In this paper we study the notion of strong non-reflection, and its contrapositive weak reflection. We say theta strongly non-reflects at lambda iff there is a function F: theta ---> lambda such that for all alpha < theta with cf(alpha)= lambda there is C club in alpha such that F restriction C is strictly increasing. We prove that it is consistent to have a cardinal theta such that strong non-reflection and weak reflection each hold on an unbounded set of cardinals less than theta .
A new large-cardinal property is introduced which enables one to give a relative consistency proof of restricted versions of the reflection principles discussed by Tait in his essay "Constructing Cardinals from Below".
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…
We describe a "slow" version of the hierarchy of uniform reflection principles over Peano Arithmetic ($\mathbf{PA}$). These principles are unprovable in Peano Arithmetic (even when extended by usual reflection principles of lower complexity) and introduce a new provably total function. At the same time the consistency of $\mathbf{PA}$ plus slow reflection is provable in $\mathbf{PA}+\operatorname{Con}(\mathbf{PA})$. We deduce a conjecture of S.-D. Friedman, Rathjen and Weiermann: Transfinite iterations of slow consistency generate a hierarchy of precisely $\varepsilon_0$ stages between $\mathb…
In this paper we investigate the consequences and consistency of the downward Löwenheim-Skolem theorem for extension of the first order logic by the Magidor-Malitz quantifier. We derive some combinatorial results and improve the known upper bound for the consistency of Chang's Conjecture at successor of singular cardinals.
Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection
The notion of stationary reflection is one of the most important notions of combinatorial set theory. We investigate weak reflection, which is, as the name suggests, a weak version of stationary reflection. This sort of reflection was introduced in [DjSh:545] (math.LO/9601219), where it was shown that weak reflection has applications to various guessing principles, in the sense that if there is no weak reflection, than a guessing principle holds, and an application dealing with the saturation of normal filters. Further investigations of weak reflection were carried in [CDSh:571] (math.LO/95042…
The weakly compact reflection principle $\text{Refl}_{\text{wc}}(κ)$ states that $κ$ is a weakly compact cardinal and every weakly compact subset of $κ$ has a weakly compact proper initial segment. The weakly compact reflection principle at $κ$ implies that $κ$ is an $ω$-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that $κ$ is $(ω+1)$-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at $κ$ then there is a forcing extension preserving this in which $κ$ is the least $ω$-weakly compact cardinal.…
We prove a weakened version of the reflection of Reinhardt cardinals by super Reinhardt cardinals: Let $M=(V^M,P)$ be a countable model of second order set theory $\mathsf{ZF}_2$ (with universe $V^M$ and classes $P$) which models "$κ$ is super Reinhardt". We show that there are unboundedly many $μ<κ$ such that there is $j$ such that $(V^M,j)$ models $\mathsf{ZF}(j)+$"$μ$ is Reinhardt, as witnessed by $j$". In particular, $j\upharpoonright X\in V^M$ for all $X\in V^M$ (but we allow $j\notin P$).
We show that square(theta) implies that there is a first countable <theta-collectionwise Hausdorff space that is not weakly theta-collectionwise Hausdorff. We also show that in the model obtained by Levy collapsing a weakly compact (supercompact) cardinal to omega_2, first countable aleph_1-collectionwise Hausdorff spaces are weakly aleph_2-collectionwise Hausdorff (weakly collectionwise Hausdorff). In the last section we show that assuming E^omega_theta, a certain theta-family of integer valued functions exists and that in the model obtained by Levy collapsing a supercompact cardinal to omega…
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…
We prove that a strong version of Chang's Conjecture, equivalent to the Weak Reflection Principle at $ω_2$, together with $2^ω=ω_2$, imply there are no $ω_2$-Aronszajn trees.
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…
For $n<ω$, we say that the $Π^1_n$-reflection principle holds at $κ$ and write $\text{Refl}_n(κ)$ if and only if $κ$ is a $Π^1_n$-indescribable cardinal and every $Π^1_n$-indescribable subset of $κ$ has a $Π^1_n$-indescribable proper initial segment. The $Π^1_n$-reflection principle $\text{Refl}_n(κ)$ generalizes a certain stationary reflection principle and implies that $κ$ is $Π^1_n$-indescribable of order $ω$. We define a forcing which shows that the converse of this implication can be false in the case $n=1$. Moreover, we prove that if $κ$ is $(α+1)$-weakly compact where $α<κ^+$, then ther…
Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle $\mathrm{SR}^-$ introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle $\mathr…
We indicate that an argument of da Costa and Doria in fact proves P=NP. This observation makes their argument appear dubious. We isolate a weak version of one of their lemmas which would already prove P=NP. We point out that even this weak version is probably false. In fact, a generalized form of this weak version is provably false.
Schmerl and Beklemishev's work on iterated reflection achieves two aims: It introduces the important notion of $Π^0_1$-ordinal, characterizing the $Π^0_1$-theorems of a theory in terms of transfinite iterations of consistency; and it provides an innovative calculus to compute the $Π^0_1$-ordinals for a range of theories. The present note demonstrates that these achievements are independent: We read off $Π^0_1$-ordinals from a Schütte-style ordinal analysis via infinite proofs, in a direct and transparent way.
In the paper we introduce a weak set theory $\mathsf{H}_{<ω}$ . A formalization of arithmetic on finite von Neumann ordinals gives an embedding of arithmetical language into this theory. We show that $\mathsf{H}_{<ω}$ proves a natural arithmetization of its own Hilbert-style consistency. Unlike some previous examples of theories proving their own consistency, $\mathsf{H}_{<ω}$ appears to be sufficiently natural. The theory $\mathsf{H}_{<ω}$ is infinitely axiomatizable and proves existence of all individual hereditarily finite sets, but at the same time all its finite subtheories have finite mo…
