Abstract
AbstractThe central result of this paper isthe small‐is‐very‐small principlefor restricted sequential theories. The principle says roughly that whenever the given theory shows that a definable property has a small witness, i.e., a witness in a sufficiently small definable cut, then it shows that the property has a very small witness: i.e., a witness below a given standard number. Which cuts are sufficiently small will depend on the complexity of the formula defining the property. We draw various consequences from the central result. E.g., roughly speaking, (i) every restricted, recursively enumerable sequential theory has a finitely axiomatized extension that is conservative with respect to formulas of complexity ; (ii) every sequential model has, for anyn, an extension that is elementary for formulas of complexity , in which the intersection of all definable cuts is the natural numbers; (iii) we have reflection for ‐sentences with sufficiently small witness in any consistent restricted theoryU; (iv) supposeUis recursively enumerable and sequential. Suppose further that every recursively enumerable and sequentialVthat locally intepretsU, globally interpretsU. Then,Uis mutually globally interpretable with a finitely axiomatized sequential theory. The paper contains some careful groundwork developing partial satisfaction predicates in sequential theories for the complexity measuredepth of quantifier alternations.
Highlights
The relevant theorem due to Friedman tells us that, if a finitely axiomatized, sequential, consistent theory A interprets a recursively enumerable theory U, A interprets U faithfully. (Cf. § 2.4 for the definition of a sequential theory.) Krajıcek’s theorem tells us that a finitely axiomatized, sequential, consistent theory cannot prove its own inconsistency on arbitrarily small cuts
We provide a series of definitions illustrative of what we need to get off the ground
We have shown above how to formulate things in order to cope with the fact that each number, set, pair, function and sequence can have several representatives
Summary
This paper is about one such proof We discovered it when searching for alternative, more syntactic, proofs of certain theorems by Friedman (discussed in [17]) and Krajıcek [9]. The relevant theorem due to Friedman tells us that, if a finitely axiomatized, sequential, consistent theory A interprets a recursively enumerable theory U , A interprets U faithfully. The syntactic argument in question is a Rosser-style argument or, a Friedman-GoldfarbHarrington-style argument. It has all the mystery of a Rosser argument: even if every step is completely clear, it still retains a feeling of magic trickery
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