Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1710.10825

Characterizations of pretameness and the Ord-cc

Peter Holy·Regula Krapf·Philipp Schlicht
View original source ↗Read source document ↗Export metadata ↓

Abstract

It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of $\mathsf{ZF}^-$, that is $\mathsf{ZF}$ without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of $\mathsf{ZF}$. We show that pretameness in fact has various other characterizations, for instance in terms of the forcing theorem, the preservation of the axiom scheme of separation, the forcing equivalence of partial orders and their dense suborders, and the existence of nice names for sets of ordinals. These results show that pretameness is a strong dividing line between well and badly behaved notions of class forcing, and that it is exactly the right notion to consider in applications of class forcing. Furthermore, for most properties under consideration, we also present a corresponding characterization of the $\mathrm{Ord}$-chain condition.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Peter Holy, Regula Krapf, Philipp Schlicht. 2017-10-30. Characterizations of pretameness and the Ord-cc. https://arxiv.org/abs/1710.10825

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bluebirds and mockingbirds cannot produce a fixed-point combinator

Let $B$ be the bluebird combinator with reduction rule $Bxyz \to_{w} x\left(yz\right)$, let $M$ be the mockingbird combinator with reduction rule $Mx \to_{w} xx$, and let $I$ be the identity bird combinator with reduction rule $Ix \to_{w} x$. A fixed-point combinator, called a sage bird by Smullyan, is a closed term $Y$ such that, for a fresh variable $x$, $Yx$ is equivalent to $x\left(Yx\right)$ under these reduction rules. For a fixed variable $x$, we construct an invariant $\mathrm{Tr}_{x}\left(u\right)$ of a $BMI$-term $u$ with respect to $\to_{w}$. This invariant traces the occurrences of $x$ in the leftmost-innermost reduction sequence of $u$. We then prove that $\mathrm{Tr}_{x}\left(Yx\right) \neq \mathrm{Tr}_{x}\left(x^{r}\left( Yx \right)\right)$ for every $x$-free $BMI$-term $Y$ and every $r\geq 1$. Consequently, there exists no fixed-point combinator in $BMI$-combinatory logic. This provides a negative answer to the problem posed by Smullyan in 1985.

math.LO↗
arXiv2026

Pointwise provable equality and the failure of composition

Montagna (1989) and Di Paola--Montagna (1991) claim that the algebraic systems $S'$ and $S'_T$, respectively, are categories. We show that the proposed composition is not independent of the choice of representatives. For every consistent recursively enumerable extension $T$ of Peano arithmetic ($\mathrm{PA}$), we exhibit two program indices that are pointwise provably equal in $T$ but yield inequivalent composites when each is run after the same program. Montagna's $S'$ is the case $T=\mathrm{PA}$. The failure already occurs for partial maps from $ω$ to itself. Weak totality and the proposed range assignment also depend on the choice of representatives. More generally, for consistent $T\supseteq\mathrm{PA}$, pointwise provable equality is a composition congruence exactly when $T$ proves every true $Π^0_1$ sentence, in which case it is extensional equality. This completeness condition fails for every consistent recursively enumerable $T\supseteq\mathrm{PA}$ by Gödel's second incompleteness theorem. For every extension $T\supseteq\mathrm{PA}$, the least composition congruence containing pointwise provable equality is extensional equality if $T$ is $Σ^0_1$-sound and the universal relation otherwise.

math.LO↗
arXiv2026

Compactness via Consistency Properties

We will use consistency properties to characterize strongly compact cardinals, first showing an adequate Model Existence Theorem for larger fragments.

math.LO↗