224 research outputs found
The Veblen functions for computability theorists
We study the computability-theoretic complexity and proof-theoretic strength
of the following statements: (1) "If X is a well-ordering, then so is
epsilon_X", and (2) "If X is a well-ordering, then so is phi(alpha,X)", where
alpha is a fixed computable ordinal and phi the two-placed Veblen function. For
the former statement, we show that omega iterations of the Turing jump are
necessary in the proof and that the statement is equivalent to ACA_0^+ over
RCA_0. To prove the latter statement we need to use omega^alpha iterations of
the Turing jump, and we show that the statement is equivalent to
Pi^0_{omega^alpha}-CA_0. Our proofs are purely computability-theoretic. We also
give a new proof of a result of Friedman: the statement "if X is a
well-ordering, then so is phi(X,0)" is equivalent to ATR_0 over RCA_0.Comment: 26 pages, 3 figures, to appear in Journal of Symbolic Logi
The strength of Ramsey Theorem for coloring relatively large sets
We characterize the computational content and the proof-theoretic strength of
a Ramsey-type theorem for bi-colorings of so-called {\em exactly large} sets.
An {\it exactly large} set is a set X\subset\Nat such that
\card(X)=\min(X)+1. The theorem we analyze is as follows. For every infinite
subset of \Nat, for every coloring of the exactly large subsets of
in two colors, there exists and infinite subset of such that is
constant on all exactly large subsets of . This theorem is essentially due
to Pudl\`ak and R\"odl and independently to Farmaki. We prove that --- over
Computable Mathematics --- this theorem is equivalent to closure under the
Turing jump (i.e., under arithmetical truth). Natural combinatorial
theorems at this level of complexity are rare. Our results give a complete
characterization of the theorem from the point of view of Computable
Mathematics and of the Proof Theory of Arithmetic. This nicely extends the
current knowledge about the strength of Ramsey Theorem. We also show that
analogous results hold for a related principle based on the Regressive Ramsey
Theorem. In addition we give a further characterization in terms of truth
predicates over Peano Arithmetic. We conjecture that analogous results hold for
larger ordinals
The Complexity Era in Economics
This article argues that the neoclassical era in economics has ended and is being replaced by a new era. What best characterizes the new era is its acceptance that the economy is complex, and thus that it might be called the complexity era. The complexity era has not arrived through a revolution. Instead, it has evolved out of the many strains of neoclassical work, along with work done by less orthodox mainstream and heterodox economists. It is only in its beginning stages. The article discusses the work that is forming the foundation of the complexity era, and how that work will likely change the way in which we understand economic phenomena and the economics profession.
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
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