ZFC and Non-Denumerability

ZFC와 열거불가능성

  • An, Yohan (Department of Philosohpy, Korea University)
  • Received : 2018.11.29
  • Accepted : 2018.12.25
  • Published : 2019.02.28

Abstract

If 1st order ZFC is consistent(has a model($M_1$)) it has a transitive denumerable model($M_2$). This leads to a paradoxical situation called 'Skolem paradox'. This can be easily resolved by Skolem's typical resolution. but In the process, we must accept the model theoretic relativity for the concept of set. This relativity can generate a situation where the meaning of the set concept, for example, is given differently depending on the two models. The problem is next. because the sentence '¬denu(PN)' which indicate that PN is not denumerable is equally true in two models, A indistinguishability problem that the concept <¬denu> is not formally indistinguishable in ZFC arise. First, I will give a detail analysis of what the nature of this problem is. And I will provide three ways of responding to this problem from the standpoint of supporting ZFC. First, I will argue that <¬denu> concept, which can be relative to the different models, can be 'almost' distinguished in ZFC by using the formalization of model theory in ZFC. Second, I will show that <¬denu> can change its meaning intrinsically or naturally, by its contextual dependency from the semantic considerations about quantifier that plays a key role in the relativity of <¬denu>. Thus, I will show the model-relative meaning change of <¬denu> concept is a natural phenomenon external to the language, not a matter of responsible for ZFC.

References

  1. 권병진 (2007), "스콜렘의 상대주의 논증과 베나세라프의 미결정성 논증", 철학적 분석, 16, pp. 99-142
  2. Abian, A. (1965), The Theory of Sets and Transfinite Arithmetic, Saunders Mathematics books.
  3. Bellotti, L. (2006), "Skolem, the Skolem 'Paradox' and Informal Mathematics", Theoria 72 (3), pp. 177-212. https://doi.org/10.1111/j.1755-2567.2006.tb00956.x
  4. Benacerraf, P. (1967), "What numbers could not be", reprinted in Benacerraf and Putnam (1983), Cambridge Press, pp. 272-294.
  5. Benacerraf, P. and Putnam, H. (1983), Philosophy of mathematics, Blackwell.
  6. Benacerraf, P. (1985), "Skolem and the Skeptic", Proceedings of the Aristotelian Society, Vol. 59, pp. 85-115. https://doi.org/10.1093/aristoteliansupp/59.1.85
  7. Boolos, G. (1971), "The Iterative Conception of Set" in Benacerraf and Putnam (1983), Cambridge Press, pp. 486-502.
  8. Bostock, D. (2009), Philosophy of Mathematics: An Introduction, Wiley-Blackwell.
  9. Cohen, P. (1966), Set Theory and the Continuum Hypothesis, New York, NY: W.A. Benjamin.
  10. Devlin, K. (1994) Joy of sets, (2nd ed.), Springer.
  11. Drake, F. (1996), Intermediate set theory, John wiley & sons.
  12. Franzen, T. (2004), Inexhaustibility: A Non-Exhaustive Treatment, Association for Symbolic Logic.
  13. Frege, G. (1980), The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number, Northwestern University Press.,
  14. Godel, K. (1947/64), "What is Cantor's continuum problem?", Benacerraf and Putnam (1983), Cambridge Press, pp. 470-485.
  15. Hallet, M. (1984), Cantorian Set Theory and Limitation of Size, Clarendon Press.
  16. Hallet, M. (1994), "Putnam and the Skolem Paradox", In M. Baghramian (ed.), Reading Putnam, Blackwell.
  17. Hallet, M. (2011), "Absoluteness and the Skolem Paradox", In Logic, Mathematics, Philosophy, Vintage Enthusiasms. Springer.
  18. Jane, I. (2001), "Reflections on Skolem's Relativity of Set-Theoretical Concepts", Philosophia Mathematica 9 (2), pp. 129-153. https://doi.org/10.1093/philmat/9.2.129
  19. Jech, T. (2000), Set theory, Springer.
  20. Klein, P.D. (1976), "Knowledge, Causality and Defeasibility", Journal of Philosophy. 73, pp. 797-8.
  21. Klenk, V. (1976), "Intended Models and the Lowenheim-Skolem Theorem", Journal of Philosophical Logic, 5, pp. 475-89. https://doi.org/10.1007/BF02109439
  22. Kunen, K. (1980), "Set Theory: An Introduction to Independence Proofs", Studies in Logic and the Foundations of Mathematics, Vol. 102. North-Holland Publishing Company, Amsterdam.
  23. Maddy, P. (1988), "Believing the Axioms I", Jouranl of Symbolic Logic, 53, pp. 481-511. https://doi.org/10.1017/S0022481200028425
  24. Maddy, P. (2011), "Set Theory as a Foundation In Foundational Theories of Classical and Constructive Mathematics", The Western Ontario Series in Philosophy of Science, Springer.
  25. McIntosh, C. (1979), "Skolem's Criticisms of Set Theory", Nous 13 (3), pp. 313-334. https://doi.org/10.2307/2215103
  26. Cook, R. (2009), A Dictionary of Philosophical Logic, Edinburgh University Press
  27. Peters, S and Westerstahl, D. (2006), "Quantifiers in Language and Logic", Oxford University Press
  28. Shapiro, S. (1991), Foundation without Foundationalism: A Case for Second-Order Logic, Oxford University Press.
  29. Shapiro, S. (1997), Philosophy of Mathematics: Structure and Ontology, Oxford University Press.
  30. Skolem, T. (1922), "Some remarks in axiomatized set theory", in Van Heijenoort, J.(ed.), (1967), From Frege to Godel, Harvard Press.
  31. Tennant, N and McCarty, C. (1987), "Skoelm's Paradox and Constructivism", Journal of Philosophical Logic 16, pp. 166-202.
  32. Wang, H. (1974), "The Concept of Set" in Benacerraf and Putnam (1983), Cambridge Press, pp. 530-570.