Is ZFC's consistency related to Skolem's paradox?

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SUMMARY

The discussion centers on the relationship between Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) and Skolem's paradox. It establishes that if ZFC is consistent, it has a countable model, which leads to the paradox where a set of all real numbers appears countable within the model. This contradiction arises from the differing contexts of "countable" in model theory versus standard set theory. The key takeaway is that the countable model of ZFC does not contain any internal bijections between its versions of natural numbers (N) and real numbers (R), highlighting the complexities of set theory.

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  • Understanding of Zermelo-Fraenkel set theory (ZFC)
  • Familiarity with model theory concepts
  • Knowledge of countability and uncountability in set theory
  • Basic comprehension of bijections and their implications in mathematics
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  • Study the implications of Skolem's paradox in depth
  • Explore the main theorem of model theory and its applications
  • Investigate the differences between internal and external bijections in models
  • Learn about the consistency proofs for ZFC and their significance
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Mathematicians, logicians, and students of set theory who are interested in the foundations of mathematics and the implications of model theory on set theory concepts.

sairalouise
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If you assume that ZFC is consistent, then by the main theorem of model theory ZFC has a model, let the model be countable.
Since ZFC proves: "there is a set consisting of all real numbers" there is a point a belonging to M such that:
M satisfies " a is the set of all real numbers"
But since M is countable there are only countably many points b belonging to M such that:
M satisfies b belonging to a, so a contains only countably many elements. But the real numbers is uncountable, what has happened? Shouldnt a be uncountable?
 
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This is called Skolem's paradox. It arises by (accidentally) equivocating the word "countable" used in two different contexts.

e.g. you'll find that your countable model of ZFC does not contain any bijection between the model's version of N and the model's version of R. Any bijections that do exist between them are "external", meaning they do not correspond to something in the model.
 

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