First order reasoning about hyperintegers can prove things about sets of integers. In the author’s paper Nonstandard Arithmetic and Reverse Mathematics, Bulletin of Symbolic Logic 12 100–125, it was shown that each of the “big five” theories in reverse mathematics, including the base theory , has a natural nonstandard counterpart. But the counterpart of has a defect: it does not imply the Standard Part Principle that a set exists if and only if it is coded by a hyperinteger. In (...) this paper we find another nonstandard counterpart, *RCA0′, that does imply the Standard Part Principle. (shrink)

We settle a number of questions concerning definability in first order logic with an extra predicate symbol ranging over semi-linear sets. We give new results both on the positive and negative side: we show that in first-order logic one cannot query a semi-linear set as to whether or not it contains a line, or whether or not it contains the line segment between two given points. However, we show that some of these queries become definable if one makes (...) small restrictions on the semi-linear sets considered. (shrink)

We show that each of the five basic theories of second order arithmetic that play a central role in reverse mathematics has a natural counterpart in the language of nonstandard arithmetic. In the earlier paper [3] we introduced saturation principles in nonstandard arithmetic which are equivalent in strength to strong choice axioms in second order arithmetic. This paper studies principles which are equivalent in strength to weaker theories in second order arithmetic.

The randomization of a complete first-order theory [Formula: see text] is the complete continuous theory [Formula: see text] with two sorts, a sort for random elements of models of [Formula: see text] and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of [Formula: see text]. We show that if [Formula: see text] has the exchange property and [Formula: see text], then [Formula: see text] has a strict (...) independence relation in the home sort, and hence is real rosy. In particular, if [Formula: see text] is o-minimal, then [Formula: see text] is real rosy. (shrink)

This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn on properties S that says "there are n components having S". We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As (...) a corollary, if the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.We introduce another operation alt on properties which has the same relationship with the Circuit Value Problem as reach has with the Directed Reachability Problem. We use alt to show that Πn⊈ FO, Σn⊈ FO, and Δn+1⊈ FOB, solving some open problems raised in [Mat98]. (shrink)

In a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, then there is a κ-saturated nonstandard (...) universe in which the hyperreal numbers have the λ-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic. (shrink)

In a nonstandard universe, the $\kappa$-saturation property states that any family of fewer than $\kappa$ internal sets with the finite intersection property has a nonempty intersection. An ordered field $F$ is said to have the $\lambda$-Bolzano-Weierstrass property iff $F$ has cofinality $\lambda$ and every bounded $\lambda$-sequence in $F$ has a convergent $\lambda$-subsequence. We show that if $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, then there is a $\kappa$-saturated nonstandard (...) universe in which the hyperreal numbers have the $\lambda$-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic. (shrink)

A general metatheorem is proved which reduces a wide class of statements about continuous time stochastic processes to statements about discrete time processes. We introduce a strong language for stochastic processes, and a concept of forcing for sequences of discrete time processes. The main theorem states that a sentence in the language is true if and only if it is forced. Although the stochastic process case is emphasized in order to motivate the results, they apply to a wider class (...) of random variables. At the end of the paper we illustrate how the theorem can be used with three applications involving submartingales. (shrink)

First order reasoning about hyperintegers can prove things about sets of integers. In the author’s paper Nonstandard Arithmetic and Reverse Mathematics, Bulletin of Symbolic Logic 12 100–125, it was shown that each of the “big five” theories in reverse mathematics, including the base theory, has a natural nonstandard counterpart. But the counterpart of has a defect: it does not imply the Standard Part Principle that a set exists if and only if it is coded by a hyperinteger. In this (...) paper we find another nonstandard counterpart, *RCA0′, that does imply the Standard Part Principle. (shrink)

In prior work [7] we considered networks of agents who have knowledge bases in first order logic, and report facts to their neighbors that are in their common languages and are provable from their knowledge bases, in order to help a decider verify a single sentence. In report complete networks, the signatures of the agents and the links between agents are rich enough to verify any deciderʼs sentence that can be proved from the combined knowledge base. This paper (...) introduces a more general setting where new observations may be added to knowledge bases and the decider must choose a sentence from a set of alternatives. We consider the question of when it is possible to prepare in advance a finite plan to generate reports within the network. We obtain conditions under which such a plan exists and is guaranteed to produce the right choice under any new observations. (shrink)

Gaifman's normal form theorem showed that every first-order sentence of quantifier rank n is equivalent to a Boolean combination of “scattered local sentences”, where the local neighborhoods have radius at most 7n−1. This bound was improved by Lifsches and Shelah to 3×4n−1. We use Ehrenfeucht–Fraïssé type games with a “shrinking horizon” to get a spectrum of normal form theorems of the Gaifman type, depending on the rate of shrinking. This spectrum includes the result of Lifsches and Shelah, with a (...) more easily understood proof and with the bound on the radius improved to 4n−1. We also obtain bounds for a normal form theorem of Schwentick and Barthelmann. (shrink)

We consider extensions of Peano arithmetic suitable for doing some of nonstandard analysis, in which there is a predicate N(x) for an elementary initial segment, along with axiom schemes approximating ω 1 -saturation. We prove that such systems have the same proof-theoretic strength as their natural analogues in second order arithmetic. We close by presenting an even stronger extension of Peano arithmetic, which is equivalent to ZF for arithmetic statements.

M. E. Rudin proved, under CH, that for each P-point p there exists a P-point q strictly RK-greater than p. This result was proved under ${\mathfrak {p}= \mathfrak {c}}$ by A. Blass, who also showed that each RK-increasing $ \omega $ -sequence of P-points is upper bounded by a P-point, and that there is an order embedding of the real line into the class of P-points with respect to the RK-ordering. In this paper, the results cited above are proved (...) under the assumption that $\mathfrak { b}=\mathfrak {c}$. A. Blass asked in 1973 which ordinals can be embedded in the set of P-points, and pointed out that such an ordinal cannot be greater than $ \mathfrak {c}^{+}$. In this paper it is proved, under $\mathfrak {b}=\mathfrak {c}$, that for each ordinal $\alpha < \mathfrak {c}^{+}$, there is an order embedding of $ \alpha $ into P-points. It is also proved, under $\mathfrak {b}=\mathfrak {c}$, that there is an embedding of the long line into P-points. (shrink)

In this paper, we study some new examples of ideals on $$\omega $$ with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order—known as the weak Rudin–Keisler order—and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin (Note Mat 11:177–214, 1991) on the Tukey order, we also show that there is an analytic P-ideal (...) above all other analytic P-ideals in the weak Rudin–Keisler order. (shrink)

A 2-affable ultrafilter has only finitely many predecessors in the Rudin-Keisler ordering of isomorphism classes of ultrafilters over the natural numbers. If the continuum hypothesis is true, then there is an ℵ 1 -sequence of ultrafilters D α such that the strict Rudin-Keisler predecessors of D α are precisely the isomorphs of the D β 's for $\beta.

In this paper, we provide a new characterization of Keisler’s order in terms of saturation of Boolean ultrapowers. To do so, we apply and expand the framework of ‘separation of variables’ recently developed by Malliaris and Shelah. We also show that good ultrafilters on Boolean algebras are precisely the ones which capture the maximum class in Keisler’s order, answering a question posed by Benda in 1974.

We show that the analysis of Keisler’s order can be localized to the study of φ-types. Specifically, if is a regular ultrafilter on λ such that and M is a model whose theory is countable, then is λ+-saturated iff it realizes all φ-types of size λ.

There is a Turing functional $\Phi $ taking $A^\prime $ to a theory $T_A$ whose complexity is exactly that of the jump of A, and which has the property that $A \leq _T B$ if and only if $T_A \trianglelefteq T_B$ in Keisler’s order. In fact, by more elaborate means and related theories, we may keep the complexity at the level of A without using the jump.

The Rudin-Keisler ordering of ultrafilters is extended to complete Boolean algebras and characterised in terms of elementary embeddings of Boolean ultrapowers. The result is applied to show that the Rudin-Keisler poset of some atomless complete Boolean algebras is nontrivial.

We prove a model theoretic generalization of an extension of the Keisler-Morley theorem for countable recursively saturated models of theories having a K-like model, where K is an inaccessible cardinal.

Keisler in [7] proved that for a strong limit cardinal κ and a singular cardinal λ, the transfer relation κ → λ holds. We analyze the λ -like models produced in the proof of Keisler's transfer theorem when κ is further assumed to be regular. Our main result shows that with this extra assumption, Keisler's proof can be modified to produce a λ -like model M with built-in Skolem functions that satisfies the following two properties: M is (...) generated by a subset C of order-type λ. M can be written as union of an elementary end extension chain 〈Ni: i < δ 〉 such that for each i < δ, there is an initial segment Ci of C with Ci ⊆ Ni, and Ni ∩ = ∅. (shrink)

An ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size.

We discuss the finite-to-one Rudin-Keisler ordering of ultrafilters on the natural numbers, which we baptize the Rudin-Blass ordering in honour of Professor Andreas Blass who worked extensively in the area. We develop and summarize many of its properties in relation to its bounding and dominating numbers, directedness, and provide applications to continuum theory. In particular, we prove in ZFC alone that there exists an ultrafilter with no Q-point below in the Rudin-Blass ordering.

We consider the question, of longstanding interest, of realizing types in regular ultrapowers. In particular, this is a question about the interaction of ultrafilters and theories, which is both coarse and subtle. By our prior work it suffices to consider types given by instances of a single formula. In this article, we analyze a class of formulas φ whose associated characteristic sequence of hypergraphs can be seen as describing realization of first- and second-order types in ultrapowers on one hand, (...) and properties of the corresponding ultrafilters on the other. These formulas act, via the characteristic sequence, as points of contact with the ultrafilter D, in the sense that they translate structural properties of ultrafilters into model-theoretically meaningful properties and vice versa. Such formulas characterize saturation for various key theories , yet their scope in Keisler’s order does not extend beyond Tfeq. The proof applies Shelah’s classification of second-order quantifiers. (shrink)

Provability, Computability and Reflection.

We prove a local normal form theorem of the Gaifman type for the infinitary logic L∞ωω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht-Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L∞ωω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form ψ, where ψ has counting quantifiers restricted to the -neighborhood of y.