Related papers: Huge Reflection
In this paper, we prove that: if $\kappa$ is supercompact and the $\mathsf{HOD}$ Hypothesis holds, then there is a proper class of regular cardinals in $V_{\kappa}$ which are measurable in $\mathsf{HOD}$. Woodin also proved this result. As…
We provide analogues of the results from [FMR11, CMMR13] in the reference list (which correspond to the case $\kappa = \omega$) for arbitrary $\kappa$-Souslin quasi-orders on any Polish space, for $\kappa$ an infinite cardinal smaller than…
Set-theoretic axioms formulated in terms of existence of a Laver-generic large cardinal were introduced in [16] and studied further in [17], [18], [20]. These axioms, let us call them Laver-genericity axioms, claim the existence of a…
A ccc-generically supercompact cardinal $\kappa$ can be smaller than or equal to the continuum. On the other hand, such a cardinal $\kappa$ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically…
We generalise Jensen's result on the incompatibility of subcompactness with square. We show that alpha^+-subcompactness of some cardinal less than or equal to alpha precludes square_alpha, but also that square may be forced to hold…
We construct an embedding G of the category of graphs into the category of abelian groups such that for graphs X and Y we have Hom(GX,GY)=Z[Hom(X,Y)], the free abelian group whose basis is the set Hom(X,Y). The isomorphism is functorial in…
Let $G$ be a reflection group acting on a vector space $V$ (over a field with zero characteristic). We denote by $S(V^*)$ the coordinate ring of $V$, by $M$ a finite dimensional $G$-module and by $\chi$ a one-dimensional character of $G$.…
There is no infinite sequence of $\Pi^1_1$-sound extensions of $\mathsf{ACA}_0$ each of which proves $\Pi^1_1$-reflection of the next. This engenders a well-founded ``reflection ranking'' of $\Pi^1_1$-sound extensions of $\mathsf{ACA}_0$.…
Answering one of the main questions of [FHK14, Chapter 7], we show that there is a tight connection between the depth of a classifiable shallow theory $T$ and the Borel rank of the isomorphism relation $\cong^\kappa_T$ on its models of size…
We obtain a characterization of left perfect rings via superstability of the class of flat left modules with pure embeddings. $\mathbf{Theorem.}$ For a ring $R$ the following are equivalent. - $R$ is left perfect. - The class of flat left…
Scaling, hyperscaling and finite-size scaling were long considered problematic in theories of critical phenomena in high dimensions. The scaling relations themselves form a model-independent structure that any model-specific theory must…
If cf(kappa) = kappa, kappa^+< cf(lambda) = \lambda, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing…
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…
This dissertation includes many theorems which show how to change large cardinal properties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new…
In this paper we investigate more characterizations and applications of $\delta$-strongly compact cardinals. We show that, for a cardinal $\kappa$ the following are equivalent: (1) $\kappa$ is $\delta$-strongly compact, (2) For every…
One of the earliest conjectures in computational learning theory-the Sample Compression conjecture-asserts that concept classes (equivalently set systems) admit compression schemes of size linear in their VC dimension. To-date this…
In mathematical logic there are two seemingly distinct kinds of principles called "reflection principles." Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic…
We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vop\v{e}nka's…
In this paper, we introduce the notion of large scale resemblance structure as a new large scale structure by axiomatizing the concept of `being alike in large scale' for a family of subsets of a set. We see that in a particular case, large…
We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $\omega_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The…