On the relative strengths of fragments of collection

Let $\mathbf{M}$ be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, $\Delta_0$-separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to $\mathbf{M}$. We focus on two common parameterisations of collection: $\Pi_n$-collection, which is the usual collection scheme restricted to $\Pi_n$-formulae, and strong $\Pi_n$-collection, which is equivalent to $\Pi_n$-collection plus $\Sigma_{n+1}$-separation. The main result of this paper shows that for all $n \geq 1$, (1) $\mathbf{M}+\Pi_{n+1}\textrm{-collection}+\Sigma_{n+2}\textrm{-induction on } \omega$ proves the consistency of Zermelo Set Theory plus $\Pi_{n}$-collection, (2) the theory $\mathbf{M}+\Pi_{n+1}\textrm{-collection}$ is $\Pi_{n+3}$-conservative over the theory $\mathbf{M}+\textrm{strong }\Pi_n \textrm{-collection}$. It is also shown that (2) holds for $n=0$ when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke-Platek Set Theory with Infinity and $V=L$) that does not include the powerset axiom.Comment: 22 page

There’s no public benefit in BBC programmes being ‘distinctive’

Question 9 in the UK Government’s recent Green Paper on the BBC Charter review asked, “Is the BBC’s content sufficiently high quality and distinctive from that of other broadcasters?”. Patrick Barwise, Visiting Senior Fellow in the Department for Media and Communications at the LSE, Emeritus Professor of Management and Marketing at London Business School, and former Chairman of Which?, provides his response

Disrupting the digital giants – advertisers and traditional media push back

The downside of digital giants like Facebook and Google includes the increase in fake news, political polarisation, the dumbing down of debate and the long-term decline in print journalism as newspapers lose readers and advertising to these platforms. But a combination of problems with digital advertising – fraud, mismeasurement, and programmatic ad placement on undesirable sites – means that advertisers and traditional media are starting to push back. The battle for advertising may be entering a new phase, with a better balance between old and new media. This post is based on a presentation by Patrick Barwise (London Business School) at the Oxford Media Convention 2017

Changing characteristics of hospital admissions but not the children admitted—a whole population study between 2000 and 2013

Funding: The data were hosted in the safe haven thanks to funding by the FARR institution. Open Access via Springer Compact Agreement.Peer reviewedPublisher PD

Letter from E. Barwise to James B. Finley

Barwise asks Finley to send a letter to Governor Bebb of Ohio concerning a possible reprieve for his son William. If it can be proved that William has reformed, his reprieve will be granted. Abstract Number - 1073https://digitalcommons.owu.edu/finley-letters/2057/thumbnail.jp

Committee, Government Diverging on Convergence?

The House of Lords Select Committee on Communications begins taking oral evidence next Tuesday in its Inquiry on Media Convergence and its Public Policy Impact. It is a broad inquiry, attempting to take in some of the high level issues that the Communications Review and the Leveson Inquiry are unwilling, or unable, to cover. It proposes to cover very general issues related to convergence of media asking whether there should be some kind of over-arching regulatory framework for broadcasting and the internet, and asks for suggestions for a ‘guiding principle’ for policy on convergence