# Tagged Questions

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### Problem 24 from Chapter 1 of Kunen's Set Theory: An Introduction to Independence Proofs

Just want to make sure I'm tracking Kunen here, and hopefully the proof I have is correct. Comments / Suggestions welcome. Thanks! Problem 24. Let T be any consistent set of axioms extending ZF. ...
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### Notation for inductive definitions of sets

Is there a formal notation for inductive definitions of sets? For example, like this: $Closure(U,C,A)$ where $U$ is a set, $C$ is a set of constructors (in a simple case, operations on the set $U$), ...
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### Introducing ordered pairs in an axiomatic way [duplicate]

Suppose that in $ZFC$ we have introduced ordered pairs not in the usual way as $(a,b) = \{\{a\}, \{a,b\}\}$ but axiomatically, by extending $ZFC$ by adding to $ZFC$ a new binary functional symbol $g$ ...
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### Transitive closure of a union

I have a quick question regarding transitive closures. In the text I'm currently reading, (Kunen - Set Theory, 2011), the transitive closure of a set $x$ is defined as trcl$(x) = \{ a : a \in^* x \}$, ...
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### Why is the continuum hypothesis believed to be false by the majority of modern set theorists?

A quote from Enderton: One might well question whether there is any meaningful sense in which one can say that the continuum hypothesis is either true or false for the "real" sets. Among those ...
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### What is the Axiom of choice [duplicate]

I was learning Set Theory for fun and I came across something called the axiom of Choice, What is the axiom of choice?
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### Prove the principle of mathematical induction in $\sf ZFC$

How does one prove the principle of mathematical induction using the standard axioms of $\sf ZFC$?
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### Other criteria for a set to be an ordinal.

I'm reading up on ordinals, and I've come across various definitions for what it means to be an ordinal. The main book that I'm following is Kunen's Set Theory book. He defines an ordinal as a set ...
I'm currently reading about ordinals, working on various exercises since I haven't formally studied the material before. I'm looking at transitive sets. A set $z$ is said to be transitive if and only ...