Nonprincipal Ultrafilter Research Articles

Multi-level nonstandard analysis and the axiom of choice

Model-theoretic frameworks for Nonstandard Analysis depend on the existence of nonprincipal ultrafilters, a strong form of the Axiom of Choice (AC). Hrbacek and Katz, APAL Volume 72 formulate axiomatic nonstandard set theories SPOT and SCOT that are conservative extensions of respectively ZF and ZF + ADC (the Axiom of Dependent Choice), and in which a significant part of Nonstandard Analysis can be developed.The present paper extends these theories to theories with many levels of standardness, called respectively SPOTS and SCOTS. It shows that Jin's recent nonstandard proof of Szemeredi's Theorem can be carried out in SPOTS and that SCOTS is a conservative extension of ZF + ADC.

DISCONTINUOUS HOMOMORPHISMS OF WITH

AbstractAssume that M is a transitive model of $ZFC+CH$ containing a simplified $(\omega _1,2)$ -morass, $P\in M$ is the poset adding $\aleph _3$ generic reals and G is P-generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$ -linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega $ , into the Esterle algebra of formal power series. Therefore it is consistent that $2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X, there exists a discontinuous homomorphism of $C(X)$ , the algebra of continuous real-valued functions on X.

Limit computability and ultrafilters

We study a class of operators on Turing degrees arising naturally from ultrafilters. Suppose U is a nonprincipal ultrafilter on ω. We can then view a sequence of sets A = ( A i ) i ∈ ω as an “approximation” of a set B produced by amalgamating the A i via U: we set lim U ( A ) = { x : { i : x ∈ A i } ∈ U }. This can be extended to the Turing degrees, by defining δ U ( a ) = { lim U ( A ) : A = ( A i ) i ∈ ω ∈ a }. The δ U – which we call “ultrafilter jumps” – resemble classical limit computability in certain ways. In particular, δ U ( a ) is always a Turing ideal containing Δ 2 0 ( a ). However, they are also closely tied to Scott sets: δ U ( a ) is always a Scott set containing a ′ . (This yields an alternate proof of the standard result in reverse mathematics that Weak Konig’s Lemma is strictly weaker than arithmetic comprehension.) Our main result is that the converse also holds: if S is a countable Scott set containing a ′ , then there is some ultrafilter U with δ U ( a ) = S. We then turn to the problem of controlling the action of an ultrafilter jump δ U on two degrees simultaneously, and for example show that there are nontrivial degrees which are “low” for some ultrafilter jump. Finally, we study the structure on the set of ultrafilters arising from the construction U ↦ δ U ; in particular, we introduce a natural preordering on this set and show that it is connected with the classical Rudin–Keisler ordering of ultrafilters. We end by presenting two directions for further research.

Exactly two and exactly three near-coherence classes

We prove that for [Formula: see text] and [Formula: see text] there is a forcing extension with exactly n near-coherence classes of non-principal ultrafilters. We introduce localized versions of Matet forcing and we develop Ramsey spaces of names. The evaluation of some of the new forcings is based on a relative of Hindman’s theorem due to Blass 1987.

On the commutant of B(H) in its ultrapower

Let $${\cal B}\left( H \right)$$ be the algebra of bounded linear operators on a separable infinite-dimensional Hilbert space H. In 2004 Kirchberg asked whether the relative commutant of $${\cal B}\left( H \right)$$ in its ultrapower is trivial. In [13] the authors have shown that under the Continuum Hypothesis the commutant of $${\cal B}\left( H \right)$$ in its ultrapower depends on the choice of the ultrafilter. We here give a combinatorial characterization of the class of non-principal ultrafilters for which this commutant is non-trivial, answering Question 5.2 of [13]. This reduces Kirchberg’s question to a purely set-theoretic question: Can the existence of non-flat ultrafilters be proven in ZFC? In addition, we introduce the notion of quasi P-points and show that for such ultrafilters, and for ultrafilters satisfying the three functions property, the relative commutant of $${\cal B}\left( H \right)$$ in its ultrapower is trivial.

Between reduced powers and ultrapowers

One of our results is a transfer principle between ultrapowers and reduced powers associated with the Fréchet ideal. Although motivated by the Elliott classification programme, this result applies to any axiomatizable category. We also show that there exists a nonprincipal ultrafilter \mathcal{U} on \mathbb{N} such that for every countable (or separable metric) structure B in a countable language the quotient map from the reduced power associated with the Fréchet ideal onto an ultrapower has a right inverse. While the transfer principle is proved without appealing to additional set-theoretic axioms, the conclusion of the latter theorem relies on the Continuum Hypothesis and it is independent of the standard axioms of set theory. We also prove that in the category of C^* -algebras, tensoring with the C^* -algebra of all continuous functions on the Cantor space preserves elementary equivalence. As a side note, neither the Jiang–Su algebra \mathcal{Z} nor any UHF algebra share this property.

Infinitesimal analysis without the Axiom of Choice

It is often claimed that analysis with infinitesimals requires more substantial use of the Axiom of Choice than traditional elementary analysis. The claim is based on the observation that the hyperreals entail the existence of nonprincipal ultrafilters over N, a strong version of the Axiom of Choice, while the real numbers can be constructed in ZF. The axiomatic approach to nonstandard methods refutes this objection. We formulate a theory SPOT in the st-∈-language which suffices to carry out infinitesimal arguments, and prove that SPOT is a conservative extension of ZF. Thus the methods of Calculus with infinitesimals are just as effective as those of traditional Calculus. The conclusion extends to large parts of ordinary mathematics and beyond. We also develop a stronger axiomatic system SCOT, conservative over ZF+ADC, which is suitable for handling such features as an infinitesimal approach to the Lebesgue measure. Proofs of the conservativity results combine and extend the methods of forcing developed by Enayat and Spector.

Special ultrafilters and cofinal subsets of $$({}^\omega \omega , <^*)$$

The interplay between ultrafilters and unbounded subsets of $${}^\omega \omega $$ with the order $$<^*$$ of strict eventual domination is studied. Among the tools are special kinds of non-principal (“free”) ultrafilters on $$\omega $$ . These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order $$\subset ^*$$ of almost inclusion. It is shown that the cofinality of such a base must be either $$\mathfrak {b}$$ , the least cardinality of $$<^*$$ -unbounded (“undominated”) set, or $$\mathfrak {d}$$ , the least cardinality of a $$<^*$$ -cofinal (“dominating”) set. The small uncountable cardinal $$\pi \mathfrak {p}$$ is introduced. Consequences of $$\mathfrak {b}< \pi \mathfrak {p}$$ and of $$\mathfrak r < \mathfrak {d}$$ are explored; in particular, both imply $$\mathfrak {b}< \mathfrak {d}$$ . Here $$\mathfrak r$$ is the reaping number, and is also the least cardinality of a $$\pi $$ -base for a free ultrafilter. Both of these inequalities are shown to occur if there exist simple P-points of different cofinalities (in other words, if $$\mathfrak {b}< \mathfrak {d}$$ and there exist simple $$P_\mathfrak {b}$$ -points and $$P_\mathfrak {d}$$ -points), but this is a long-standing open problem. Six axioms on nonprincipal ultrafilters on $$\omega $$ and the relationships between them are discussed along with various models of set theory in which one or more are known to hold (or are known to fail). The strongest of these, Axiom 1, is that for every free ultrafilter $$\mathcal U$$ and for every $$<^*$$ -unbounded $$<^*$$ -chain C of increasing functions in $${}^\omega \omega $$ , C is also unbounded in the ultraproduct $${}^\omega \omega , <_\mathcal U$$ . The other axioms replace one or both quantifiers with “there exists.” The negation of Axiom 3 in a model provides a family of normal sequentially compact spaces whose product is not countably compact. The question of whether such a family exists in ZFC, even with “normal” weakened to “regular”, is a famous unsolved problem of set-theoretic topology, known as the Scarborough–Stone problem.

Chromatic homotopy theory is asymptotically algebraic

Inspired by the Ax–Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. More precisely, we show that the ultraproduct of the E(n, p)-local categories over any non-principal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by Franke. This shows that chromatic homotopy theory at a fixed height is asymptotically algebraic.

Asymptotic Orthogonalization of Subalgebras in II$_1$ Factors

Let M be a II _1 factor with a von Neumann subalgebra Q\subset M that has infinite index under any projection in Q'\cap M (e.g., if Q'\cap M is diffuse, or if Q is an irreducible subfactor with infinite Jones index). We prove that given any separable subalgebra B of the ultrapower II _1 factor M^\omega , for a nonprincipal ultrafilter \omega on \mathbb{N} , there exists a unitary element u\in M^\omega such that uBu^* is orthogonal to Q^\omega .

The $\omega$-Borel invariant for representations into $\mathrm{SL}(n,\mathbb{C}_\omega)$

Let \Gamma be the fundamental group of a complete hyperbolic 3 -manifold M with toric cusps. By following [3] we define the \omega -Borel invariant \beta_n^\omega(\rho_\omega) associated to a representation \rho_\omega\colon \Gamma \rightarrow \mathrm{SL}(n,\mathbb C_\omega) , where \mathbb C_\omega is a field introduced by [18] which can be constructed as a quotient of a suitable subset of \mathbb C^\N with the data of a non-principal ultrafilter \omega on \N and a real divergent sequence \lambda_l such that \lambda_l \geq 1 . Since a sequence of \omega -bounded representations \rho_l into \mathrm{SL}(n,\mathbb C) determines a representation \rho_\omega into \operatorname{SL}(n,\mathbb C_\omega) , for n=2 we study the relation between the invariant \beta^\omega_2(\rho_\omega) and the sequence of Borel invariants \beta_2(\rho_l) . We conclude by showing that if a sequence of representations \rho_l\colon \Gamma \rightarrow \mathrm{SL}(2,\mathbb C) induces a representation \rho_\omega\colon \Gamma \rightarrow \operatorname{SL}(2,\mathbb C_\omega) which determines a reducible action on the asymptotic cone C_\omega(\mathbb{H}^3,d/\lambda_l,O) with non-trivial length function, then it holds \beta^\omega_2(\rho_\omega)=0 .

Discontinuous homomorphisms, selectors, and automorphisms of the complex field

We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$. This shows that a nonprincipal ultrafilter on the integers is not sufficient to construct a discontinuous automorphism of the complex field, confirming a conjecture of Simon Thomas. This is an improved version of our paper in the Proceedings of the American Mathematical Society, which used a weak version of the Axiom of Choice for the same result.

HIGH DIMENSIONAL ELLENTUCK SPACES AND INITIAL CHAINS IN THE TUKEY STRUCTURE OF NON-P-POINTS

AbstractThe generic ultrafilter${\cal G}_2 $forced by${\cal P}\left( {\omega \times \omega } \right)/\left( {{\rm{Fin}} \otimes {\rm{Fin}}} \right)$was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters ([1]), but it was left open where exactly in the Tukey order it lies. We prove${\cal G}_2 $that is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each${\cal G}_2 $, the collection of all nonprincipal ultrafilters Tukey reducible to the generic ultrafilter${\cal G}_k $forced by${\cal P}\left( {\omega ^k } \right)/{\rm{Fin}}^{ \otimes k} $forms a chain of lengthk. Essential to the proof is the extraction of a dense subsetεkfrom (Fin⊗k)+which we prove to be a topological Ramsey space. The spacesεk,k≥ 2, form a hierarchy of high dimensional Ellentuck spaces. New Ramsey-classification theorems for equivalence relations on fronts on εkare proved, extending the Pudlák–Rödl Theorem for fronts on the Ellentuck space, which are applied to find the Tukey and Rudin–Keisler structures below${\cal G}_k $.

THE NEXT BEST THING TO A P-POINT

AbstractWe study ultrafilters on ω2 produced by forcing with the quotient of ${\cal P}$(ω2) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]&lt;ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters.

Model theory and the QWEP conjecture

We observe that Kirchberg’s QWEP conjecture is equivalent to the statement that C∗(F) is elementarily equivalent to a QWEP C∗ algebra. We also make a few other model-theoretic remarks about WEP and LLP C∗ algebras. For the sake of simplicity, all C∗ algebras in this note are assumed to be unital. Suppose that B is a C∗ algebra and A is a subalgebra. We say that A is relatively weakly injective in B if there is a u.c.p. map φ : B → A∗∗ such that φ|B = idA; such a map is referred to as a weak conditional expectation. (We view A as canonically embedded in its double dual.) A C∗ algebra A is said to have the weak expectation property (or be WEP) if it is relatively weakly injective in every extension and A is said to be QWEP if it is the quotient of a WEP algebra. Kirchberg’s QWEP Conjecture states that every separable C∗ algebra is QWEP. In the seminal paper [9], Kirchberbg proved that the QWEP Conjecture is equivalent to the Connes Embedding Problem (CEP), namely that every finite von Neumann algebra embeds into an ultrapower of the hyperfinite II1 factor. If F is the free group on countably many generators, then using the fact that C∗(F) is surjectively universal, we see that the QWEP conjecture is equivalent to the statement that C∗(F) is QWEP. The main point of this note is to point to an a priori weaker equivalent statement of the QWEP conjecture: Theorem 1. The QWEP conjecture is equivalent to the statement that C∗(F) is elementarily equivalent to a QWEP C∗ algebra. Here, two C∗ algebras A and B are elementarily equivalent if they have the same first-order theories in the sense of model theory. (Here, we work in the signature for unital C∗ algebras.) The next lemma is probably well known to the experts but since we could not locate it in the literature we include a proof here. Proposition 2. Let A be a C∗ algebra and ω a nonprincipal ultrafilter on some (possibly uncountable) index set. The work here was partially supported by NSF CAREER grant DMS-1349399.

A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 2

Motivated by Tukey classification problems and building on work in Part 1, we develop a new hierarchy of topological Ramsey spaces R α \mathcal {R}_{\alpha } , α &gt; ω 1 \alpha &gt;\omega _1 . These spaces form a natural hierarchy of complexity, R 0 \mathcal {R}_0 being the Ellentuck space, and for each α &gt; ω 1 \alpha &gt;\omega _1 , R α + 1 \mathcal {R}_{\alpha +1} coming immediately after R α \mathcal {R}_{\alpha } in complexity. Associated with each R α \mathcal {R}_{\alpha } is an ultrafilter U α \mathcal {U}_{\alpha } , which is Ramsey for R α \mathcal {R}_{\alpha } , and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on R α \mathcal {R}_{\alpha } , 2 ≤ α &gt; ω 1 2\le \alpha &gt;\omega _1 . These form a hierarchy of extensions of the Pudlak-Rödl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our Ramsey-classification theorems to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U α \mathcal {U}_{\alpha } , for each 2 ≤ α &gt; ω 1 2\le \alpha &gt;\omega _1 : Every nonprincipal ultrafilter which is Tukey reducible to U α \mathcal {U}_{\alpha } is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of rapid p-points. Moreover, we show that the Tukey types of nonprincipal ultrafilters Tukey reducible to U α \mathcal {U}_{\alpha } form a descending chain of rapid p-points of order type α + 1 \alpha +1 .

Ultrafilters in reverse mathematics

We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA0, ATR0, and [Formula: see text].

Model theory of operator algebras II: model theory

We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on ℕ are isomorphic even when the Continuum Hypothesis fails.

A new class of Ramsey-Classification Theorems and their Applications in the Tukey Theory of Ultrafilters, Parts 1 and 2

Abstract Motivated by Tukey classification problems, we develop a new hierarchy of topological Ramsey spaces R α , α ω 1 . These spaces form a natural hierarchy of complexity, R 0 being the Ellentuck space [Erik Ellentuck, A new proof that analytic sets are Ramsey, Journal of Symbolic Logic 39 (1974), 163–165], and for each α ω 1 , R α + 1 coming immediately after R α in complexity. Associated with each R α is an ultrafilter U α , which is Ramsey for R α , and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on R α , 1 ⩽ α ω 1 . These form a hierarchy of extensions of the Pudlak-Rodl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our Ramsey-classification theorems to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U α , for each 1 ⩽ α ω 1 : Every nonprincipal ultrafilter which is Tukey reducible to U α is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of rapid p-points. Moreover, we show that the Tukey types of nonprincipal ultrafilters Tukey reducible to U α form a descending chain of rapid p-points of order type α + 1 .

Strongly complete almost maximal left invariant topologies on groups

Let G be a countably infinite group. A topology on G is left invariant if left translations are continuous. A left invariant topology is strongly complete if it is regular and for every partition {Un:n<ω} of G into open sets, there is a neighborhood V of 1 such that for every x∈G, {n<ω:(xV)∩Un≠∅} is finite. We show that assuming MA, for every n∈N, there is a strongly complete left invariant topology T on G with exactly n nonprincipal ultrafilters converging to 1, and in the case G=⨁ωZ2, T can be chosen to be a group topology. We also show that it is consistent with ZFC that if G can be embedded algebraically into a compact group, then there are no such topologies on G.