Tagged Questions

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Absoluteness of $\forall x (x=x)$

Is there any kind of (set-theoretic) absoluteness result for the formula $\forall x (x=x)$? And what about for $\exists x (x=x)$? I know $x=x$ is absolute given that it's $\Delta_0$. Also, if I'm ...
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Is the “Most Important Property a Set S has” Necessary and Sufficient to Define a Paradox-Free Notion of Set?

About a year and a half ago, while I was looking on the Web for papers regarding the Russell paradox, I chanced to find an interesting concept. This concept was contained in what (for want of a ...
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What is the meaning/purpose of finding the “foundations of mathematics”?

I've read in a lot of places how there was a "foundational crisis" in defining the "foundations of mathematics" in the 20th century. Now, I understand that mathematics was very different then, I ...
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Is it Theoretically Impossible to Demonstrate that Set Theories Are Consistent?

I have to present on the main realist and non-realist arguments for/against set theory. According to one of my sources, it remains a matter of debate as to whether any of the set theories' (ZF, NF, ...
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Set theoretic realism

What are the main contemporary arguments for and against realism about set theory?
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A question regarding Worldly Cardinals and L

For some $L_\kappa$ in the constructible hierarchy, does there exist a $\kappa$ such that $\kappa$ is a worldly cardinal and that $L_\kappa$ contains all of the constructible reals? The motivation ...
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Difference between impredicative and predicative version of separation axiom

What is the difference between an impredicative and a predicative version of the separation axiom in ZFC: $$\forall x \exists y \forall z ( z\in y \leftrightarrow (z \in x \wedge \phi (z))$$ What ...
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Are there other approaches for the foundations of mathematics, other than logic and set theory?

Are there other approaches for the foundations of mathematics, other than logic and set theory? And why does set theory begin talking about objects and groups of objects. Is it proven somewhere that ...
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Consistency of ZFC and the key assumption [closed]

I recently read this answer to a MathOverflow question that got me thinking. Very roughly, here's what the author of that answer says: Gödel's second incompleteness theorem implies that if there ...
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Does the concept of predicativity need to be formalized to go beyond Feferman-Schutte ordinal?

Feferman-Schütte ordinal is sometimes said to be: ....first impredicative ordinal, though this is controversial, partly because there is no generally accepted precise definition of "predicative". ...
592 views

Difference between undecidable statements in set-theory and number theory?

Do all statements about the integers have a definite truth value? For instance: Goodstein's theorem is clearly true, otherwise we could find a finite counterexample thus it would be possible to ...
2k views

What are natural numbers?

What are the natural numbers? Is it a valid question at all? My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational ...
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Consequences of solving the Halting problem

What impact would a device (ie super-computer or relativistic computer or other method) that solves the halting problem have on math? Would there be any mathematical problems left to solve? What ...