (from the proof).
A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi
Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry
This paper surveys results related to well-known works of B. Plotkin and V.
Remeslennikov on the edge of algebra, logic and geometry. We start from a brief
review of the paper and motivations. The first sections deal with model theory.
In Section 2.1 we describe the geometric equivalence, the elementary
equivalence, and the isotypicity of algebras. We look at these notions from the
positions of universal algebraic geometry and make emphasis on the cases of the
first order rigidity. In this setting Plotkin's problem on the structure of
automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of
categories is pretty natural and important. Section 2.2 is dedicated to
particular cases of Plotkin's problem. Section 2.3 is devoted to Plotkin's
problem for automorphisms of the group of polynomial symplectomorphisms. This
setting has applications to mathematical physics through the use of model
theory (non-standard analysis) in the studying of homomorphisms between groups
of symplectomorphisms and automorphisms of the Weyl algebra. The last two
sections deal with algorithmic problems for noncommutative and commutative
algebraic geometry. Section 3.1 is devoted to the Gr\"obner basis in
non-commutative situation. Despite the existence of an algorithm for checking
equalities, the zero divisors and nilpotency problems are algorithmically
unsolvable. Section 3.2 is connected with the problem of embedding of algebraic
varieties; a sketch of the proof of its algorithmic undecidability over a field
of characteristic zero is given.Comment: In this review we partially used results of arXiv:1512.06533,
arXiv:math/0512273, arXiv:1812.01883 and arXiv:1606.01566 and put them in a
new contex
Evitable iterates of the consistency operator
Let's fix a reasonable subsystem of arithmetic; why are natural
extensions of pre-well-ordered by consistency strength? In previous work,
an approach to this question was proposed. The goal of this work was to
classify the recursive functions that are monotone with respect to the
Lindenabum algebra of . According to an optimistic conjecture, roughly,
every such function must be equivalent to an iterate of
the consistency operator in the limit.
In previous work the author established the first case of this optimistic
conjecture; roughly, every recursive monotone function is either as weak as the
identity operator in the limit or as strong as in the limit.
Yet in this note we prove that this optimistic conjecture fails already at the
next step; there are recursive monotone functions that are neither as weak as
in the limit nor as strong as in the limit.
In fact, for every , we produce a function that is cofinally as strong
as yet cofinally as weak as
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