Home Philosophy Advances in Proof-Theoretic Semantics

## SetsFrom an extensional perspective viewing sets as collections of given sets, the notion of an elementary set connects to hierarchies of what we somehow can visualize, i.e., low levels of the cumulative hierarchy. From an intensional point of view, where the act of abstraction with respect to a given defining property/function is in focus, a natural notion of an elementary set must build on characteristics of the definition. The Levy hierarchy [8] of course shows strong connections between both perspectives for more open set theories. It is for instance clear that a set such as very elementary set with respect to its defining function. Let us say that a set • all sets • Both ## Foundational IssuesIt is clear that consistency is not an explicit issue in the present context. Falsity (i.e., Stating that there are functions to axioms of set theory, and proving or believing that the theory The idea of reduction is somehow inherent in the notion of foundations, i.e., that we build on elementary foundations. Although we evidently just walk around in ontological circles, this idea of reduction is not meaningless. A very clear and conceptually elementary model provides a reduction in the sense that we see clearly why given axioms make sense. The argument that the idea of a reduction is an illusion since the construction of the model involves all the power of the axioms themselves does not make for a strong case. It is the suggestive simplicity and clearness of the picture the model paints that is important, i.e., that we really can Can we The definition itself is in some sense elementary, the proofs defining reasoning in theories Since set
## References1. Aczel, P.: Frege structures and the notions of proposition, truth and set. In: J. Barwise, H.J. Keisler, K. Kunen (eds.) The Kleene Symposium. North-Holland (1980) 2. Cantor, G.: Contributions to the Founding of the Theory of Transfinite Numbers. Dover Publications (1955) 3. Ekman, J.: Normal proofs in set theory. Ph.D. thesis, Department of Computing Science, Chalmers University of Technology (1994) 4. Fraenkel, A.A., Bar-Hillel, Y., Levy, A.: Foundations of Set Theory. North-Holland, Amsterdam (1973) 5. Hallnäs, L.: On normalization of proofs in set theory. Ph.D. Thesis, Dissertationes Mathematicae CCLXI, Warszawa (1988) 6. Hallnäs, L.: Partial inductive definitions. Theor. Comput. Sci. 7. Hallnäs, L., Schroeder-Heister, P.: A proof-theoretic characterization of logic programming II: programs as definitions. J. Log. Comput. 8. Levy, A.: A hierarchy of formulas in set theory. In: Memoirs of the American Mathematical Society, vol. 57. American Mathematical Society, Providence (1965) 9. Prawitz, D.: Natural Deduction. A Proof-Theoretical Study. Almqvist & Wiksell, Stockholm (1965) 10. Prawitz, D.: Ideas and results in proof theory. In: Fenstad, J.E. (ed.) Proceedings of the Second Scandinavian Logic Symposium. North-Holland (1971) 11. Prawitz, D.: Towards a general proof theory. In: Suppes, P. (ed.) Logic. Methodology and the Philosophy of Science IV. North-Holland, Amsterdam (1973) 12. Prawitz, D.: On the idea of a general proof theory. Synthese 13. Schroeder-Heister, P.: Rules of definitional reflection. In: Proceedings of the 8th Annual IEEE Symposium on Logic in Computer Science. Los Alamitos, Montreal (1993) 14. Schroeder-Heister, P.: Paradoxes and structural rules. In: Novaes, C.D., Hjortland, O.T. (eds.) Insolubles and Consequences: Essays in honour of Stephen Read. College Publications, London (2012) |

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