Zermelo's Axiom of Separation is: Exist x: Forall y: (y in x <==> y in a & E(y)) with definite(E) and parameter a. Thoralf Skolem suggested to characterize the terminus "definite" by "the property E should be representable by a FOL formula". But that is trivial. "definite" must mean more. The author claims that "definite" means "in accordance with the theory of definitions of logic". In this case the theorem of Cantor is no longer a theorem, but a undecidable sentence, and has to be established explicitly as axiom. This is not done by the community, but it is made a silent assumption that we can drop the appendix "definite(E)" from the axiom of separation at all. But this is a convention (even when it is silent) and it is nothing else than an axiom.
Much recent work in cardinal characteristics has focused on generalizing results about $ω$ to uncountable cardinals by studying analogues of classical cardinal characteristics on the generalized Baire and Cantor spaces $κ^κ$ and $2^κ$. In this note I look at generalizations to other function spaces, focusing particularly on the space of functions $f:ω^ω\to ω^ω$. By considering classical cardinal invariants on Baire space in this setting I derive a number of "higher dimensional" analogues of such cardinals, ultimately introducing 18 new cardinal invariants, alongside a framework that allows for…
We study the values of the higher dimensional cardinal characteristics for sets of functions $f:ω^ω\to ω^ω$ introduced by the second author. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg\mathsf{CH}$ such as the Cohen, random and Sacks models and, as a byproduct show that, with possibly one exception, for the bounding numbers there are no $\mathsf{ZFC}$ relations between them beyond those in the higher dime…
We introduce and axiomatize the notion of a reflective cardinal, use it to give semantics to higher order set theory, and explore connections between the notion of reflective cardinals and large cardinal axioms.
We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of $L_{κ^+,κ^+}$-indescribability and $Π^1_ξ$-indescribability of a cardinal $κ$ for all $ξ<κ^+$. In this context, universal $Π^1_ξ$ formulas exist, there is a normal ideal associated to $Π^1_ξ$-indescribability and the notions of $Π^1_ξ$-indescribability yield a strict hierarchy below a measurable cardinal. Additionally, given…
Technical University of Košice（科希策理工大学）；Pavol Jozef Šafárik University in Košice（科希策帕沃尔·约瑟夫·沙法里克大学）；Slovak Academy of Sciences（斯洛伐克科学院）；Technische Universität Wien（维也纳工业大学）；University of Vienna（维也纳大学）
We prove the consistency, assuming an ineffable cardinal, that any two normal countably closed $ω_2$-Aronszajn trees are club isomorphic. This work generalizes to higher cardinals the property of Abraham-Shelah that any two normal $ω_1$-Aronszajn trees are club isomorphic, which follows from $\textsf{PFA}$. The statement that any two normal countably closed $ω_2$-Aronszajn trees are club isomorphic implies that there are no $ω_2$-Suslin trees, so our proof also expands on the method of Laver-Shelah for obtaining the $ω_2$-Suslin hypothesis.
A question dating to Sibe Mardešić and Andrei Prasolov's 1988 work Strong homology is not additive, and motivating a considerable amount of set theoretic work in the ensuing years, is that of whether it is consistent with the ZFC axioms for the higher derived limits $\mathrm{lim}^n$ $(n>0)$ of a certain inverse system $\mathbf{A}$ indexed by ${^ω}ω$ to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all $n$-coherent families of functions indexed by ${^ω}ω$ to be trivial. In this paper, we prove that, in any forcing extension given by ad…
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$λ$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq λ$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
We show that generalized eventually narrow sequences on a strongly inaccessible cardinal $κ$ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on $κ$. Moreover assuming GCH, $κ^{<κ}=κ$, we show that: (1) if $κ$ is strongly unfoldable, $κ^+\leqβ=\hbox{cf}(β)\leq \hbox{cf}(δ)\leqδ\leqμ$ and $\hbox{cf}(μ)>κ$,then there is a cardinal preserving generic extension in which $$\mathfrak{s}(κ)=κ^+\leq\mathfrak{b}(κ)=β\leq\mathfrak{d}(κ)=δ\leq 2^κ=μ.$$ (2) if $κ$ is strongly inaccessible, $λ>κ^+$, then in the generic extension obtained as the $<κ$-support iterat…
For a regular uncountable cardinal kappa, we discuss the order relationship between the unbounding and dominating numbers on kappa and cardinal invariants of the higher meager ideal M_kappa. In particular, we obtain a complete characterization of add(M_kappa) and cof(M_kappa) in terms of cov(M_kappa) and non(M_kappa) and unbounding and dominating numbers, and we provide models showing that there are no restrictions on the value of non(M_kappa) in the degenerate case 2^{<kappa} > kappa except 2^{<kappa} leq non(M_kappa) leq 2^kappa. The corresponding question for cof(M_kappa) remains open. Our …
We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (boldface Sigma^0_2 and, under suitable restrictions, Pi^0_2) are shown to have pleasant properties, related to Baire category. We construct models of set theory where (unrestricted) boldface Pi^0_2-characteristics behave quite chaotically and no new characteristics appear at higher complexity levels. We also discuss some characteristics associated with partition theorems and we present, in an appendix, a simplified p…
We present a coherent collection of finite mathematical theorems some of which can only be proved by going well beyond the usual axioms for mathematics. The proofs of these theorems illustrate in clear terms how one uses the well studied higher infinities of abstract set theory called large cardinals in an essential way in order to derive results in the context of the natural numbers. The findings raise the specific issue of what consitutes a valid mathematical proof and the general issue of objectivity in mathematics in a down to earth way. Large cardinal axioms, which go beyond the usual axi…
We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo-Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals. This class includes the example of unordered trees of any arity.
We introduce bounded category forcing axioms for well-behaved classes $Γ$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{λ_Γ^+}$ modulo forcing in $Γ$, for some cardinal $λ_Γ$ naturally associated to $Γ$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $λ_Γ=ω$--to classes $Γ$ with $λ_Γ>ω$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions o…
We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that for every $X\subseteq\mathbb R$ with $|X|=|\mathbb R|$, and every colour $γ\in\mathbb Q$, there are two distinct elements $x_0,x_1$ of $X$ for which $c(x_0+x_1)=γ$. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah. Theorem 2: For every Abelian group $G$, there exists a colouring $c:G\rightarrow\mathbb Q$ such that…
This is a report on state-of-the-art on the question of developing higher analogues of the forcing axiom PFA. Recently there have been several attempts to develop forcing axioms analogous to the proper forcing axiom (PFA) for cardinals of the form aleph_n where n > 1. We investigate the difficulties of doing this and survey some of the successes
