For T any completion of Peano Arithmetic and for n any positive integer, there is a model of T of size $\beth_n$ with no (n + 1)-length sequence of indiscernibles. Hence the Hanf number for omitting types over T, H(T), is at least $\beth_\omega$ . (Now, using an upper bound previously obtained by Julia Knight H (true arithmetic) is exactly $\beth_\omega$ ). If T ≠ true arithmetic, then $H(T) = \beth_{\omega1}$ . If $\delta \not\rightarrow (\rho)^{ , then any completion of (...) Peano Arithmetic has a model of size δ with no set of indiscernibles of size ρ. There are similar results for theories strongly resembling Peano Arithmetic, e.g., ZF + V = L. (shrink)

We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: let [Formula: see text] is the Turing degree of a [Formula: see text] set J ≥T0″}. Let [Formula: see text] such that [Formula: see text] is upward closed in [Formula: see text]. Then there is an ℒ property [Formula: see text] such that [Formula: see text] if and only if there is an A where A ≡T F and [Formula: see text]. (...) A corollary of this is that, for all n ≥ 2, the high n computably enumerable degrees are invariant in the computably enumerable sets. Our work resolves Martin's Invariance Conjecture. (shrink)

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.We will work in the structure of the computably enumerable sets. The language is just inclusion, (...) ⊆. This structure is called ε.All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted. Our notation and definitions are standard and follow Soare [1987]; however we will warm up with some definitions and notation issues so the reader need not consult Soare [1987]. Some historical remarks follow in Section 2.1 and throughout Section 3.We will also consider the quotient structure ε modulo the ideal of finite sets, ε*. ε* is a definable quotient structure of ε since “Χ is finite” is definable in ε; “Χ is finite” iff all subsets of Χ are computable. We use A* to denote the equivalent class of A under the ideal of finite sets. (shrink)

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is ${\Sigma _1^1 }$ -complete. This result and proof have a number of nice corollaries: the Scott rank of ε is $\omega _1^{{\rm{CK}}}$ + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly $\Delta (...) _\alpha ^0 $ orbit (from the proof). (shrink)

We show that if A and $\widehat{A}$ are automorphic via Φ then the structures $S_{R}(A)$ and $S_{R}(\widehat{A})$ are $\Delta_{3}^{0}-isomorphic$ via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.