Mathematics

Recall that I 0,λ is the assertion that λ is a limit ordinal and there is an elementary embedding j:L( V λ+1 )→L( V λ+1 ) with critical point <λ . This hypothesis is usually studied assuming ZFC holds in the full universe V , but we assume only ZF. We show, assuming ZF+ I 0,λ , that there is a proper class transitive inner model M containing V λ+1 and modelling the theory ZF+ I 0,λ +"there is an elementary embedding j: V λ+2 → V λ+2 ". By employing the results of the papers \emph{Periodicity in the cumulative hierarchy} and \emph{Even ordinals and the Kunen inconsistency}, we also show that this generalizes to all even ordinals λ . In the case that λ is a limit and λ -DC holds in V , then the model M constructed also satisfies λ -DC. We also show that if ZFC+ I 0,λ is consistent, then it does not imply the existence of V # λ+1 . Likewise, if ZF+" λ is an even ordinal and j:L( V λ+1 )→L( V λ+1 ) is elementary with critical point <λ " is consistent, then it does not imply the existence of V # λ+1 . We show that, however, this theory does imply that A # exists for every A∈ V λ+1 . We also make some further obserations on L( V λ+1 ) under such hypotheses.

In this paper, we use Gödel's incompleteness theorem as a case study for investigating mathematical depth. We take for granted the widespread judgment by mathematical logicians that Gödel's incompleteness theorem is deep, and focus on the philosophical question of what its depth consists in. We focus on the methodological study of the depth of Gödel's incompleteness theorem, and propose three criteria to account for its depth: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of Gödel's incompleteness theorem.

In this paper we study the notion of Salem set from the point of view of descriptive set theory. We first work in the hyperspace K([0,1]) of compact subsets of [0,1] and show that the closed Salem sets form a Π 0 3 -complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in K([0,1 ] d ) . We then generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still Π 0 3 -complete when we endow F( R d ) with the Fell topology. A similar result holds also for the Vietoris topology. We apply our results to characterize the Weihrauch degree of the functions computing the Hausdorff and Fourier dimensions.

We investigate a model theoretic invariant κ m srd (T) , which was introduced by Shelah in his famous book, and prove that κ m srd (T) is sub-additive. When κ m srd (T) is infinite, this gives the equality κ m srd (T)= κ 1 srd (T) , answering a question by Shelah. We apply the same proof method to analyze another invariant κ m ird (T) , and show that it is also sub-additive, improving a result in the book.

We present a model where \omega_1 is inaccessible by reals, Silver measurability holds for all sets but Miller and Lebesgue measurability fail for some sets. This contributes to a line of research started by Shelah in the 1980s and more recently continued by Schrittesser and Friedman, regarding the separation of different notions of regularity properties of the real line.

The uncountability of R (Cantor, 1874) is famous like few other theorems are; we investigate the logical and computational properties of NIN (resp. NBI ) the statement there is no injection (resp. bijection) from [0,1] to N . While intuitively weak, NIN (and similar for NBI ) is classified as rather strong on the 'normal' scale, both in terms of which comprehension axioms prove NIN and which discontinuous functionals compute (Kleene S1-S9) the real numbers from NIN from the data. Indeed, full second-order arithmetic is essential in each case. To obtain a classification in which NIN and NBI are weak, we explore the 'non-normal' scale based on (classically valid) continuity axioms and non-normal functionals, going back to Brouwer. In doing so, we derive NIN and NBI from basic theorems, like Arzelà's convergence theorem for the Riemann integral (1885) and central theorems from Reverse Mathematics formulated with the standard definition of `countable set' involving injections or bijections to N . Thus, the uncountability of R is a corollary to basic mainstream mathematics; NIN and NBI are (among) the weakest principles on the non-normal scale, which serendipitously reproves many of our previous results. Moreover, the Bolzano-Weierstrass theorem for countable sets in Cantor space is weak, but gives rise to Π 1 2 - CA 0 when combined with higher-order Π 1 1 - CA 0 , i.e. the Suslin functional. Finally, NIN and NBI allow us to showcase to a wide audience the techniques (like Gandy selection) used in our ongoing project on the logical and computational properties of the uncountable.

We prove that the value group of the field of transseries is isomorphic to the additive reduct of the field.

A wide Aronszajn tree is a tree of size and height ω 1 with no uncountable branches. We prove that under MA( ω 1 ) there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and Väänänen from 1994. We also prove that under the same assumption there is no universal Aronszajn tree, improving a result of Todorčevi{ć} from 2007 who proved the same under the assumption of BPFA for posets of size ℵ 1 . Finally, we prove that under MA( ω 1 ) , every wide Aronszajn tree weakly embeds in an Aronszajn tree.

We introduce the notion of a coordinate k -algebra scheme and the corresponding notion of a B -operator. This class of operators includes endomorphisms and derivations of the Frobenius map, and it also generalizes the operators related to D -rings from [15]. We classify the (coordinate) k -algebra schemes for a perfect field k and we also discuss the model-theoretic properties of fields with B -operators.

Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. Finally, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures.