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Constructivism and Proof Theory
, 2003
"... Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. F ..."
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Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. For constructive (intuitionistic)
arithmetic, Kleene’s realizability interpretation is given; this provides an example
of the possibility of a constructive mathematical practice which diverges from classical
mathematics. The crucial notion in intuitionistic analysis, choice sequence, is
briefly described and some principles which are valid for choice sequences are discussed.
The second half of the article deals with some aspects of proof theory, i.e.,
the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions
are outlined: his introduction of the socalled Gentzen systems which use
sequents instead of formulas and his result on firstorder arithmetic showing that
(suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in
firstorder arithmetic.
CONSTRUCTIVE POINTFREE TOPOLOGY ELIMINATES NONCONSTRUCTIVE REPRESENTATION THEOREMS FROM RIESZ SPACE THEORY
, 807
"... Abstract. In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost falgebras are commutative. The proof is obtained relat ..."
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Abstract. In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost falgebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree StoneYosida representation theorem by Coquand and Spitters. The StoneYosida representation theorem for Riesz spaces [LZ71, Zaa83] shows how to embed every Riesz space into the Riesz space of continuous functions on its spectrum. Theorem 1. [StoneYosida] Let R be an Archimedean Riesz space (vector lattice) with unit. Let Σ be its (compact Hausdorff) space of representations. Define the continuous function ˆr(σ): = σ(r) on Σ. Then r ↦ → ˆr is a Riesz embedding of R into C(Σ,). The theorem is very convenient, but sometimes better avoided, since it leads out of the theory of Riesz spaces. To quote Zaanen [Zaa97]:
TERM EXTRACTION AND RAMSEY’S THEOREM FOR PAIRS
"... Abstract. In this paper we study with prooftheoretic methods the function(al)s provably recursive relative to Ramsey’s theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functionals provably recursive from RCA0 + COH + Π0 1CP are primitive recursive. This ..."
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Abstract. In this paper we study with prooftheoretic methods the function(al)s provably recursive relative to Ramsey’s theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functionals provably recursive from RCA0 + COH + Π0 1CP are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that WKL0 + Π0 1CP + COH is Π0 2conservative over PRA. Recent work of the first author showed that Π0 1CP + COH is equivalent to a weak variant of the BolzanoWeierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey’s theorem for pairs and two colors (RT2 2) we obtain the upper bounded that the type 2 functionals provable recursive relative to RCA0 + Σ0 2IA+RT2 2 are in T1. This is the fragment of Gödel’s system T containing only type 1 recursion — roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T1bounds from proofs that use RT2 2. Moreover, this yields a new proof of the fact that WKL0 + Σ0 2IA + RT2 2 is Π0 3conservative over RCA0 + Σ0 2IA. The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel’s functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using Π0 1comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard’s ordinal analysis of bar recursion (for RT2 2). 1.