# Questions tagged [set-theory]

This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.

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### Cardinality of a totally ordered union

Let $(X_{\alpha})_{\alpha \in A}, (Y_{\alpha})_{\alpha \in A}$ be families of sets such that for $\{X_{\alpha} \mid \alpha \in A \}$ and $\{Y_{\alpha} \mid \alpha \in A \}$ are totally ordered by ...
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### Question on An Explicit Enumeration of Ordinals

Question: For any countable ordinal $X$, is there an ordered triplet of ordinals $(a,b,c)$ in the class of all ordered triplets of ordinals such that $X \in f((a,b,c))$ (where $f((a,b,c))$ is defined ...
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### Why in ZFC any expressible set also exists?

I would like to ask you a few questions about the following quote from Wang (On Denumerable Bases, 1955): "Zermelo's set theory is, for example, a system in which every number set, if ...
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### What's the meaning of "the set of isomorphic classes of extensions of $\mathcal F$ by $\mathcal G$"?

I think an isomorphic class of extensions of $\mathcal F$ by $\mathcal G$ is a proper class, but a proper class cann't be an element of a set, so what's the meaning of "the set of isomorphic classes ...
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### Cores in Boolean valued models

The following is taken from Bell's Set Theory: Boolean-Valued Models and Independence Proofs. $V^{(B)}$ is a Boolean valued model, where $B$ is some (complete) Boolean algebra. I'm stuck in verifying ...
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### Isomorphism of $V_\lambda$ and the ultraproduct of $V_{\lambda_{\mathrm{otp}(x)}}, x \in \mathcal{P}_\kappa({\lambda})$ by a normal fine measure.

This question is actually exercise (20.5) from Set Theory by Thomas Jech. The original statement is: Let $\lambda \ge \kappa$ and let $U$ be a normal measure on $\mathcal{P}_\kappa({\lambda})$. The ...
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### Can $(X, τ)$ be a nontrivial topological space such that $(τ,X)$ is also a topological space and a metric space under the same metric?

Can $(X, τ)$ be a nontrivial topological space such that $(τ,X)$ is also a topological space and a metric space under the same metric? I'm looking at the Collatz graph as a surjection from a subset ...
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### Another ordinals question $\sum_{i\in\omega +3} i\cdot\omega + \omega\cdot i$

So I have to calculate sum $$\sum_{i\in\omega +3} i\cdot\omega + \omega\cdot i$$ If I got it right, $1\cdot\omega+2\cdot\omega+\dots+(\omega-1)\cdot\omega = \omega\cdot(\omega-1)$ Then I have to ...
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### What is the definition of belonging in axiomatic set theory?

In Hungerford's Algebra it says that Intuitively we consider a class to be a collection $A$ of objects such that given any object $x$ it is possible to determine whether or not $x$ is a member (or ...
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### Hungerford's statement in what is class in Gödel-Bernays axiomatic set theory. [duplicate]

For a collection $C$ define an object $b \in C$ iff $b$ is a class and $b \notin b$ and otherwise $b \in C$. Then we can decide any object whether it belongs to $C$. In Hungerford's Algebra p.2 it ...
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### Defining functions of arbitrary arity and higher abstraction in set theory

I am wondering what is the best / agreed upon way to define functions of arbitrary arities in formal ZFC set theory, allowing also for higher levels of abstraction (e.g. arbitrary sequemces of ...
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### Inequality with cardinal numbers

I want to check if the following statement holds: If $a, b, c$ are cardinal numbers and $a<b$ then $a^c<b^c$. If the statement would hold, given that there is a bijective funcion $f: a \to b$ ...
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### Is any proper subclass of a proper class a set?

Am I allowed in ZFC / other ZFC-style systems to write: (1) C ≝ { x | x ∉ x} (2) S ⊂ C ∧ S ≠ C And use S as a set? As far as I see, Russell's paradox doesn't go through for S, since it is not ...
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### Reference request: definition of class

I am looking for a reference on mathematically strict definition of class. For example, how do you construct the class of all ZFC-sets?