PA is Peano Arithmetic. Pr(x) is the usual Σ1-formula representing provability in PA. A strong provability predicate is a formula which has the same properties as Pr(·) but is not Σ1. An example: Q is ω-provable if PA + ¬ Q is ω-inconsistent (Boolos [4]). In [5] Dzhaparidze introduced a joint provability logic for iterated ω-provability and obtained its arithmetical completeness. In this paper we prove some further modal properties of Dzhaparidze's logic, e.g., the fixed point property and the Craig (...) interpolation lemma. We also consider other examples of the strong provability predicates and their applications. (shrink)

We say that two arithmetical formulas A, B have the Σ1-interpolation property if they have an ‘interpolant’ σ, i.e., a Σ1 formula such that the formulas A→σ and σ→B are provable in Peano Arithmetic PA. The Σ1-interpolability predicate is just a formalization of this property in the language of arithmetic.Using a standard idea of Gödel, we can associate with this predicate its provability logic, which is the set of all formulas that express arithmetically valid principles in the modal language with (...) two modal operators, □ for provability in PA and for Σ1-interpolability. In this paper we determine this provability logic .It turns out that this problem appears to be quite similar to the problem of such modal description of relative interpretability, a problem, solved by Berarducci and Shavrukov. However, the present case is sometimes easier. In particular, we prove in this paper the fixed point property and the Craig interpolation property for our modal system ELH. (shrink)

In this paper I explore white attempts to move toward a proactive position against racism that will amount to more than self-criticism in the following three ways: by assessing the debate within feminism over white women's relation to whiteness; by exploring "white awareness training" methods developed by Judith Katz and the "race traitor" politics developed by Ignatiev and Garvey, and; a case study of white revisionism being currently attempted at the University of Mississippi.

For any ordinal $\Lambda$, we can define a polymodal logic $\mathsf{GLP}_\Lambda$, with a modality $[\xi]$ for each $\xi < \Lambda$. These represent provability predicates of increasing strength. Although $\mathsf{GLP}_\Lambda$ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted $\mathsf{GLP}^0_\omega$. Later, Icard defined a topological model for $\mathsf{GLP}^0_\omega$ which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for (...) arbitrary $\Lambda$. More generally, for each $\Theta,\Lambda$ we build a Kripke model $\mathfrak I^\Theta_\Lambda$ and a topological model $\mathfrak T^\Theta_\Lambda$, and show that $\mathsf{GLP}^0_\Lambda$ is sound for both of these structures, as well as complete, provided $\Theta$ is large enough. (shrink)

Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories: iterate adding consistency of some weak base theory until you “hit” the target theory. Turing progressions based on n-consistency give rise to a \ proof-theoretic ordinal \ also denoted \. As such, to each theory U we can assign the sequence of corresponding \ ordinals \. We call this sequence a Turing-Taylor expansion or spectrum of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories (...) of Peano Arithmetic to Ignatiev’s universal model for the closed fragment of the polymodal provability logic \. In particular, we observe that each point in the Ignatiev model can be seen as Turing-Taylor expansions of formal mathematical theories. Moreover, each sub-theory of Peano Arithmetic that allows for a Turing-Taylor expansion will define a unique point in Ignatiev’s model. (shrink)