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Assume all independent reals that are added by random real forcing. Take enumeration of each independent real. Is the family of all enumerations dominating?

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    $\begingroup$ Could you clarify exactly what you mean by your assumption? What is an "indepenent" real? Do you mean to assume that the universe was obtained by forcing to add $\kappa$ many random reals, for some uncountable $\kappa$? Which $\kappa$? $\endgroup$ Oct 29, 2013 at 13:42
  • $\begingroup$ independent real in extension is subset of natural numbers which interesects every subset of natural numbers from the groundmodel in infinite set ... and the universe was obtained by adding one random real $\endgroup$
    – Jan Grebik
    Oct 29, 2013 at 19:07
  • $\begingroup$ if someone was interested the answer is no. $\endgroup$
    – Jan Grebik
    Dec 14, 2013 at 15:28
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    $\begingroup$ I'm afraid that I still don't understand the question. What does the first sentence mean? $\endgroup$ Dec 14, 2013 at 16:13
  • $\begingroup$ Random real forcing adds independent real and not just one, so take enumeration of each of them. My queastion was if this set of enumerations is dominationg $\endgroup$
    – Jan Grebik
    Dec 15, 2013 at 17:20

2 Answers 2

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As noted by Jan Grebik in the comments, the answer is no. Here's a sketch: Let $r$ be random over $V$. Let $Y$ be any infinite set in $V[r]$ whose enumeration map is pointwise above $n \mapsto 2^n$. Let $p_k$ be the probability that $k \in Y$ (measure of the boolean value of $k \in \mathring{Y}$). Since $n/2^n \to 0$, we can find an increasing sequence $\langle k_j: j < \omega \rangle$ such that $\sum_j p_{k_j} < 1/10$. It follows that with probability $> 9/10$, $Y$ does not meet the set $\{k_j : j < \omega\}$. So it is not an "independent" set.

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Since Random real forcing is $\omega^\omega$-bounding, every new function is dominated by a ground-model one. Hence it is impossible for a family of "new" functions to be dominating.

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    $\begingroup$ No single new function can be dominating, but I think the OP was interested in a family that's jointly dominating. That is, my best guess as to what the question means is: In the extension by one random real, is every function $f:\omega\to\omega$ dominated by the increasing enumeration of some independent $x\subseteq\omega$ (where $x$ can depend on $f$)? $\endgroup$ Sep 26, 2018 at 14:29

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