Some modifications of Scott's theorem on injective spaces

D. Scott in his paper [5] on the mathematical models for the Church-Curry -calculus proved the following theorem.A topological space X. is an absolute extensor for the category of all topological spaces iff a contraction of X. is a topological space of Scott's open sets in a continuous lattice.In this paper we prove a generalization of this theorem for the category of , -closure spaces. The main theorem says that, for some cardinal numbers , , absolute extensors for the category of , -closure spaces are exactly , -closure spaces of , -filters in , >-semidistributive lattices (Theorem 3.5).If = and = we obtain Scott's Theorem (Corollary 2.1). If = 0 and = we obtain a characterization of closure spaces of filters in a complete Heyting lattice (Corollary 3.4). If = 0 and = we obtain a characterization of closure space of all principial filters in a completely distributive complete lattice (Corollary 3.3).     

Generalization of Scott's formula for retractions from generalized Alexandroff's cube

In the paper [2] the following theorem is shown: Theorem (Th. 3,5, [2]), If α=0 or δ=∞ or α?δ, then a closure space X is an absolute extensor for the category of 〈α, δ〉 -closure spaces iff a contraction of X is the closure space of all 〈α, δ〉-filters in an 〈α, δ〉-semidistributive lattice. In the case when α=ω and δ=∞, this theorem becomes Scott's theorem: Theorem ([7]). A topological space X is an absolute extensor for the category of all topological spaces iff a contraction of X is a topological space of “Scott's open sets” in a continuous lattice. On the other hand, when α=0 and δ=ω, this theorem becomes Jankowski's theorem: Theorem ([4]). A closure space X is an absolute extensor for the category of all closure spaces satisfying the compactness theorem iff a contraction of X is a closure space of all filters in a complete Heyting lattice. But for separate cases of α and δ, the Theorem 3.5 from [2] is proved using essentialy different methods. In this paper it is shown that this theorem can be proved using, for retraction, one uniform formula. Namely it is proved that if α= 0 or δ= ∞ or α ? δ and \(F_{\alpha ,\delta } \left( L \right) \subseteq B_{\alpha ,\delta }^\mathfrak{n} \) and if L is an 〈α, δ〉-semidistributive lattice, then the function $$r:{\text{ }}B_{\alpha ,\delta }^\mathfrak{n} \to F_{\alpha ,\delta } \left( L \right)$$ such that for x ε ? ( \(\mathfrak{n}\) ): (*) $$r\left( x \right) = inf_L \left\{ {l \in L|\left( {\forall A \subseteq L} \right)x \in C\left( A \right) \Rightarrow l \in C\left( A \right)} \right\}$$ defines retraction, where C is a proper closure operator for \(B_{\alpha ,\delta }^\mathfrak{n} \) . It is also proved that the formula (*) defines retraction for all 〈α, δ〉, whenever L is an 〈α, δ〉 -pseudodistributive lattice. Moreover it is proved that when α=ω and δ=∞, the formula (*) defines identical retraction to the formula given in [7], and when α = 0 and δ=ω, the formula (*) defines identical retraction to the formula given in [4].     

On the extension of intuitionistic propositional logic with Kreisel-Putnam's and Scott's schemes

LetSKP be the intermediate prepositional logic obtained by adding toI (intuitionistic p.l.) the axiom schemes:S = ((? ?α→α)→α∨ ?α)→ ?α∨ ??α (Scott), andKP = (?α→β∨γ)→(?α→β)∨(?α→γ) (Kreisel-Putnam). Using Kripke's semantics, we prove:

1. SKP has the finite model property;
2. SKP has the disjunction property.

In the last section of the paper we give some results about Scott's logic S = I+S.     

Floating with Kim Scott's Benang: Vertigo,settler-colonial mobilities and levitation's geographies

In this paper I explore the strange figure of the levitator within Kim Scott's (1999) Miles Franklin Prize winning novel Benang: From the Heart. Through different genres of art and creative practice, mysticism, religion, science-fiction, magic, and even civil disobedience, the levitator is a poorly acknowledged mobile subject who seems to refuse scholarly enquiry. Levitation is more readily understood as a maligned form of fraud, fakery and social frippery, a figure of esoteric interest. Building on recent attempts to resurrect and reconsider levitators (Adey, 2017; Young, 2018), as well as floating, lighter-than-air atmospheres and elements (McCormack, 2018; Engelmann, 2015), this paper argues that floatations like levitation provide a crucial addition to critical and radical thinking in mobilities, affective life and studies of settler-colonialism. Through Benang, Kim Scott's vast historical and yet intimate novel, the paper works with the floating encounters woven through the politics and ecologies of Western Australia and its racist policies which sought to regulate the Noongar people, an Aboriginal people recognised as the traditional owners of south west Western Australia. Within Benang, levitation, lightness and detachment, become expressions of vertiginous (post)colonial distance, fear, anxiety and escape.     

Commentary on Jansen's Paper

Karl Jansen raises a fundamental and exciting question: Is humankind's consciousness the result of neuronal function, or are there extra-cerebral aspects as well? While his neurotransmitter model of near-death experiences (NDEs) is well described, I find his supporting evidence weak. Methodological differences between studies of ketamine hallucinations and near-death experiences (NDEs) raise doubts about how similar those experiences are phenomenologically. While Jansen's model has electrifying implications, the data required to support his conclusions do not yet exist.