Gabriel Fernandes, Miguel Moreno, Assaf Rinot
We introduce a generalization of stationary set reflection which we call "filter reflection", and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We prove the independence of filter reflection from ZFC, and present applications of filter reflection to the study of canonical equivalence relations of the higher Cantor and Baire spaces.
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The Axiom of Full Reflection at a measurable cardinal has been conjectured to be equiconsitent with the existence of a coherent sequence of measures with a repeat point. However we prove that the Axiom of Full Reflection at a measurable cardinal is surprisingly equiconsistent just with the existence of a measurable cardinal. We also generalize the result to larger cardinals as strong or supercompact. Hence we can conclude that the Axiom of Full Reflection at large cardinals weaker than measurable, e.g. as n-Mahlo, does push the consistency strength up, but does not push the consistency strengt…
Forcing axioms, approachability, and stationary set reflection
本文证明平稳集反射与内部可逼近性的多种定理，指出Fuchino-Usuba蕴含不可逆，加强并简化Krueger结果，并借鉴Woodin思想统一力迫公理保持论证。
In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2^omega = omega_2 and that L(P(omega_1)) satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that combinatorial principle Square(kappa) fails for all regular kappa > omega_1.
Reflection Principles in ZFU
ZFU中分离了集合原理、反射原理和部分反射原理，并探讨了它们在不同公理假设下的等价性与独立性。
For an arbitrary forcing class $Γ$, the $Γ$-fragment of Todorcevic's strong reflection principle SRP is isolated in such a way that (1) the forcing axiom for $Γ$ implies the $Γ$-fragment of SRP, (2) the stationary set preserving fragment of SRP is the full principle SRP, and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. Along the way, some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity are explored, and some limitations to the extent to which fragments of SRP may capture the effects of their correspond…
William Tait and Peter Koellner have written on the topic of which reflection principles are intrinsically justified. Phillip Welch and Sam Roberts have recently sought to motivate much stronger reflection principles. This work also had some similarities with previous work of Victoria Marshall's on reflection principles, as may be seen from considerations which we will seek to articulate in what follows. The statement of our three philosophical theses shall be given in the Introduction. We will present a set of technical results in support of three philosophical theses which will be articulate…
A formula $ϕ$ is called \emph{$n$-provable} in a formal arithmetical theory $S$ if $ϕ$ is provable in $S$ together with all true arithmetical $Π_{n}$-sentences taken as additional axioms. While in general the set of all $n$-provable formulas, for a fixed $n>0$, is not recursively enumerable, the set of formulas $ϕ$ whose $n$-provability is provable in a given r.e.\ metatheory $T$ is r.e. This set is deductively closed and will be, in general, an extension of $S$. We prove that these theories can be naturally axiomatized in terms of progressions of iterated local reflection principles. In parti…
We recently formulated a new large-cardinal axiom of strength intermediate between a totally indescribable cardinal and an $ω$-Erdős cardinal, positing the existence of what we called an "extremely reflective cardinal", and we showed that the property of being extremely reflective was in fact equivalent to the property of being remarkable, and we sought to argue that this axiom should be seen as intrinsically justified. This built on related earlier work in which the notion of an $α$-reflective cardinal was formulated. Then Welch and Roberts put forward a family of reflection principles, Welch…
We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original system of arithmetic. Much stronger systems, however, are obtained by adding either induction axioms or reflection axioms on top of them. Theories of this kind can interpret some well-known predicatively reducible fragments of second-order arithmetic such as iterated arithmetical comprehension. We obtain sharp results on the proof-theoretic strength of thes…
The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega = {a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.
In this note we axiomatize the $Π_{k+1}$-consequences in the set theory ${\sf KP}Π_{N}$ for $Π_{N}$-reflecting universes in terms of iterations of $Π_{i}$-recursively Mahlo operations for $1\leq k\leq i<N$.
Generalized ordinal analysis and reflection principles in set theory
针对自然公理系统良序层级的质疑，本文基于集合论反射原理提出证明论强度概念，证明其与Pohlers广义序数分析的理论序一致，生成良序层级，并推广至其他内模型。
Reflection and Recurrence
本文研究结构性反射原则与递归公理对 ZFC 的扩展，提出其极大一致组合 Laver-generic Maximum（LGM），将连续统确定为 aleph_2 并整合多种已知独立陈述。
These notes are extracted from the lectures on forcing axioms and applications held by professor Matteo Viale at the University of Turin in the academic year 2011-2012. Our purpose is to give a brief account on forcing axioms with a special focus on some consequences of them (SRP, OCA, PID). These principles were first isolated by Todor\v cević and interpolate most consequences of MM and PFA, thus providing a useful insight on the combinatorial structure of the theory of forcing axioms. In the first part of this notes we will give a brief account on forcing axioms, introducing some equivalent …
Set Theory with Urelements
本论文系统研究含本元的集合论，围绕其公理化、力迫法、类理论等方向展开，提出力迫新方法并构造相关模型，完善了该领域的理论体系。
We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss and prove that they follow from stationary and semi stationary set reflection augmented with a weak form of Martin's Axiom. We also show that there are some differences between the two reflection principles which suggest that stationary set reflection is analogous to supercompactness whereas semi-stationary set r…
We show that induction over $Δ(\mathbb R)$-definable well-founded classes is equivalent to the reflection principle which asserts that any true formula of first order set theory with real parameters holds in some transitive set. The equivalence is proved in primitive recursive set theory (which is weaker than Kripke-Platek set theory) extended by the axiom of dependent choice.
Plenitudinous Urelements and the Definability of Cardinality
本文研究了完备性公理与基数定义在包含 urelements 的 ZF 集合论中的相互关系及推导效果。
Rado's Conjecture is a compactness/reflection principle that says any nonspecial tree of height $ω_1$ has a nonspecial subtree of size $\leq \aleph_1$. Though incompatible with Martin's Axiom, Rado's Conjecture turns out to have many interesting consequences that are consequences of forcing axioms. In this paper, we obtain consistency results concerning Rado's Conjecture and its Baire version. In particular, we show a fragment of PFA, that is the forcing axiom for \emph{Baire Indestructibly proper forcings}, is compatible with the Baire Rado's Conjecture. As a corollary, Baire Rado's Conjectur…
Approximation of hyperarithmetic analysis by $ω$-model reflection
本文提出依赖选择公理的新变体并证明其属于超算术分析，同时证明基于 $ω$-模型反射公理的理论类 $\text{RFN}^{-1}(\text{ATR}_0)$ 可部分逼近超算术分析。
Axiomatization and Forcing in Set Theory with Urelements
本文系统研究含本原元素集合论中的公理层级，并提出新的力迫名称定义以证明力迫基本定理，进而揭示力迫对公理的破坏与恢复以及地面模型可定义性的失效现象。
A stationary subset S of a regular uncountable cardinal kappa reflects fully at regular cardinals if for every stationary set T subseteq kappa of higher order consisting of regular cardinals there exists an alpha in T such that S cap alpha is a stationary subset of alpha. We prove that the Axiom of Full Reflection which states that every stationary set reflects fully at regular cardinals, together with the existence of n-Mahlo cardinals is equiconsistent with the existence of Pi^1_n-indescribable cardinals. We also state the appropriate generalization for greatly Mahlo cardinals.
It is well-known that the square principle $\square_λ$ entails the existence of a non-reflecting stationary subset of $λ^+$, whereas the weak square principle $\square^*_λ$ does not. Here we show that if $μ^{\mathrm{cf}(λ)} < λ$ for all $μ< λ$, then $\square^*_λ$ entails the existence of a non-reflecting stationary subset of $E^{λ^+}_{\mathrm{cf}(λ)}$ in the forcing extension for adding a single Cohen subset of $λ^+$. It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplet…
Continuing the previous paper, we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary logic and their variations. It has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters down to $<\aleph_2$ is equivalent to the conjunction of CH and Cox's Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak second-order parameters down to $<2^{\aleph_0}$ implies that the size of the continuum is $\aleph_2$. In contrast, an internal interpretation of the stationary logic can satisf…
This paper presents the logic QRC$_1$, which is a strictly positive fragment of quantified modal logic. The intended reading of the diamond modality is that of consistency of a formal theory. Predicate symbols are interpreted as parametrized axiomatizations. We prove arithmetical soundness of the logic QRC$_1$ with respect to this arithmetical interpretation. Quantified provability logic is known to be undecidable. However, the undecidability proof cannot be performed in our signature and arithmetical reading. We conjecture the logic QRC$_1$ to be arithmetically complete. This paper takes the …
A model of the Axiom of Determinacy in which every set of reals is universally Baire
本文基于导出模型构造，结合泛型迭代方法，证明了满足“每个实数集都是泛Baire集”的确定性公理模型的一致性强度对应存在武丁基数与强基数的极限基数的ZFC系统。
We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The upwards reflection theorems are obtained in the presence of a forcing axiom, while most of the downwards reflection results use large cardinal assumptions. The combinatorial content of arguments showing that a given space is a $D$-space, can be formulated using the concept of discrete families. We note the connection between non-reflection arguments involving dis…
Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal
本文探讨了带原子的二阶集合论中二阶反射的强度，证明了带丰富原子公理的二阶反射与超紧致基数存在性可互译且等一致。
We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The upwards reflection theorems are obtained in the presence of a forcing axiom, while most of the downwards reflection results use large cardinal assumptions. The combinatorial content of arguments showing that a given space is a $D$-space, can be formulated using the concept of discrete families. We note the connection between non-reflection arguments involving dis…
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。
Tree Rewriting Calculi for Strictly Positive Logics
本文提出严格正逻辑的树重写演算，通过公式与模态树的对应建立充分性和完备性，并给出规范化复杂度（指数或双指数上界），深化其证明论与计算应用。
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