# Tagged Questions

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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### Formulations of Singular Cardinals Hypothesis

I have stumbled on a few different formulations of the Singular Cardinals Hypothesis. The most common are: SCH1: $\quad 2^{cf(\kappa)}<\kappa \ \Longrightarrow \ \kappa^{cf(\kappa)}=\kappa^+$ for ...
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### For each of the following sets, determine its cardinality (ω, 2ω, or something else) and prove that your answer is correct

(a) A1 = {f ∈ (ω → ω) : ∀n,m ∈ ω (n < m ⇒ f(n) < f(m))}. (b) A2 = {f ∈ (ω → ω) : ∃n ∈ ω∀m ∈ ω f(m) ≤ n}. (c) A3 = {f ∈ (ω → ω) : ∃n ∈ ω∀m ∈ ω (n ≤ m ⇒ f(n) = f(m))}. a) A1 = {f ∈ (ω → ω) : ∀n,...
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### Show $\mathbb{N}^{\{0,1\}}$ is uncountable with a hint

Let $\mathbb{N}^{\{0,1\}} :=\{f: \mathbb{N} \to \{0,1\}\}$ is uncountable I have never heard of the table approach, and all the proofs say uncountability of $\mathcal {P}(\mathbb{N})$ I have seen so ...
### The usual bijection between $[0,1]$ and $[0,1]\times[0,1]$
While trying to explain to someone else how you can have a bijection between $[0,1]$ and $[0,1]\times[0,1]$, I found an issue in the usual bijection that we use. The usual bijection that I'm talking ...