The Axiom of Determinacy Implies Dependent Choices in L(R)

We prove the following Main Theorem: ZF+AD+V=L(R)⇒DC. As a corollary we have that Con(ZF+AD)⇒Con(ZF+AD+DC). Combined with the result of Woodin that Con(ZF+AD)⇒Con(ZF+AD+¬AC^ω) it follows that DC (as well as AC^ω) is independent relative to ZF+AD. It is finally shown (jointly with H. Woodin) that ZF+AD+¬DC_R, where DC_R is DC restricted to reals, implies the consistency of ZF+AD+DC, in fact implies R^# (i.e. the sharp of L(R)) exists

The consistency strength of long projective determinacy

We determine the consistency strength of determinacy for projective games of length omega(2). Our main theorem is that Pi(1)(n+1)-determinacy for games of length omega(2) implies the existence of a model of set theory with omega + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that M-n(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies A = R and the Axiom of Determinacy. Then we argue how to obtain a model with omega + n Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length omega(2) with payoff in (sic)(R)Pi(1)(1) or with sigma-projective payoff

On the Relative Consistency Strength of Determinacy Hypothesis

For any collection of sets of reals C, let C-DET be the statement that all sets of reals in C are determined. In this paper we study questions of the form: For given C ⊆ C', when is C'-DET equivalent, equiconsistent or strictly stronger in consistency strength than C-DET (modulo ZFC)? We focus especially on classes C contained in the projective sets

The Ramsey property and higher dimensional mad families

Suppose every set has the Ramsey property and Ramsey-co-null uniformization, as well as the Principle of Dependent Choice hold. Then there is no infinite $\mathcal I$-mad family, for any ideal $\mathcal I$ in smallest class of ideals containing the Fr\^echet ideal and closed under taking Fubini sums. In fact, we show a local form of this theorem which in turn has many consequences, improving and unifying the proofs of several results which were already known for classical mad families. These results were previously announced in Proceedings of the National Academy of Sciences of the U.S.A. We show as a contrasting result that there is a co-analytic infinite mad family in the Laver extension of $L$.Comment: 29 page

Agency is molecular: moved by being moved to moving or co-constitution in intra-active knowledge production

This practice-based PhD aims to intertwine theoretical research and artistic practice on the basis of knowledge production by conceptually thinking through motion, with movement informing the methodological counterpart in performative research settings. I argue that movement and the concept of motion, in their immanent potential for in/determinancy, transport possibilities of transversality that have been neglected in western Modernity. Both offer the means of moving beyond the bifurcated exceptionalism of Modernity's epistemology. The project interrogates its own positioning from within by affirming embodied ways of knowing, which are marginalised within the rationalised epistemes in European Universalisms (Wallerstein). In doing so it also takes a stand against appropriation. From a feminist position, new materialism's situatedness (Haraway) and relational objectivity (Barad) are particularly suitable tools for a shift from within. The apparatus definitions of Agential Realism gather insights through agential cuts that provide a transient exteriority-within, allowing modifying the bounds of knowing from within. The primary chapters examine the impact of practicing through theory and coalesce into a final experiment that reverses the process. Applied to the path of thoughts, movement's induction of changes to matter initiates an essential process of creating space for delinking (Mignolo/Walsh) and unlearning (Singh). The foundation of both practice- and theory-based approaches is Barad's notion of intra-active doing-being, which provides an understanding of agential intertwinement by approaching matter through and with interferences. In experiments, electronic devices were set to receive techno-sound-reverberations as diffractional concerns (noise), that transposed mattering (meaning) from co-constitutional forms. These 'voices', enacted in material-discursive experiments of various entangled engagements in different molecular matterings (body-mind, nature-culture, non-human-human, other-self) are typically ignored, denied, or misunderstood by the notorious bifurcation of the western metaphysical matrix (Jackson). Listening to matter’s iterative performativity (Barad) disclosed uneven levels of capacity (Wilderson) within such non-interrogated generalisations as the flattening to 'we' of the Anthropocene discourse. This awareness of interferential reverberations demands a multidirectional pluriverse of capabilities, which compromises any one-world (Law) exceptionality

A program logic for fresh name generation

This thesis introduces a program logic for an extension of the call-by-value simply typed λ-calculus (STLC), with a mechanism for the generation of fresh names via gensym, which is an adaptation of Pitts and Stark s ν-calculus 52 . Names can be compared for equality and inequality, producing programs with subtle observable properties. Hidden names, produced by interactions between name generation and λ-abstraction, are captured logically with a new restricted quantification. The restrictions require only derived values from previously derived terms, ensuring hidden names are not revealed. The concept of derivation is extended to type contexts and models, ensuring hidden names are not revealed at later stages. Type contexts are adapted to include an order and the ability to represent future extensions. The logic quantifies over future extensions, using a second-order quantification over future type contexts. This quantification names the future context to allow for them to be reasoned about within the logic. A new model construction is introduced to replicate the order in which names and values are produced with potentially hidden names. The semantics of the logic in the new model are used to prove each axiom and rule sound and as such the soundness of the logic. A proof that the logic is an extension of the STLC logic is given alongside a sketch of the proof that the extension is conservative. Usage of the logic is illustrated through reasoning about numerous examples. These ex- amples range from simple STLC and ν-calculus examples to well-known difficult programs from the literature