Erdős problem 1119 · problems.science

Let $\mathfrak{m}$ be an infinite cardinal with $\aleph_0 < \mathfrak{m} < \mathfrak{c} = 2^{\aleph_0}$ . Let $\{f_\alpha\}$ be a family of entire functions such that, for every $z_0 \in \mathbb{C}$ , there are at most $\mathfrak{m}$ distinct values of $f_\alpha(z_0)$ . Must $\{f_\alpha\}$ have cardinality at most $\mathfrak{m}$ ?

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5 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

1119.lean

Retained formal statement4 of 5

Erdős's theorem [Er64g], answering a question of Wetzel: if $\mathfrak{c} > \aleph_1$ , then every family of entire functions taking only countably many distinct values at each point $z_0 \in \mathbb{C}$ is itself countable.

FormalConjectures/ErdosProblems/1119.leanErdos1119.erdos_1119.variants.erdos_wetzel2 linesExact file

1Cardinal.aleph 1 < Cardinal.continuum →2  ∀ (F : Set (ℂ → ℂ)), (∀ f ∈ F, Differentiable ℂ f) → (∀ (z₀ : ℂ), {y | ∃ f ∈ F, f z₀ = y}.Countable) → F.Countable

SolvedStatement only, no proof

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