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A New Approach to Predicative Set Theory
"... We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an a ..."
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We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domainindependence from database theory, and Godel notion of absoluteness from set theory. The language of PZF is typefree, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of PZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1
Ancestral Arithmetic and Isaacson’s Thesis
, 2007
"... So the question naturally arises: what kinds of sentences belonging to PA’s language LA can we actually establish to be true even though they are unprovable in PA? There are two familiar classes of cases. First, there are sentences like the canonical Gödel sentence for PA. Second, there are sentence ..."
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So the question naturally arises: what kinds of sentences belonging to PA’s language LA can we actually establish to be true even though they are unprovable in PA? There are two familiar classes of cases. First, there are sentences like the canonical Gödel sentence for PA. Second, there are sentences like the arithmetization of Goodstein’s Theorem. In the first sort of case, we can come to appreciate the truth of the Gödelian undecidable sentences by reflecting on PA’s consistency or by coming to accept the instances of the Π1 reflection schema for PA. And those routes involve deploying ideas beyond those involved in accepting PA as true. To reason to the truth of the Gödel sentence, we need not just to be able to do basic arithmetic, but to be able to reflect on our practice. In the second sort of case, we come to appreciate the truth of the sentences which are undecidable in PA by deploying transfinite induction or other infinitary ideas. So the reasoning again involves ideas which go beyond what’s involved in grasping basic
1. Isaacson’s Thesis stated
"... arises: what kinds of sentences belonging to PA’s language L A can we actually establish to be true even though they are unprovable in PA? There are two familiar classes of cases. First, there are sentences like the canonical Gödel sentence for PA. Second, there are sentences like the arithmetizatio ..."
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arises: what kinds of sentences belonging to PA’s language L A can we actually establish to be true even though they are unprovable in PA? There are two familiar classes of cases. First, there are sentences like the canonical Gödel sentence for PA. Second, there are sentences like the arithmetization of Goodstein’s Theorem. In the first sort of case, we can come to appreciate the truth of the Gödelian undecidable sentences by reflecting on PA’s consistency or by coming to accept the instances of the Π 1 reflection schema for PA. And these routes involve deploying ideas beyond those involved in accepting PA as true. To reason to the truth of the Gödel sentence, we need not just be able to do basic arithmetic, but need to be able to reflect on our practice. In the second sort of case, we come to appreciate the truth of the sentences which are undecidable in PA by deploying transfinite induction or other infinitary ideas. So the reasoning again involves ideas which go beyond what’s involved in grasping basic arithmetic.