On notions of compactness, object classifiers, and weak Tarski universes

Tarski On Logical Notions

We try to explain Tarski’s conception of logical notions, as it emerges from a lecture of his, delivered in 1966 and published posthumously in 1986 (History and Philosophy of Logic, 7, pp. 143-154), a conception based on the idea of invariance. The evaluation of Tarski’s proposal leads us to consider an interesting (and neglected) reply to Skolem in which Tarski hints at his own point of view o...

Efficient Boosted Weak Classifiers for Object Detection

This paper accelerates boosted nonlinear weak classifiers in boosting framework for object detection. Although conventional nonlinear classifiers are usually more powerful than linear ones, few existing methods integrate them into boosting framework as weak classifiers owing to the highly computational cost. To address this problem, this paper proposes a novel nonlinear weak classifier named Pa...

On Box, Weak Box and Strong Compactness

One of the most important goals of set theorists over the last few years has been to re-prove old results which previously had used very strong assumptions from hypotheses which, at least prima facie, are weaker. Examples of these abound, including, but certainly not limited to, the work of Woodin and Cummings (see [3]) on the Singular Cardinals Problem, in which results previously obtained by ...

On the Strength of Weak Compactness

We study the logical and computational strength of weak compactness in the separable Hilbert space `2. Let weak-BW be the statement the every bounded sequence in `2 has a weak cluster point. It is known that weak-BW is equivalent to ACA0 over RCA0 and thus that it is equivalent to (nested uses of) the usual Bolzano-Weierstraß principle BW. We show that weak-BW is instance-wise equivalent to Π2-...

Weak Compactness of the Set of ε-Extensions