On first-order expressibility of satisfiability in submodels

Let $\kappa,\lambda$ be regular cardinals, $\lambda\le\kappa$, let $\varphi$ be a sentence of the language $\mathcal L_{\kappa,\lambda}$ in a given signature, and let $\vartheta(\varphi)$ express the fact that $\varphi$ holds in a submodel, i.e., any model $\mathfrak A$ in the signature satisfies $\vartheta(\varphi)$ if and only if some submodel $\mathfrak B$ of $\mathfrak A$ satisfies $\varphi$. It was shown in [1] that, whenever $\varphi$ is in $\mathcal L_{\kappa,\omega}$ in the signature having less than $\kappa$ functional symbols (and arbitrarily many predicate symbols), then $\vartheta(\varphi)$ is equivalent to a monadic existential sentence in the second-order language $\mathcal L^{2}_{\kappa,\omega}$, and that for any signature having at least one binary predicate symbol there exists $\varphi$ in $\mathcal L_{\omega,\omega}$ such that $\vartheta(\varphi)$ is not equivalent to any (first-order) sentence in $\mathcal L_{\infty,\omega}$. Nevertheless, in certain cases $\vartheta(\varphi)$ are first-order expressible. In this note, we provide several (syntactical and semantical) characterizations of the case when $\vartheta(\varphi)$ is in $\mathcal L_{\kappa,\kappa}$ and $\kappa$ is $\omega$ or a certain large cardinal

Set Theory and its Place in the Foundations of Mathematics:a new look at an old question

This paper reviews the claims of several main-stream candidates to be the foundations of mathematics, including set theory. The review concludes that at this level of mathematical knowledge it would be very unreasonable to settle with any one of these foundations and that the only reasonable choice is a pluralist one

A multiverse perspective on the axiom of constructibility

I shall argue that the commonly held V ≠= L via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V = L inside larger set-Theoretic universes

Bericht ueber die Taetigkeit der Bergbehoerden des Landes Nordrhein-Westfalen

SIGLEAvailable from TIB Hannover: ZO 4042(1990) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

ZFC proves that the class of ordinals is not weakly compact for definable classes

In ZFC, the class Ord of ordinals is easily seen to satisfy the definable version of strong inaccessibility. Here we explore deeper ZFC-verifiable combinatorial properties of Ord, as indicated in Theorems A & B below. Note that Theorem A shows the unexpected result that Ord is never definably weakly compact in any model of ZFC. Theorem A. Let be any model of ZFC. (1)The definable tree property fails in : There is an -definable Ord-tree with no -definable cofinal branch. (2)The definable partition property fails in : There is an -definable 2-coloring for some -definable proper class X such that no -definable proper classs is monochromatic for f. (3)The definable compactness property for fails in : There is a definable theory in the logic (in the sense of ) of size Ord such that every set-sized subtheory of is satisfiable in , but there is no -definable model of . Theorem B. The definable ⋄Ord principle holds in a model of ZFC iff carries an -definable global well-ordering. Theorems A and B above can be recast as theorem schemes in ZFC, or as asserting that a single statement in the language of class theory holds in all ‘spartan’ models of GB (Gödel-Bernays class theory); where a spartan model of GB is any structure of the form , where and is the family of -definable classes. Theorem C gauges the complexity of the collection GBspa of (Gödel-numbers of) sentences that hold in all spartan models of GB. Theorem C. GBspa is -complete

Bi-interpretation in weak set theories

In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory ZFC− without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC− that are bi-interpretable, but not isomorphic—even hHω, ∈i and hHω2, ∈i can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC−. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails