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

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In summary, If ZFC is consistent, then it has a countable model which satisfies the existence of a set consisting of all real numbers. However, due to Skolem's paradox, this countable model does not contain a bijection between its version of natural numbers and real numbers, leading to the contradiction that the real numbers should be uncountable. This paradox arises from the use of the term "countable" in two different contexts.
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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.
 

1. What does it mean to assume ZFC is consistent?

Assuming ZFC is consistent means accepting the foundational axioms of the Zermelo-Fraenkel set theory, which is commonly used as the basis for modern mathematics. This assumption allows mathematicians to build upon a consistent and well-defined framework for their proofs and theories.

2. Why is it important to assume ZFC is consistent?

If ZFC is not consistent, then it is possible for contradictory statements to be proven, leading to a breakdown of the entire mathematical system. By assuming ZFC is consistent, mathematicians can ensure that their work is built upon a solid foundation.

3. Is there any evidence that supports the consistency of ZFC?

There is no definitive proof that ZFC is consistent, but there is a vast amount of evidence that suggests it is. Many of the most important and fundamental theorems in mathematics have been proven within the framework of ZFC, providing strong support for its consistency.

4. What would happen if ZFC were proven to be inconsistent?

If ZFC were proven to be inconsistent, it would have a significant impact on the entire field of mathematics. It would mean that some of the most fundamental and widely accepted mathematical concepts, such as the existence of infinite sets, would no longer be valid. Mathematicians would have to re-evaluate and potentially rebuild their entire body of work.

5. Are there any alternative set theories that could be used instead of ZFC?

Yes, there are alternative set theories, such as the von Neumann-Bernays-Gödel set theory and the Morse-Kelley set theory. However, these theories are less commonly used and have not gained as much acceptance as ZFC. They also have their own set of axioms and assumptions, and their consistency is still an open question.

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