# Tagged Questions

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### Need some help with this Cardinality/sets question.

I've got this problem about sets, and cardinality. I don't really understand it other than cardinality is the number of elements within each set, I don't understand a lot of the signs used within the ...
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### Partial order on cardinalities without the axiom of choice

Cardinality can still be defined without choice, e.g. as equivalence class of equipotent sets, see Defining cardinality in the absence of choice. Injections define partial order on cardinalities by ...
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### Are there ordinals other than the set of natural numbers which satisfy this property?

Let $\alpha$ be an ordinal. We say that $\alpha$ is good iff for every $\beta\in \alpha$, there exists $\gamma\in \alpha$ such that $|\scr{P}(\beta)|\leq |\gamma|$. Question: Is the set of natural ...
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### Does cardinality really have something to do with the number of elements in a infinite set?

I've seen some videos and read some texts (non-rigorous ones) that explained the concept of cardinality, and sometimes I see someone asking if there are more numbers between the reals in $[0,1]$ then ...
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### Jech's proof of Silver's Theorem on SCH

Jech's textbook proves Silver's Theorem–that if the Singular Cardinals Hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals–by breaking up the ...
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### How can I quantify over the class of all cardinalities?

I'd like to quantify over all cardinalities of sets. My end goal is to make a category-theoretic arguement: For all cardinalities of sets, in the category of sets with maps as morphisms: the ...
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### Prove that transcendental numbers exist: Are there less paniful ways of doing it?

I've found this exercise on Boolos' Logic and Computability: A real number $x$ is called algebraic if it is a solution to some equation of the form: c_{\small d}x^{\small ...
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### Does $\mathfrak{(p<q)\land(r<s)}$ imply $\mathfrak{p^r<q^s}$?

Does $\mathfrak{(p<q)\land(r<s)}$ imply $\mathfrak{p^r<q^s}$, where $\mathfrak{ p,q,r,s}$ are cardinal numbers? Is it possible to prove in $\mathsf{ZFC}$ that there is a counterexample?
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### Is the expressive power of infinitary logic language $L(\infty,\infty)$ larger, the same or smaller than that of ZFC+large cardinal axioms?

In a previous question I learned that the power of statements of the form $\Pi_m^n$ or $\Sigma_m^n$ for arbitrary positive $m$ and $n$, is smaller than that of ZFC. For instance, the GCH cannot be ...
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### Do you need the Axiom of Choice to accept Cantor's Diagonal Proof?

Math people: It is my understanding that set theorists/logicians compare cardinalities of sets using injections rather than surjections. Wikipedia defines countable sets in terms of injections. ...
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### Uncountable models for a language $L_Q$

$L$ is first-order language with identity and $L_Q$ a language obtained by adding to $L$ the quantifier $Q$. Definition of $Q$: If $P$ is a formula and $x$ a variable, $QxP$ is a formula of ...
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