We prove that for every antichain A in the poset (langle [omega ]^{<omega },subseteq rangle ) the set of maximal antichains which extend A is either finite or has the size of the continuum. As a consequence we prove a conjecture of de Jongh and Vargas-Sandoval about nepfi families of finite languages [2, 10].
We show that a measurable cardinal is enough in order to construct two distinct countably closed mutually embeddable models. This answers a question from Eskew, M., et al.: Annals of Pure and Applied Logic, Vol. 175, Issue 1, Part B, 103325 (2024).
We describe specific fragments of the immune system using relational systems (Kripke frames) that represent the semantics of the novel logic EŁ(_{P})-Modal Epistemic Łukasiewicz logic. The language of EŁ(_{P}) extends the infinitely valued Łukasiewicz logic Ł by introducing a unary connective, interpreted as a modal epistemic operator that denotes knowledge and quasi-knowledge. This paper demonstrates that theorems within this logic hold for the immune system model. Furthermore, we propose conjectures related to the model that are unresolved by immune scientists, striving to prove or refute these conjectures as theorems. Additionally, we investigate the decidability and unification problems of the corresponding logic and its admissible rules.
We study the relationship between non-trivial values of generalized cardinal invariants at an inaccessible cardinal (kappa ) and compactness principles at (kappa ^+) and (kappa ^{++}). Let (textsf {TP}(kappa ^{++})), (textsf {SR}(kappa ^{++})) and (lnot textsf {wKH}(kappa ^+)) denote the tree property and stationary reflection on (kappa ^{++}) and the negation of the weak Kurepa Hypothesis on (kappa ^+), respectively. We show that if the existence of a supercompact cardinal (kappa ) with a weakly compact cardinal (lambda ) above (kappa ) is consistent, then the following are consistent as well (where (mathfrak {t}(kappa )) and (mathfrak {u}(kappa )) are the tower number and the ultrafilter number, respectively): (i) There is an inaccessible cardinal (kappa ) such that (kappa ^+< mathfrak {t}(kappa )= mathfrak {u}(kappa )< 2^kappa ) and (textsf {SR}(kappa ^{++})) hold, and (ii) There is an inaccessible cardinal (kappa ) such that (kappa ^+ = mathfrak {t}(kappa )< mathfrak {u}(kappa )< 2^kappa ) and (textsf {SR}(kappa ^{++}), textsf {TP}(kappa ^{++})) and (lnot textsf {wKH}(kappa ^+)) hold. The cardinals (mathfrak {u}(kappa )) and (2^kappa ) can have any reasonable values in these models. We obtain these results by combining the forcing construction from [4] due to Brooke-Taylor, Fischer, Friedman and Montoya with the Mitchell forcing and with (new and old) indestructibility results related to (textsf {TP}(kappa ^{++})), (textsf {SR}(kappa ^{++})) and (lnot textsf {wKH}(kappa ^+)). Apart from (mathfrak {u}(kappa )) and (mathfrak {t}(kappa )) we also compute the values of (mathfrak {b}(kappa )), (mathfrak {d}(kappa )), (mathfrak {s}(kappa )), (mathfrak {r}(kappa )), (mathfrak {a}(kappa )), (textrm{cov}({mathcal {M}}_kappa )), (textrm{add}({mathcal {M}}_kappa )), (textrm{non}({mathcal {M}}_kappa )), (textrm{cof}({mathcal {M}}_kappa )) which will all be equal to (mathfrak {u}(kappa )). In (ii), we compute (mathfrak {p}(kappa ) = mathfrak {t}(kappa ) = kappa ^+) by observing that the (kappa ^+)-distributive quotient of the Mitchell forcing adds a tower of size (kappa ^+). Finally, as a corollary of the construction, we observe that items (i) and (ii) hold also for the traditional invariants on (kappa = omega ), using Mitchell forcing up to a weakly compact cardinal; in this case we also obtain the disjoint stationary sequence property (textsf {DSS}(omega _2)), which implies the negation of the app
This article provides a language-theoretic rendering of Herbrand’s theorem. To each first-order proof is associated a higher-order recursion scheme that abstracts the computation of Herbrand sets obtained through Gentzen-style multicut elimination. The representation extends previous results in this area by lifting the prenex restriction on cut formulas and relaxing the cut-elimination strategies. Features of the new approach are the interpretation of cut as simple composition and contraction as ‘call with current continuation’.
I study definability notions in the framework of affine continuous logic. Some general results concerning types and definable relations in affinely complete theories are proved. If the theory has a first order model, its extremal theory is a complete first order theory and first order definable sets are affinely definable. If it has a compact model, definable sets are exactly the end-sets of definable predicates. As an example, it is proved in the theory of probability algebras that one dimensional definable sets are exactly the intervals [a, b].
For mass problems (P,Qsubseteq {mathbb {N}^mathbb {N}}) (Baire space), P is Medvedev reducible to Q ((Ple _sQ)) if for some Turing funcional (Phi ), (Phi (Q)subseteq P), and Medvedev equivalent to Q if also (Qle _sP). Shafer asked if every closed problem P is Medvedev equivalent to a closed problem Q with (Qsubseteq 2^mathbb {N}) (Cantor space). We show that this is not the case.
We prove that a countable direct sum of chains has one, countably many or else continuum many isomorphism classes of siblings. This proves Thomassé’s conjecture for such structures. Further, we show that a direct sum of chains of any cardinality has one or infinitely many siblings, up to isomorphism.
We consider cardinal invariants determined by Hausdorff measures. We separate many cardinal invariants of Hausdorff measure 0 ideals using two models that separate many cardinal invariants of Yorioka ideals at once from earlier work. Also, we show the uniformity numbers of s-dimensional Hausdorff measure 0 ideals for (0< s < 1) and of the Lebesgue null ideal can be separated using the Mathias forcing.
There is no infinite sequence of (Pi ^1_1)-sound extensions of (textsf{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 (textsf{ACA}_0). For any (Pi ^1_1)-sound theory T extending (textsf{ACA}^+_0), the reflection rank of T equals the proof-theoretic ordinal of T. This provides an alternative characterization of the notion of “proof-theoretic ordinal,” which is one of the central concepts of proof theory. We provide an alternative proof of this theorem using cut-elimination for infinitary derivations.
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