Vladimir Kanovei, Michael Reeken
We study the structure of effective cardinals in the nonstandard set universe of Hrbacek set theory HST. Some results resemble those known in descriptive set theory in the domain of Borel reducibility of equivalence relations.
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Nonstandard Universes
本讲义面向初学者，系统整理非标准分析中集合论全域的存在性与基本性质，并在附录补充一阶逻辑预备知识。
Isomorphism property in nonstandard extensions of ZFC universe
我们研究模型的HST，一个非标准的集合论，其中包括，特别是，ZFC替换和分离图式的语言包含的成员资格和标准谓词，以及内部集的有序家庭的饱和度。这个理论承认同构性质IP的一个适当的公式：“任何两个基本等价的内部呈现结构的wellorderable语言是同构的。例如，IP意味着所有无限内部集合都是等数的，并且存在标准实数的唯一（模同构）内部初等扩张。我们证明了IP是独立的HST（使用类的所有集构造从内部集）和HST（使用通用扩展的$\HST$模型由足够数量的通用同构）一致。
We observe that the nonstandard finite cardinality of a definable set in a strongly minimal pseudofinite structure D is a polynomial over the integers in the nonstandard finite cardinality of D. We conclude that D is unimodular, hence also locally modular. We also deduce a regularity lemma for graphs definable in strongly minimal pseudofinite structures. The paper is elementary, and the only surprising thing about it is that the results were not explicitly noted before. In the new version we add a comment on relations to work of Macpherson and Steinhorn, as well as on the limited nature of the…
We introduce the notion of "functional extension" of a set X, by means of two natural algebraic properties of the operator * on unary functions. We study the connections with ultrapowers of structures with universe X, and we give a simple characterization of those functional extensions that correspond to limit ultrapower extensions. In particular we obtain a purely algebraic proof of Keisler's characterization of nonstandard (= complete elementary) extensions.
Currently the two popular ways to practice Robinson's nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach. It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets. We show that internal frameworks provide successful accounts of nonstandard hulls and Loeb measures. The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe.
We use a reverse Easton forcing iteration to obtain a universe with a definable well-ordering, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle diamond star at a proper class of successor cardinals.
No cardinal correct inner model elementarily embeds into the universe
该研究针对Caicedo提出的相关问题，证明不存在从基数正确的内模型M到全集V的非平凡保基数初等嵌入，推进了集合论中内模型与初等嵌入方向的研究。
Set theoretical analogues of the Barwise-Schlipf theorem
该研究在集合论领域给出Barwise-Schlipf定理的类似结果，分别刻画了满足对应类理论扩张条件的任意基数ZF非标准模型与可数ZFC非标准模型。
While many inner model theoretic combinatorial principles are incompatible with large cardinal axioms, on some rare occasions, large cardinals actually imply that the structure of the universe of sets is analogous to the canonical inner models. This note provides two new instances of this phenomenon.
A transitive model $M$ of ZFC is called a ground if the universe $V$ is a set forcing extension of $M$. We show that the grounds of $V$ are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) $V$ has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.
We establish some similarities/analogies between uncountable cardinals or powersets and the class $V$ of all sets. They concern mainly the Boolean algebras ${\cal P}(κ)$, for a regular cardinal $κ$, and ${\cal C}(V)$ (the class of subclasses of the universe $V$), endowed with some ideals, especially the ideal $[κ]^{<κ}$ for ${\cal P}(κ)$, and the ideal of sets $V$ for ${\cal C}(V)$.
We prove that if there is an elementary embedding from the universe to itself, then there is a proper class of measurable successor cardinals.
Recognizable sets and Woodin cardinals: Computation beyond the constructible universe
本文研究了在构造宇宙外通过序数时间 Turing 机进行计算的可识别集，证明了可计算集属于 L，而无限时间计算的可识别对象可达 Woodin 数字的层次。
We show that if the universe is self-iterable and $κ$ is an inaccessible limit of Woodin cardinal then $AD_R + "Θ$ is regular" holds in the derived model at $κ$. The proof is fine-structure free, and only assumes basic knowledge of iteration trees and iteration strategies. Our proof can be viewed as the fine-structure free version of the well-known fact that $AD_R + "Θ$ is regular" is true in the derived models of hod mice that have inaccessible limit of Woodin cardinals (see for example [6]). However, the proof uses a different set of ideas and is more general.
It is shown that the existence of a measurable cardinal is equiconsistent to a model of ZFC in which there is no ordinal-definable, stationary, costationary subset of $ω_1$
Andrew Wiles' Proof of Fermat's Last Theorem, As Expected, Does Not Require a Large Cardinal Axiom. A Discussion of Colin McLarty's "The Large Structures of Grothendieck Founded on Finite-Order Arithmetic"
本文探讨了 Fermat 大定理的证明是否依赖大基数公理，指出 Colin McLarty 证明了 Grothendieck 的大工具可建立在有限阶算术强度的 ZFC 片段上，从而消除了该证明依赖强于 ZFC 基础的表象。
We show: There are pairs of universes V_1 subseteq V_2 and there is a notion of forcing P in V_1 such that the change mentioned in the title occurs when going from V_1[G] to V_2[G] for a P-generic filter G over V_2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.
Splendid extensions
研究在特定条件下，后继基数κ相关的PA模型扩展问题，通过强制构造全域的方法，使得大小为κ的PA模型扩展为M且无辉煌扩张，若有不可表达基数，在ZFC中该结论在其下某基数处成立。
The strength of the isomorphism property
本文第一节利用模型论中某些二序型的实现刻画了非标准宇宙的同构性质。在第2节中，给出了几个应用。其中一个应用程序回答了D. Ross关于无限Loeb测度空间
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many stationary sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some stationary set, does hold for s…
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.
Reinhardt Cardinals and Eventually Dominating Functions
本文证明了关于集合论宇宙到自身的初等嵌入与最终支配函数的结果，并据此证明了满足严格条件的初等嵌入的存在性，刻画了相关大基数的本质，最后给出了 Kunen 不一致性的新证明。
Large cardinals beyond HOD
本文研究exacting与ultraexacting大基数，证明其与I0等一致，并给出其强度界及新刻画，表明它们与HOD假设相容。
We define a potentialist system of ZF-structures, that is, a collection of possible worlds in the language of ZF connected by a binary accessibility relation, achieving a potentialist account of the full background set-theoretic universe $V$. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just ZF. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theory S4.2. Moreover, we characterize the worlds sat…
We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $MM^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $MM^{++}$ collapsing a superhuge cardinal to become the second uncoun…
This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate \(L\), providing some evidence for the Ultimate \(L\) Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality…
Vopěnka's Alternative Set Theory can be viewed both as an evolution and as a revolution: it is based on his previous experience with nonstandard universes, inspired by Skolem's construction of a nonstandard model of arithmetic, and its inception has been explicitly mentioned as an attempt to axiomatize Robinson's Nonstandard Analysis. Vopěnka preferred working in an axiomatic theory to investigating its individual models; he also viewed other areas of nonclassical mathematics through this prism. This article is a contribution to the mapping of the mathematical neighbourhood of the Alternative …
A Universe with large Continuum, global $Σ$-Uniformization and a projective Well-Order of its Reals
本文构造集合论模型，使连续统大小为任意正则基数 $κ$，同时对所有 $n\ge2$ 满足 $Σ^1_n$-一致化，并具备实数的 $Δ^1_3$-可定义良序。
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…
Existence Theorem for Cumulative Universe Towers and Its Applications
本文构建了 Grothendieck 宇宙的累积塔，通过归纳-递归定义宇宙编码与秩，证明其在依赖积、依赖和等下的封闭性及严格累积性，并在 ZF 集合论上确立了高阶类型论的可靠集合论模型存在性定理。
Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call ${\rm GM}^+(ω_3,ω_1)$ holds. This principle implies ${\rm ISP}(ω_2)$ and ${\rm ISP}(ω_3)$, and hence the tree property at $ω_2$ and $ω_3$, the Singular Cardinal Hypothesis, and the failure of the weak square principle $\square(ω_2,λ)$, for all regular $λ\geq ω_2$. In addition, it implies that the restriction of the approachability ideal $I[ω_2]$ to the set of ordinals of cofinality $ω_1$ is the non stationary ideal on this set. The consistency of this last statement was pre…
Forcing Axioms and the Definabilty of the Nonstationary Ideal on $ω_1$
该数理逻辑研究探讨力迫公理下$\omega_1$非驻定理想的可定义性，证明$\text{BMM}$加武丁基数、$(*)$-公理下其不可被$\Sigma_1(\omega_1)$定义，同时构造出$\text{BPFA}$、$\text{MA}$等公理下其可定义甚至饱和的集合论宇宙。
The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on omega. We treat the problem to characterize those sets A which, in some forcing extension of the universe, can be the mad spectrum. We solve this problem to some extent. What remains open is the possible values of min(A) and max(A).
A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Gr…
The Coding Conception of Set
该研究提出序数和集合的编码概念，以康托尔原则为基础生成有界序数集及理论$SC^{reg}$。将其扩展到全集理论产生两个不一致的一阶集合理论，证明幂集二分法不可避免，得出$ZFC^ - +$“每个基数都有后继”是无争议片段，幂集公理地位待决。
Adding highly generic subsets of $ω_2$
本研究从GCH出发构造保基数与GCH的集合论泛型扩张，得到满足特定Cohen泛型性质的$\omega_2$子集，推进了无穷组合泛型集相关研究。
A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Gr…
On $Σ_1$-Definable Closed Unbounded Sets
该文聚焦集合论中基数$κ$的$Σ_1$-可定义闭无界集，研究其对完全闭无界滤子的逼近程度及对定义复杂度的依赖，发现逼近能力差异大且依赖底层集合论宇宙性质。
Set theory with a proper class of indiscernibles
本文研究带真类不可分辨元的ZFC扩展集合论，证明其纯集合论推论与添加Levy模式的ZFC系统的定理等价，明确了两类集合论系统在纯集合论层面的一致性。
The main theorem of this article is that every countable model of set theory M, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random Q-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acy…
Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem
该研究围绕集合论全域V与HOD的紧性现象，证明了奇异强极限基数的幂集分歧特性，回答了Hayut的问题，并揭示Magidor覆盖定理的最优性。
We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $φ(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $φ(a)$ true in $V$ is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are…
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