Week 8, 4 March 2014 (17:00-19:00 Watson 310)

Speaker: Benedict Eastaugh (Bristol)
Computational reverse mathematics and foundational analysis

Reverse mathematics studies which natural subsystems of second order arithmetic are equivalent to key theorems of ordinary or non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of various weak foundations for mathematics (finitistic reductionism, predicativism, predicative reductionism) in a formally precise manner.

Recursion theorist Richard Shore has proposed an alternative framework in which to conduct reverse mathematics, called computational reverse mathematics. Instead of provable equivalence over the weak base theory RCA0, the equivalence relation used in computational reverse mathematics is model-theoretic: two statements are computably equivalent iff they have the same Turing ideals (omega-models closed under Turing reducibility and recursive joins).

Despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, since the computable entailment relation it employs is Pi^1_1-complete. By internalising the proof of this theorem within the weak base theory RCA0 we can see that the existence of the truth set for computable entailment is equivalent to Pi^1_1 comprehension: a strong system which outstrips the resources which proof-theoretically weaker foundational programmes can draw upon.

About the seminar

The Midlands Logic Seminar was founded in 2011 and aims to cover all areas of mathematical logic, as well as related areas of theoretical computer science, and philosophy of mathematics.


All meetings for Term 2, 2013-14 will be Tuesday from 17:00-19:00 in room 310 of Watson Building (School of Mathematics) on the campus of the University of Birmingham.
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    To get to campus, take the train from Birmingham New Street to the University stop (7 minutes).


    Dr Richard Kaye
    School of Mathematics
    University of Birmingham


    Dr Walter Dean
    Department of Philosophy
    University of Warwick