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About the Strange Tree Paradox and Possible Inconsistency of Set Theory
Mountain View, CA, USA
- 1 Mountain View, CA, USA
Advances in Pure Mathematics·Volume 13 (2023)·Pages 694–713·Published 10 October 2023·DOI10.4236/apm.2023.1310048
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Abstract
The existence of “strange trees” is proven and their paradoxical nature is discussed, due to which set theory is suspected of being contradictory. All proofs rely on informal set-theoretic reasoning, but without using elements that were prohibited in axiomatic set theories in order to overcome the difficulties encountered by Cantor’s naive set theory. Therefore, in fact, the article deals with the possible inconsistency of existing axiomatic set theories, in particular, the ZFC theory. Strange trees appear when uncountable cardinals appear.
KeywordsSet TheoryInconsistencyTreeStrange TreeThrough WayAlmost Through WayIsomorphismAlmost Isomorphism
- Patrick Suppes (1972) Axiomatic Set Theory. Dover Publications, Inc., New York.
- Shoenfield, J.R. (1967) Mathematical Logic. Addison-Wesley Pub. Co., New York.
- Hao, V., and McNaughton, R. (1953) Les systemes axiomatiquies de la theorie des ensembles. Gauthler-Villars, Paris.
- Kleene, S.C. (1952) Introduction to Metamathematics. American Elsevier Publishing Company, Inc., New York.
- Kolmogorov, A.N. and Dragalin, A.G (2004) Mathematical Logic. URSS, Moscow. (In Russian)
- Davis, D. (2005) Wither Mathematics? Notices of the AMS, 52, 1350-1356.
- Volin, Y.M. (2013) About New Paradoxes in Set Theory and Topological Approach to Their Investigation. Actual Problems of Contemporary Science, No. 3, 183-205. (In Russian)
- Volin, Y.M. (2018) About Inconsistency of Theory ZFC. Actual Problems of Contemporary Science, No. 6, 63-69. (In Russian)
- Volin, Y.M. (2007) On New Results Related to Gödel’s Self-Referential Sentences. Actual Problems of Contemporary Science, No. 6, 89-96. (In Russian)
- Volin, Y.M. (2015) On Normal Inference in Arithmetic and the Problem of Inconsistency. Actual Problems of Contemporary Science, No. 6, 146-159. (In Russian)