27 research outputs found
Capturing sets of ordinals by normal ultrapowers
We investigate the extent to which ultrapowers by normal measures on
can be correct about powersets for . We
consider two versions of this questions, the capturing property
and the local capturing property
. holds if there is
an ultrapower by a normal measure on which correctly computes
. is a weakening of
which holds if every subset of is
contained in some ultrapower by a normal measure on . After examining
the basic properties of these two notions, we identify the exact consistency
strength of . Building on results of Cummings,
who determined the exact consistency strength of
, and using a forcing due to Apter and Shelah, we
show that can hold at the least measurable
cardinal.Comment: 20 page
Graph Relations and Constrained Homomorphism Partial Orders
We consider constrained variants of graph homomorphisms such as embeddings,
monomorphisms, full homomorphisms, surjective homomorpshims, and locally
constrained homomorphisms. We also introduce a new variation on this theme
which derives from relations between graphs and is related to
multihomomorphisms. This gives a generalization of surjective homomorphisms and
naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs.
Both R-cores and R-cocores of graphs are unique up to isomorphism and can be
computed in polynomial time.
The theory of the graph homomorphism order is well developed, and from it we
consider analogous notions defined for orders induced by constrained
homomorphisms. We identify corresponding cores, prove or disprove universality,
characterize gaps and dualities. We give a new and significantly easier proof
of the universality of the homomorphism order by showing that even the class of
oriented cycles is universal. We provide a systematic approach to simplify the
proofs of several earlier results in this area. We explore in greater detail
locally injective homomorphisms on connected graphs, characterize gaps and show
universality. We also prove that for every the homomorphism order on
the class of line graphs of graphs with maximum degree is universal
(Extra)ordinary equivalences with the ascending/descending sequence principle
We analyze the axiomatic strength of the following theorem due to Rival and
Sands in the style of reverse mathematics. "Every infinite partial order of
finite width contains an infinite chain such that every element of is
either comparable with no element of or with infinitely many elements of
." Our main results are the following. The Rival-Sands theorem for infinite
partial orders of arbitrary finite width is equivalent to over . For each fixed , the
Rival-Sands theorem for infinite partial orders of width is
equivalent to over . The Rival-Sands theorem for
infinite partial orders that are decomposable into the union of two chains is
equivalent to over . Here
denotes the recursive comprehension axiomatic system,
denotes the induction scheme, denotes the
ascending/descending sequence principle, and denotes the stable
ascending/descending sequence principle. To our knowledge, these versions of
the Rival-Sands theorem for partial orders are the first examples of theorems
from the general mathematics literature whose strength is exactly characterized
by , by , and by
. Furthermore, we give a new purely combinatorial result by
extending the Rival-Sands theorem to infinite partial orders that do not have
infinite antichains, and we show that this extension is equivalent to
arithmetical comprehension over
Calibrating the complexity of combinatorics: reverse mathematics and Weihrauch degrees of some principles related to Ramsey’s theorem
In this thesis, we study the proof-theoretical and computational strength of some combinatorial principles related to Ramsey's theorem: this will be accomplished chiefly by analyzing these principles from the points of view of reverse mathematics and Weihrauch complexity.
We start by studying a combinatorial principle concerning graphs, introduced by Bill Rival and Ivan Sands as a form of ``inside-outside'' Ramsey's theorem: we will determine its reverse mathematical strength and present the result characterizing its Weihrauch degree. Moreover, we will study a natural restriction of this principle, proving that it is equivalent to Ramsey's theorem.
We will then move to a related result, this time concerning countable partial orders, again introduced by Rival and Sands: we will give a thorough reverse mathematical investigation of the strength of this theorem and of its original proof. Moreover, we will be able to generalize it, and this generalization will itself be presented in the reverse mathematical perspective.
After this, we will focus on two forms of Ramsey's theorem that can be considered asymmetric. First, we will focus on a restriction of Ramsey's theorem to instances whose solutions have a predetermined color, studying it in reverse mathematics and from the point of view of the complexity of the solutions in a computability theoretic sense. Next, we move to a classical result about partition ordinals, which will undergo the same type of analysis.
Finally, we will present some results concerning a recently introduced operator on the Weihrauch degrees, namely the first-order part operator: after presenting an alternative characterization of it, we will embark on the study the result of its applications to jumps of Weak Kőnig's Lemma
Preserving levels of projective determinacy by tree forcings
We prove that various classical tree forcings -- for instance Sacks forcing,
Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve
the statement that every real has a sharp and hence analytic determinacy. We
then lift this result via methods of inner model theory to obtain
level-by-level preservation of projective determinacy (PD). Assuming PD, we
further prove that projective generic absoluteness holds and no new equivalence
classes classes are added to thin projective transitive relations by these
forcings.Comment: 3 figure
Maximality in the ⍺-C.A. Degrees
In [4], Downey and Greenberg define the notion of totally ⍺-c.a. for appropriately small ordinals ⍺, and discuss the hierarchy this notion begets on the Turing degrees. The hierarchy is of particular interest because it has already given rise to several natural definability results, and provides a definable antichain in the c.e. degrees. Following on from the work of [4], we solve problems which are left open in the aforementioned relating to this hierarchy. Our proofs are all constructive, using strategy trees to build c.e. sets, usually with some form of permitting. We identify levels of the hierarchy where there is absolutely no collapse above any totally ⍺-c.a. c.e. degree, and construct, for every ⍺ ≼ ε0, both a totally ⍺-c.a. c.e. minimal cover and a chain of totally ⍺-c.a. c.e. degrees cofinal in the totally ⍺-c.a. c.e. degrees in the cone above the chain's least member
Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH
This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions from Baire space to Baire space. I prove that these cardinals can be organized into two diagrams analogous to the standard Cichoń diagram show several independence results and investigate their relation to cardinal invariants on omega. In the second half of the thesis I look at forcing axioms compatible with CH. First I consider Jensen\u27s subcomplete and subproper forcing. I generalize these notions to larger classes which are (apparently) much more nicely behaved structurally. I prove iteration and preservation theorems for both classes and use these to produce many new models of the subcomplete forcing axiom. Finally I deal with dee-complete forcing and its associated axiom DCFA. Extending a well-known result of Shelah, I show that if a tree of height omega one with no branch can be embedded into an omega one tree, possibly with uncountable branches, then it can be specialized without adding reals. As a consequence I show that DCFA implies there are no Kurepa trees, even if CH fails