Peter Koepke, Martin Koerwien
We propose a natural theory SO axiomatizing the class of sets of ordinals in a model of ZFC set theory. Both theories possess equal logical strength. Constructibility theory in SO corresponds to a natural recursion theory on ordinals.
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This is the author's 2004 Master's thesis at Iowa State University, done under the supervision of Roger D. Maddux. It provides a background in relation algebras. Three results from the literature are demonstrated in full: (i.) RRA is a variety. (ii.) RRA is not finitely based. (iii.) Any equational basis for RRA has infinitely many variables. We also give an elementary proof of Birkoff's variety theorem.
Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. Our axiom sets have been formalized in the Isabelle/HOL interactive proof assistant, and this formalization utilizes a semantically correct embedding of free logic in classical higher-order logic. The modeling and formal analysis of our axiom sets has been significantly supported by series of experiments with automated reasoning tools integrated with Isabelle/HOL. We also address the relation of our axiom systems to…
It is true in the Solovay model that every countable ordinal-definable set of sets of reals contains only ordinal-definable elements.
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 consider a notion of "numerosity" for sets of tuples of natural numbers, that satisfies the five common notions of Euclid's Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a notion, we show that, contrasting to cardinal arithmetic, the natural "Cantorian" definitions of order relation and arithmetical operations provide a very good algebraic structure. In fact, numerosities can be taken as the non-negative part of a discretely ordered ring, namely the quotient of a formal power series ring modulo a suitable ("gauge") ideal. In particular, speci…
It is consistent that for every n >= 2, every stationary subset of omega_n consisting of ordinals of cofinality omega_k where k = 0 or k <= n-3 reflects fully in the set of ordinals of cofinality omega_{n-1}. We also show that this result is best possible.
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。
We introduce several new set-theoretic axioms formulated in terms of coloring of ordinals by reals. We show that these axioms generalize the axioms considered by I.Juhasz, L.Soukup and Z.Szentmiklossy, and give a class of p.o.s including Cohen p.o.-sets which force the axioms. In appearance these axioms are somewhat similar to OCA but it appears that they are inconsistent with OCA.
Recall that the Mouse Set Conjecture says that under AD++V=L(P(R)), a real is ordinal definable if and only if it belongs to an iterable mouse. The Mouse Set Conjecture for sets of reals says that under the same theory, a set of reals is ordinal definable from a real if and only if it belongs to a mouse over the reals. We prove that the Mouse Set Conjecture implies the Mouse Set Conjecture for sets of reals.
We investigate the extent to which ultrapowers by normal measures on $κ$ can be correct about powersets $\mathcal{P}(λ)$ for $λ>κ$. We consider two versions of this questions, the capturing property $\mathrm{CP}(κ,λ)$ and the local capturing property $\mathrm{LCP}(κ,λ)$. $\mathrm{CP}(κ,λ)$ holds if there is an ultrapower by a normal measure on $κ$ which correctly computes $\mathcal{P}(λ)$. $\mathrm{LCP}(κ,λ)$ is a weakening of $\mathrm{CP}(κ,λ)$ which holds if every subset of $λ$ is contained in some ultrapower by a normal measure on $κ$. After examining the basic properties of these two notio…
In set theory without the axiom of regularity, we consider a game in which two players choose in turn an element of a given set, an element of this element, etc.; a player wins if its adversary cannot make any next move. Sets that are winning, i.e. have a winning strategy for a player, form a natural hierarchy with levels indexed by ordinals. We show that the class of hereditarily winning sets is an inner model containing all well-founded sets, and that all four possible relationships between the universe, the class of hereditarily winning sets, and the class of well-founded sets are consisten…
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。
Clarifying ordinals
本研究通过可允许集合上的力迫法，证明ω₁的club子集内任意序数都存在同构不可由各拷贝对应完备Π¹₁集的并计算的拷贝，并分别在V=L与大基数假设下探讨了结论的最优性与推广性。
In this article we present an axiomatic definition of sets with individuals and a definition of natural numbers and ordinals. We use the axioms pairs, union, power, regularity and separation. We define the equality of sets and of individuals. There is no empty set. A single shoe is not a singleton set but an individual, a pair of shoes is a set. We call limit ordinals first numbers, that is a first number of the Peano axioms. An axiom of infinity is postulated and we prove the Peano axioms for ordinals with a first number up to a first $ω^ω$-number. Then we prove a first $ω^ω$-number notequal …
We investigate higher-dimensional $Δ$-systems indexed by finite sets of ordinals, isolating a particular definition thereof and proving a higher-dimensional version of the classical $Δ$-system lemma. We focus in particular on systems that consist of sets of ordinals, in which case useful order-theoretic uniformities can be ensured. We then present three applications of these higher-dimensional $Δ$-systems to problems involving the interplay between forcing and partition relations on the reals.
The strongest type of coloring of pairs of countable ordinals, gotten by Todorcevic from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of reals of size $\aleph_1$. In the other direction, it is shown that the existence of both a strongly Luzin set and a coherent Souslin tree is compatible with the existence of a countable partition of pairs of countable ordinals such that no coloring is strong over it. This clarifies the interaction between a gallery of coloring assertions going back to Luzin and Sierpinski a hundred years ago.
ZFC has sentences that quantify over all sets or all ordinals, without restriction. Some have argued that sentences of this kind lack a determinate meaning. We propose a set theory called TOPS, using Natural Deduction, that avoids this problem by speaking only about particular sets.
Weak compactness cardinals for strong logics and subtlety properties of the class of ordinals
该研究通过序数类组合性质刻画抽象逻辑弱紧基数存在性，证明其不蕴含强不可达基数存在，并给出其与真类微妙基数的一致性等价条件及HOD内模型中的相关推论。
Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses so strong as to contradict the Axiom of Choice -- we solve several variants of the generalized continuum problem and identify structural features of the levels $V_α$ of the cumulative hierarchy of sets that are eventually periodic, alternating according to the parity of the ordinal $α$. For example, if there is an elementary embedding from the universe of sets to itself, then for sufficiently large ordinals $α$, the supremum of the lengths of all wellfounded relations on $V_α$ is a strong limit cardinal if and only if…
A new mathematical object called a skand is introduced, which turns out in general to be a non-well-founded set. Skands of finite lengths are ordinary well-founded sets, and skands of very long length (like the hyper-skand of all ordinals) are hyper-classes. Self-similar skands are also considered, and they clarify the reflexivity of sets, i.e., the meaning of the relation X is a member of X; in particular, self-similar skands considered as non-well-founded sets are always reflexive, but not vice versa. The existence of self-similar skands shows at once that Russell's well-known paradox is not…
On Petr Novikov's problem of ordered systems of uniform sets
论文证明了每个小于ω₂的序数都是某种均匀Borel集的序型，解决了尼古拉斯·卢津1935年提出的问题。
For a countable ordinal epsilon we construct a Sigma^0_2 subset of the Cantor space for which one may force aleph_epsilon translations with intersections of size 2i, but such that it has no perfect set of such translations in any ccc extension. These sets have uncountably many translations with intersections of size 2i in ZFC, so this answers Problem 3.4 of arxiv:1711.04058 .
Effective Reducibility for Statements of Arbitrary Quantifier Complexity with Ordinal Turing Machines
本文扩展了利用序数图灵机处理任意量词复杂度的陈述的有效可约性概念，用于比较集合论基本原理及基数相关原理的可约性。
We consider classical representations of integers: Church's function iterators, cardinal equivalence classes of sets, ordinal equivalence classes of totally ordered sets. Since programs do not work on abstract entities and require formal representations of objects, we effectivize these abstract notions in order to allow them to be computed by programs. To any such effectivized representation is then associated a notion of Kolmogorov complexity. We prove that these Kolmogorov complexities form a strict hierarchy which coincides with that obtained by relativization to jump oracles and/or allowan…
Cohen's first model is a model of Zermelo--Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal $κ$. In the case that we force the function to be injective, it turns out that the resulting model is the same as adding $κ$ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that …
Ehrenfeucht's lemma (1973) asserts that whenever one element of a model of Peano arithmetic is definable from another, then they satisfy different types. We consider here the analogue of Ehrenfeucht's lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and in particular, Ehrenfeucht's lemma holds fully for models of set theory satisfying $V=HOD$. We show that the lemma can fail, however, in models of set theory with $V\neq HOD$, and it necessarily fails in the forcing extension to add a generic Cohen real. We go o…
Let V be the universe of sets and V_α the sets of rank \leqα. We develop some axiom schemata for set theory based on the following three assumptions: 1. V \models ZFC 2. V is large with respect to the class of ordinals 3. V is large with respect to each of the V_α
Realizing the totally unordered structure of ordinals
本文提出了用于分析经典集合论可实现性模型中序数的工具，通过定义 reish 算子构建递归名以计算序数，并证明 ε-完全有序集无法完全表示序数，同时提供了保留基数的方法。
We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets, that is, HOA = HOD. Moreover, we show that every (pointwise) algebraic model of ZF is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp---an algebraic analogue of the constructible universe---which is obtained by iteratively adding not only the sets that are definable over wh…
Set-Theoretic and Type-Theoretic Ordinals Coincide
本文证明构造性集合论与同伦类型论中的序数定义在 Aczel 解释下等价，并推广类型论序数以覆盖所有集合，所有结果在 Agda 中形式化。
We show the equivalence between the existence of winning strategies for $G_{δσ}$ (also called $Σ^{0}_{3}$) games in Cantor or Baire space, and the existence of functions generalized-recursive in a higher type-2 functional. (Such recursions are associated with certain transfinite computational models.) We show, inter alia, that the set of indices of convergent recursions in this sense is a complete $\Game Σ_{3}^{0}$ set: as paraphrase, the listing of those games at this level that are won by player I, essentially has the same information as the `halting problem' for this notion of recursion. Mo…
Reduced Set Theory
本文提出了一种简化集合论的片段，用于纯集合和在给定传递集合内迭代幂集，展示了该系统在数学形式化中的应用。
There exists a family $\{B_α\}_{α<ω_1}$ of sets of countable ordinals such that o $\max B_α=α$, o if $α\in B_β$ then $B_α\subseteq B_β$, o if $λ\leq α$ and $λ$ is a limit ordinal then $B_α\capλ$ is not in the ideal generated by the $B_β$, $β<α$, and by the bounded subsets of $λ$, o there is a partition $\{A_n\}_{n=0}^{\infty}$ of $ω_1$ such that for every $α$ and every $n,$ $B_α\cap A_n$ is finite.
We introduce axiomatically the ring $\bf{Z}_κ$ of the Euclidean integers, that can be viewed as the ``integral part" of the field $\mathbb{E}$ of Euclidean numbers of [4], where the transfinite sum of ordinal indexed $κ$-sequences of integers is well defined. In particular any ordinal might be identified with the transfiite sum of its characteristic function, preserving the so called natural operations. The ordered ring $\bf{Z}_κ$ may be obtained as an ultrapower of $\mathbb{Z}$ modulo suitable ultrafilters, thus constituting a \it{ring of nonstandard integers.} Most relevant is the \it{algebr…
The strange world of transfinite Melodies -- Recognizability for weak and strong infinite time $α$-register machines
本文研究指数封闭序数α上α-(w)ITRM的可识别性，刻画了失落旋律存在条件及可计算集与可识别集之间随α剧烈变化的关系。
We study Ramsey's theorem for pairs and two colours in the context of the theory of $α$-large sets introduced by Ketonen and Solovay. We prove that any $2$-colouring of pairs from an $ω^{300n}$-large set admits an $ω^n$-large homogeneous set. We explain how a formalized version of this bound gives a more direct proof, and a strengthening, of the recent result of Patey and Yokoyama [Adv. Math. 330 (2018), 1034--1070] stating that Ramsey's theorem for pairs and two colours is $\forallΣ^0_2$-conservative over the axiomatic theory $\mathsf{RCA}_0$ (recursive comprehension).
A set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD $\subseteq$ HNT $\subseteq$ V holds. Solving a problem recently proposed by Tzouvaras, a generic extension L$[a,x]$ of L, by two reals $a,x$, is presented in which L=HOD $\subsetneqq$ L$[a]$=HNT $\subsetneqq$ V=L$[a,x]$, so that HNT is a model of ZFC strictly between HOD and the universe.
In his dissertation, Wadge defined a notion of guessability on subsets of the Baire space and gave two characterizations of guessable sets. A set is guessable iff it is in the second ambiguous class (boldface Delta^0_2), iff it is eventually annihilated by a certain remainder. We simplify this remainder and give a new proof of the latter equivalence. We then introduce a notion of guessing with an ordinal limit on how often one can change one's mind. We show that for every ordinal alpha, a guessable set is annihilated by alpha applications of the simplified remainder if and only if it is guessa…
We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of computation on the reals and concepts of decidability and semi-decidability for sets of reals as well as individual reals. Every Pi^1_1 set, for example, is decidable by such machines, and the semi-decidable sets form a portion of the Delta^1_2 sets. Our oracle concept leads to a notion of relative computability for reals and sets of reals and a rich degree str…
By Tzouvaras, a set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD$\subseteq$HNT$\subseteq$V holds. Several questions about the nature of such sets, recently proposed by Tzouvaras, are solved in this paper. In particular, a model of ZFC is presented in which HOD$\subsetneqq$HNT$\subsetneqq$V, and another model of ZFC, in which HNT does not satisfy the axiom of choice.
Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive definitions, subsystems of analysis and finite type systems. With the pioneering work of Gerhard Jaeger in the late 1970s and early 1980s, the focus switched to set theories, furnishing ordinal-theoretic proof theory with a uniform and elegant framework. More recently it was shown that these tools can even sometimes be adapted to the context of strong axioms such as the powerset axiom, where one does not attain complete cut elimination but can nevertheless extract witnessing information and charact…
The barrier Ramsey theorem
本文提出了一种基于屏障大小的Ramsey定理扩展方法，通过定义Ramsey序数并利用Veblen函数迭代来计算其复杂度。
Constructive Ordinal Exponentiation
本文证明一般构造性序数指数运算不可行，并给出分别适用于正底数和含三歧最小元底数的两种近似构造，验证其等价性、代数律及 Agda 机械化。
In this paper we explore a connection between descriptive set theory and inner model theory. From descriptive set theory, we will take a countable, definable set of reals, A. We will then show that A is equal to the reals of M, where M is a canonical model from inner model theory. In technical terms, M is a ''mouse''. Consequently, we say that A is a mouse set. For a concrete example of the type of set A we are working with, let OD(n) be the set of reals which are Sigma-n definable over the omega-first level of the model L(R), from an ordinal parameter. In this paper we will show that for all …
This article is concerned with classifying the provably total set-functions of Kripke-Platek set theory, KP, and Power Kripke-Platek set theory, KP(P), as well as proving several (partial) conservativity results. The main technical tool used in this paper is a relativisation technique where ordinal analysis is carried out relative to an arbitrary but fixed set x. A classic result from ordinal analysis is the characterisation of the provably recursive functions of Peano Arithmetic, PA, by means of the fast growing hierarchy [10]. Whilst it is possible to formulate the natural numbers within KP,…
The strength of Ramsey's theorem for $α$-large sets
本文研究了Ramsey定理对有限颜色的特定自然数子集的反向数学强度，通过分析RT!α_k原理，揭示了可数不可分解的Γ0以下序数与自然Ramsey型定理之间的细粒度分类。
Let xi be a non-null countable ordinal. We study the Borel subsets of the plane that can be made $\bormxi$ by refining the Polish topology on the real line. These sets are called potentially $\bormxi$. We give a Hurewicz-like test to recognize potentially $\bormxi$ sets.
A generalized set theory (GST) is like a standard set theory but also can have non-set structured objects that can contain other structured objects including sets. This paper presents Isabelle/HOL support for GSTs, which are treated as type classes that combine features that specify kinds of mathematical objects, e.g., sets, ordinal numbers, functions, etc. GSTs can have an exception feature that eases representing partial functions and undefinedness. When assembling a GST, extra axioms are generated following a user-modifiable policy to fill specification gaps. Specialized type-like predicate…
Let $ξ\geq 1$ be a countable ordinal. We study the Borel subsets of the plane that can be made ${\bfΠ}^0_ξ$ by refining the Polish topology on the real line. These sets are called potentially ${\bfΠ}^0_ξ$. We give a Hurewicz-like test to recognize potentially ${\bfΠ}^0_ξ$ sets.
Sets with dependent elements: A formalization of Castoriadis' notion of magma
本文通过引入依赖关系和偏序结构，正式化了Castoriadis提出的magmas概念，并构建了magmatic层级体系。
Generalized fusible numbers and their ordinals
本文推广可熔数概念，研究由n元函数生成的良序集的序型，证明线性函数生成的序型至多为φ_{n-1}(0)，并构造连续函数使其序型趋近小Veblen序数。
Dense-set dependence in the Katětov order for uncountable coordinate ideals
该研究在Katětov序框架下，针对不可数坐标理想Conv(A,D)，在ZFC及CH公理下探讨其等价性与归约关系，证明CH下可数非小坐标界的严格性及非归约性
The rates of growth in a hyperbolic group
本文研究双曲群及其子群、子半群在所有有限生成集下的增长率集合，证明这些集合良序并确定极限群情形的序数为 $ω^ω$。
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