Periodic Reporting for period 4 - BeyondA1 (Set theory beyond the first uncountable cardinal)
Période du rapport: 2023-04-01 au 2024-09-30
Mathematics in its pure form is largely devoted to the understanding of the infinite and its field of Set Theory has the mission to understand the different infinities (calibrated by cardinals Aleph0, Aleph1, Aleph2, etc’). In the last 50 years or so, there has been an incredible advance in our understanding of the first two infinities — Aleph0 and Aleph1— and this insight led to the solution of many important mathematical problems, both inside and outside of Set Theory. In contrast, and despite crucial discoveries, many core questions concerning the combinatorial nature of the next infinite, being Aleph2, remain open ever since the 1970’s.
While still a mystery, understanding the combinatorial nature of the second uncountable cardinal is a necessity. Indeed, by an argument that goes back to a fundamental discovery of Cantor from the end of the 19th century, structure theorems for uncountable objects tend to be incompatible with the continuum being equal to the first uncountable cardinal.
The project concentrates on four core areas: Transfinite trees, Ramsey theory on ordinals, Club-guessing principles and Forcing Axioms. At the level of the first uncountable cardinal the theory of each of these areas is rich and well-established. Our ultimate goal is to reach a similar breadth of understanding at the higher infinite.
Just like Club-guessing found applications in the classification of linear orders, and Forcing Axioms found applications in the study of operator algebras, higher analogues will pave the way to solving open problems concerning objects of the next level.
While still a mystery, understanding the combinatorial nature of the second uncountable cardinal is a necessity. Indeed, by an argument that goes back to a fundamental discovery of Cantor from the end of the 19th century, structure theorems for uncountable objects tend to be incompatible with the continuum being equal to the first uncountable cardinal.
The project concentrates on four core areas: Transfinite trees, Ramsey theory on ordinals, Club-guessing principles and Forcing Axioms. At the level of the first uncountable cardinal the theory of each of these areas is rich and well-established. Our ultimate goal is to reach a similar breadth of understanding at the higher infinite.
Just like Club-guessing found applications in the classification of linear orders, and Forcing Axioms found applications in the study of operator algebras, higher analogues will pave the way to solving open problems concerning objects of the next level.
Advances on core area 1: Historically, a major theme in the study of the consistency of the existence of Souslin trees of various forms was finding combinatorial guessing principles sufficient for these constructions. This typically involved the diamond principle and some ladder-system axiom. Brodsky and Rinot managed to eliminate the need for the diamond principle. Rinot managed to reduce the ladder-system axiom to an assertion close to the well-known upper bound (a square). Shalev and Rinot took the unconventional path of deriving a successful combinatorial guessing principle from the existence of a Souslin tree, and gave applications to topology. In settling a question raised by Gilton, Levine and Stejskalova, Rinot proved that for every integer n such that b<2^{w_n}=w_{n+1}, square(w_{n+1}) yields an w_{n+1}-Souslin tree. In settling a question left open by Shelah’s solution to Kunen’s question, You, Yadai and Rinot developed the theory of vanishing levels of trees and proved that successor cardinals (as small as the second uncountable cardinal) may carry full Souslin trees. Brodsky and Rinot improved the lower bound (from a Mahlo cardinal to a weakly compact) for non-existence of a Souslin tree with an ascent path. Answering a question of Cummings, Eisworth and Moore, Shalev proved the consistency of the existence of a minimal non-sigma-scatter linear order of size an inaccessible cardinal. His proof goes through the construction of a particular sort of tree and provide a maximal family of pairwise-far such trees. This work also sheds a new light on a question of Baumgartner from the 1980’s.
Advances on core area 2: Historically, the study of Ramsey theory at the level of the second uncountable cardinal has always focused on negative partition relations. For instance, by a celebrated result of Shelah from 1997, Pr1(w2,w2,w2,chi) holds for chi=w0, and whether the same is true for chi=w1 remained open ever since. Zhang and Rinot proved that square(w2) yields an affirmative answer to Shelah's question. Thus, a counterexample (if exists) requires at least a weakly compact cardinal. In a series of papers, Kojman, Steprans and Rinot developed the theory of Ramsey theory over partitions. One notable result completed a long line of research initiated by Luzin and Sierpinski a hundred years ago. Another one is a *positive* partition relation for w2 which is a rather rare phenomenon. By a striking theorem of Todorcevic from 1994, there exists a coloring of triples in w2 to w0 that takes on every possible color on the cube of any uncountable set. Feldman and Rinot extended this theorem to make it applicable to questions from additive Ramsey theory, obtaining the following failure of an Hindman-type proposition: If the continuum hypothesis fails, then for every Abelian group of size w2, there exists a coloring c of the group to w0 such that for every uncountable subset X of the group, for every possible color n, there are x,y,z in X such that c(x+y+z)=n. Carvalho, Fernandes and Junqueira showed that an anti-Ramsey assertion for topological spaces is compatible with the failure of the continuum hypothesis, and then Carvalho and Rinot found such an example in ZFC, answering a question of Komjath and Weiss.
Advances on core area 3: Variations of Jensen’s diamond principle were studied. Kostana, Shelah and Rinot studied a new notion of diamond on Kurepa trees and established its independence. Zhang and Ben-Neria studied a diamond principle for product of cardinals, obtained a positive result for weakly compact cardinals and negative consistency results for others. Carvalho, Inamdar and Rinot proved that strong forms of diamond on ladder systems hold in ZFC and used it to solve a problem posed by Leiderman and Szeptycki concerning countably metacompact topological spaces. Inamdar and Rinot pushed the theory of club-guessing significantly especially with regards to partitioned forms of which. They also obtained a theorem that provides multiple club-guessing features of Shelah at the same time. It is a conjecture of Rudin from 1990 that a Dowker space of size w1 exists. Shalev, Todorcevic and Rinot proved that the very weak guessing principle known as `stick’ is sufficient.
Advances on core area 4: In a series of papers, Poveda, Sinapova and Rinot developed a novel iteration scheme for Prikry-type forcing that allow various forcing axioms at the level of successors of singular cardinals, most notably, they proved the consistency of the failure of the singular cardinals hypothesis at Aleph_omega together with the reflection of all stationary subsets of its successor, thereby showing that two classical results of Magidor (from 1977 and 1982) can hold simultaneously. Many negative results for forcing axioms and merely iterations at the level of higher cardinals were obtained. Inamdar and Rinot solved a question of Moore, proving that if there exists an w2-Aronszajn line, then there exists one without an w2-Countryman subline. Replacing w2 by w1, one arrives at Moore’s Basis Theorem which he famously proved to follow from the Proper Forcing Axiom (PFA). Our finding demonstrates a major obstruction for an higher analog of PFA. Lambie-Hanson, Zhang and Rinot verified that a conjecture of Silver holds assuming PFA: every cardinal carrying a unifrom ultrafilter that is indecomposable is either measurable or the limit of countably many measurable cardinals. In a series of papers, Lambie-Hanson and Rinot developed the theory of unbounded colorings and used it to produce many counterexamples in this vein, including an answer to a question of Cox in the form of an w2-stationarily layered poset whose infinite power does not even satisfy the w2-chain-condition.
Advances on core area 2: Historically, the study of Ramsey theory at the level of the second uncountable cardinal has always focused on negative partition relations. For instance, by a celebrated result of Shelah from 1997, Pr1(w2,w2,w2,chi) holds for chi=w0, and whether the same is true for chi=w1 remained open ever since. Zhang and Rinot proved that square(w2) yields an affirmative answer to Shelah's question. Thus, a counterexample (if exists) requires at least a weakly compact cardinal. In a series of papers, Kojman, Steprans and Rinot developed the theory of Ramsey theory over partitions. One notable result completed a long line of research initiated by Luzin and Sierpinski a hundred years ago. Another one is a *positive* partition relation for w2 which is a rather rare phenomenon. By a striking theorem of Todorcevic from 1994, there exists a coloring of triples in w2 to w0 that takes on every possible color on the cube of any uncountable set. Feldman and Rinot extended this theorem to make it applicable to questions from additive Ramsey theory, obtaining the following failure of an Hindman-type proposition: If the continuum hypothesis fails, then for every Abelian group of size w2, there exists a coloring c of the group to w0 such that for every uncountable subset X of the group, for every possible color n, there are x,y,z in X such that c(x+y+z)=n. Carvalho, Fernandes and Junqueira showed that an anti-Ramsey assertion for topological spaces is compatible with the failure of the continuum hypothesis, and then Carvalho and Rinot found such an example in ZFC, answering a question of Komjath and Weiss.
Advances on core area 3: Variations of Jensen’s diamond principle were studied. Kostana, Shelah and Rinot studied a new notion of diamond on Kurepa trees and established its independence. Zhang and Ben-Neria studied a diamond principle for product of cardinals, obtained a positive result for weakly compact cardinals and negative consistency results for others. Carvalho, Inamdar and Rinot proved that strong forms of diamond on ladder systems hold in ZFC and used it to solve a problem posed by Leiderman and Szeptycki concerning countably metacompact topological spaces. Inamdar and Rinot pushed the theory of club-guessing significantly especially with regards to partitioned forms of which. They also obtained a theorem that provides multiple club-guessing features of Shelah at the same time. It is a conjecture of Rudin from 1990 that a Dowker space of size w1 exists. Shalev, Todorcevic and Rinot proved that the very weak guessing principle known as `stick’ is sufficient.
Advances on core area 4: In a series of papers, Poveda, Sinapova and Rinot developed a novel iteration scheme for Prikry-type forcing that allow various forcing axioms at the level of successors of singular cardinals, most notably, they proved the consistency of the failure of the singular cardinals hypothesis at Aleph_omega together with the reflection of all stationary subsets of its successor, thereby showing that two classical results of Magidor (from 1977 and 1982) can hold simultaneously. Many negative results for forcing axioms and merely iterations at the level of higher cardinals were obtained. Inamdar and Rinot solved a question of Moore, proving that if there exists an w2-Aronszajn line, then there exists one without an w2-Countryman subline. Replacing w2 by w1, one arrives at Moore’s Basis Theorem which he famously proved to follow from the Proper Forcing Axiom (PFA). Our finding demonstrates a major obstruction for an higher analog of PFA. Lambie-Hanson, Zhang and Rinot verified that a conjecture of Silver holds assuming PFA: every cardinal carrying a unifrom ultrafilter that is indecomposable is either measurable or the limit of countably many measurable cardinals. In a series of papers, Lambie-Hanson and Rinot developed the theory of unbounded colorings and used it to produce many counterexamples in this vein, including an answer to a question of Cox in the form of an w2-stationarily layered poset whose infinite power does not even satisfy the w2-chain-condition.
52 research papers were written by this project’s group members, all go beyond the state of the art. See https://802756.assafrinot.com(s’ouvre dans une nouvelle fenêtre)