A Transition to Advanced Mathematics promotes the goals of a ``bridge'' course in mathematics, helping to lead students from courses in the calculus sequence to theoretical upper-level mathematics courses. The text simultaneously promotes the goals of a ``survey'' course, describing the intriguing questions and insights fundamental to many diverse areas of mathematics.

We examine the relationship between two different notions of a structure being Scott set saturated and identify sufficient conditions which guarantee that a structure is uniquely Scott set saturated. We also consider theories representing Scott sets; in particular, we identify a sufficient condition on a theory T so that for any given countable Scott set there exists a completion of T that is saturated with respect to the given Scott set. These results extend Scott's characterization of countable Scott sets via (...) models and completions of Peano arithmetic. (shrink)

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these “generic” structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will “represent” this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity (...) types in the Scott set. For example, if ? is a nonstandard model of PA, then ? represents the Scott set ? = n∈ω | ?⊧“the nth prime divides a” | a∈?.The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set.The second provides a sufficient condition for the existence of a structure ? such that ? represents a countable jump ideal and ? does not compute an enumeration of a given family of sets ?. This second result is of particular interest when the family of sets which cannot be enumerated is ? = Rep[Th(?)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory. (shrink)

We generalize a result on True Arithmetic by Lachlan and Soare to certain other completions of Peano Arithmetic. If $\mathscr{T}$ is a completion of $\mathscr{PA}$, then Rep denotes the family of sets $X \subseteq \omega$ for which there exists a formula $\varphi$ such that for all $n \in \omega$, if $n \in X$, then $\mathscr{T} \vdash \varphi})$ ) and if $n \not\in X$, then $\mathscr{T} \vdash \neg\varphi})$. We show that if $\mathscr{S,J} \subseteq \mathscr{P}$ such that $\mathscr{S}$ is a Scott set, (...) $\mathscr{J}$ is a jump ideal, $\mathscr{S} \subset \mathscr{J}$ and for all $X \in \mathscr{J}$, there exists $C \in \mathscr{S}$ such that C is a "coding" set for the family of subtrees of 2$^{<\omega}$ computable in X, and if $\mathscr{T}$ is a completion of $\mathscr{PA}$ such that Rep = $\mathscr{S}$, then there exists a model $\mathscr{A}$ of $\mathscr{T}$ such that $\mathscr{J}$ is the Scott set of $\mathscr{A}$ and no enumeration of Rep is computable in $\mathscr{A}$. The model $\mathscr{A}$ of $\mathscr{T}$ is obtained via a new notion of forcing. Before proving our main result, we demonstrate the existence of uncountably many different pairs satisfying the conditions of our theorem. This involves a new characterization of 1-generic sets as coding sets for the computable subtrees of 2$^{<\omega}$. In particular, $C \subseteq \omega$ is a coding set for the family of subtrees of 2$^{<\omega}$ computable in X if and only if for all trees T $\subseteq 2^{<\omega}$ computable in X, if $\chi_C$ is a path through T, then there exists $\sigma \in T$ such that $\sigma \subset \chi_C$ and every extension of $\sigma$ is in T. Jockusch noted a connection between 1-generic sets and coding sets for computable subtrees of 2$^{<\omega}$. We show they are identical. (shrink)