We show that in the [Formula: see text] extension of a certain Chang-type model of determinacy, if [Formula: see text], then the restriction of the club filter on [Formula: see text] to [Formula: see text] is an ultrafilter in [Formula: see text]. This answers Question 4.11 of [O. Ben-Neria and Y. Hayut, On [Formula: see text]-strongly measurable cardinals, Forum Math. Sigma 11 (2023) e19].

Let [Formula: see text] be an integer. We show that the set of real numbers that are Poisson generic in base b is [Formula: see text]-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base b and not Poisson generic in base b is complete for the class given by the differences between [Formula: see text] sets. We also show that the effective versions of these results hold (...) in the effective Borel hierarchy. (shrink)

We address the question of consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions [Benhamou and Gitik, Ann. Pure Appl. Logic 173 (2022) 103107; Questions 7.8, 7.9], [Benhamou et al., J. Lond. Math. Soc. 108(1) (2023) 190–237; Question 5] and improve theorem [Benhamou et al., J. Lond. Math. Soc. 108(1) (2023) 190–237; Theorem 2.3].

We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over an algebraically closed field (...) of characteristic 0 give rise to triangular rings without the CBP. This also applies to Baudisch’s 2-step nilpotent Lie algebras, which yields the existence of a 2-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP. (shrink)

We answer a question of Slaman and Steel by showing that a version of Martin’s conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin’s conjecture, consists of showing that we can always reduce to the case of a continuous function.

We identify a particular mouse, [Formula: see text], the minimal ladder mouse, that sits in the mouse order just past [Formula: see text] for all n, and we show that [Formula: see text], the set of reals that are [Formula: see text] in a countable ordinal. Thus [Formula: see text] is a mouse set. This is analogous to the fact that [Formula: see text] where [Formula: see text] is the sharp for the minimal inner model with a Woodin cardinal, and (...) [Formula: see text] is the set of reals that are [Formula: see text] in a countable ordinal. More generally [Formula: see text]. The mouse [Formula: see text] and the set [Formula: see text] compose the next natural pair to consider in this series of results. Thus we are proving the mouse set theorem just past projective. Some of this is not new. [Formula: see text] was known in the 1990s. But [Formula: see text] was open until Woodin found a proof in 2018. The main goal of this paper is to give Woodin’s proof. (shrink)

According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal [Formula: see text] and nontrivial elementary embedding [Formula: see text]. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered. [Formula: see text] is the assertion, introduced by Hugh Woodin, that [Formula: see text] is an ordinal and there is an elementary embedding [Formula: see text] with critical point [Formula: see text]. And [Formula: see text] asserts that (...) [Formula: see text] holds for some [Formula: see text]. The axiom [Formula: see text] is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe V (in which case [Formula: see text] must be a limit ordinal), but we assume only ZF. We prove, assuming [Formula: see text] “[Formula: see text] is an even ordinal”, that there is a proper class transitive inner model M containing [Formula: see text] and satisfying [Formula: see text] “there is an elementary embedding [Formula: see text]”; in fact we will have [Formula: see text], where j witnesses [Formula: see text] in M. This result was first proved by the author under the added assumption that [Formula: see text] exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also [Formula: see text] is a limit ordinal and [Formula: see text] holds in V, then the model M will also satisfy [Formula: see text]. We show that [Formula: see text] “[Formula: see text] is even” [Formula: see text] implies [Formula: see text] exists for every [Formula: see text], but if consistent, this theory does not imply [Formula: see text] exists. (shrink)

We prove the preservation theorems for NATP; many of them extend the previously established preservation results for other model-theoretic tree properties. Using them, we also furnish proper examples of NATP theories which are simultaneously TP2 and SOP. First, we show that NATP is preserved by the parametrization and sum of the theories of Fraïssé limits of Fraïssé classes satisfying strong amalgamation property. Second, the preservation of NATP for two kinds of dense/co-dense expansions, i.e. the theories of lovely pairs and of (...) H-structures for geometric theories, and dense/co-dense expansion on vector spaces is proved. Next, we prove the preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also the preservation of NTP1 and NTP2 is considered. Finally, we present some proper examples of NATP theories using the results proved in this paper, including the parametrization of DLO and the expansion of an algebraically closed field by adding a generic linear order. In particular, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP. (shrink)

Halin in 1965 proved that if a graph has n many pairwise disjoint rays for each n then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those results, while not computable, are relatively simple. They only use arithmetic procedures or, equivalently, (...) finitely many iterations of the Turing jump. We show that several Halin-type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán’s Open Questions in Reverse Mathematics in 2011. Some of these theorems including ones in Halin in 1965 are also shown to have unusual proof theoretic strength as well. (shrink)

Given a structure [Formula: see text] and a stably embedded [Formula: see text]-definable set Q, we prove tameness preservation results when enriching the induced structure on Q by some further structure [Formula: see text]. In particular, we show that if [Formula: see text] and [Formula: see text] are stable (respectively, superstable, [Formula: see text]-stable), then so is the theory [Formula: see text] of the enrichment of [Formula: see text] by [Formula: see text]. Assuming simplicity of T, elimination of hyperimaginaries and (...) a further condition on Q related to the behavior of algebraic closure, we also show that simplicity and NSOP1 pass from [Formula: see text] to [Formula: see text]. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of [Formula: see text]. More generally, we show that any stable (respectively, superstable, simple, NIP, NTP2, NSOP1) countable graph can be defined in a stable (respectively, superstable, simple, NIP, NTP2, NSOP1) expansion of [Formula: see text] by some unary predicate [Formula: see text]. (shrink)

We show that it is relatively consistent with ZFC that there is a non-meager set of reals X such that for every non-meager [Formula: see text], there exist distinct [Formula: see text] such that z is computable from the Turing join of x and y.

The aim of Reverse Mathematics (RM for short) is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five (...) systems together are called the ‘Big Five’. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalizations and variations of the aforementioned third-order theorems fall far outside of the Big Five. (shrink)

Bagaria and Magidor introduced the notion of almost strong compactness, which is very close to the notion of strong compactness. Boney and Brooke-Taylor asked whether the least almost strongly compact cardinal is strongly compact. Goldberg gives a positive answer in the case [Formula: see text] holds from below and the least almost strongly compact cardinal has uncountable cofinality. In this paper, we give a negative answer for the general case. Our result also gives an affirmative answer to a question of (...) Bagaria and Magidor. (shrink)