12
$\begingroup$

At the end of the following paper, Erdős asked if there is a family $F$ of entire functions of size continuum such that for every $z \in \mathbb{C}$, $\{f(z) : f \in F\}$ has size less than continuum. He also showed how to construct such a family under CH.

Did someone solve it?

$\endgroup$
10
  • 4
    $\begingroup$ 1. What's the full citation of the Erdős paper? 2. Have you checked MathSciNet or a similar database for papers that cite this one? $\endgroup$ Jul 5, 2015 at 13:52
  • 1
    $\begingroup$ Isn'it the property for which Konsevitch proved: - under CH, there exists an entire function satisfying the property, - under the negation of CH, such an entire function does not exist ? $\endgroup$ Jul 5, 2015 at 15:51
  • $\begingroup$ I do not understand what you are saying. What property? $\endgroup$ Jul 5, 2015 at 16:03
  • 1
    $\begingroup$ A proof of this can be found in Proofs from the Book by Aigner and Ziegler. See Theorem 5, p.103. $\endgroup$ Jul 5, 2015 at 22:42
  • 2
    $\begingroup$ We are probably talking about An interpolation problem associated with the continuum hypothesis, Michigan Math J 11 (1964) 9-10, MR0168482 (29 #5744). $\endgroup$ Jul 5, 2015 at 23:36

2 Answers 2

11
$\begingroup$

The following (negative answer to Erdos' question) will appear in a joint work with Shelah.

Claim: Suppose $V \models 2^{\aleph_0} = \lambda > \kappa = \aleph_1$. Let $P$ add $\kappa$ Cohen reals. Then in $V^{P}$, there is no such family.

Proof: Let $r \in {}^{\kappa}2$ be the Cohen generic sequence. Clearly $V[r] \models 2^{\aleph_0} = \lambda$. Suppose $\langle f_{\alpha} : \alpha < \lambda \rangle$ is a sequence of pairwise distinct analytic functions in $V[r]$. Choose $X \in [\lambda]^{\lambda}$, $\xi_{\star} < \kappa$ such that for each $\alpha \in X$, $f_{\alpha}$ is coded in $V[r \upharpoonright \xi_{\star}]$. Let $z_{\star} \in \mathbb{C}$ be Cohen over $V[r \upharpoonright \xi_{\star}]$. Since two distinct analytic functions only agree on a countable set, it follows that $\langle f_{\alpha}(z_{\star}) : \alpha \in X \rangle$ are pairwise distinct.

Update: Shelah and I showed that the answer to Erdos' question is independent of ZFC + not CH so the other direction also holds.

An interesting question that remains open is: Is the following consistent: $2^{\aleph_0} = \aleph_2$ and there exists $U \in [\mathbb{C}]^{\aleph_1}$ such that for every $A \in [\mathbb{C}]^{\aleph_1}$, there is a non constant entire map that sends $A$ into $U$?

$\endgroup$
9
$\begingroup$

Maybe following theorem help us to answer the question.

Theorem: The continuum hypothesis is true if and only if there is an uncountable family $\mathcal{F}$ of entire analytic functions such that for each $z\in\mathbb{C}$ the set of values $\{f(z):z\in\mathbb{C}\}$ is countable.

Please see Theorem 14.4 of Forcing for Mathematicians by Weaver for the proof

$\endgroup$
2
  • 2
    $\begingroup$ I don't see how. Can you elaborate? $\endgroup$
    – Ashutosh
    Jul 5, 2015 at 14:14
  • 1
    $\begingroup$ @Ashutosh I'm sorry, I understood wrongly. I tried to answer it, but I couldn't. I edited the answer, maybe the Theorem helps others to answer your question. $\endgroup$
    – Rahman. M
    Jul 5, 2015 at 14:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.