I’m reading Kunen’s book Foundations of mathematics. My question is whether a countable union of $\Sigma_1$ sets in $HF$ is also $\Sigma_1$ or not. I wonder if we can think $\Sigma_1$ sets as open sets in HF.
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What is HF? ${}$ – Wojowu Jul 03 '20 at 10:51
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1@Wojowu Hereditarily finite. – Michael Greinecker Jul 03 '20 at 11:10
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What is an open set of HF? Is a countable union of countable sets of HF finite? – Hanul Jeon Jul 03 '20 at 15:35
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1I suppose you mean whether a countable union of $\Sigma_1$ subsets of $HF$ is again a $\Sigma_1$ subset of $HF$? A set that's actually in HF must be finite and so it can't be a nontrivial countable union of anything. – Nate Eldredge Jul 03 '20 at 16:03
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1What have you tried? For instance, you could edit Kunen's definition of $\Sigma_1$ into your question and explain where you got stuck in trying to prove that it's satisfied. – Nate Eldredge Jul 03 '20 at 16:04
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6Singletons are $\Sigma_1$ subsets of HF; contemplate countable unions of them. – Andreas Blass Jul 03 '20 at 16:44
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$\Sigma_1$ subsets of $\mathit{HF}$ are the recursively enumerable ones, as you might have read in Kunen's book. The intuition of being “open” sets is not completely incorrect, if you consider them effectively open.
With this in mind, it can be concluded that the union of an effectively enumerated family of $\Sigma_1$ sets is again $\Sigma_1$. But, following Andreas' comment above, any subset of $\omega$ can be obtained a countable union of $\Sigma_1$ sets; and there are indeed such sets that are not $\Sigma_1$— for instance, the set of (codes for) non halting Turing machines.
Pedro Sánchez Terraf
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Thank you for answering my question. When we think about "effective" topology by effectively open sets, how we can define compactness of a subset? – minerva Jul 04 '20 at 01:48
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I wouldn't be sure what's the right answer here. Effective descriptive set theory provides corresponding definitions for concepts analogous to Borel and projective pointsets, and in particular lightface $\Pi_1^0$ correspond to complements of effective open set (thus they are not r.e. sets). But in the field of synthetic topology, computability-aware versions of several concepts (in particular, compactness) are provided. – Pedro Sánchez Terraf Jul 04 '20 at 12:59
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@minerva I found another reference that addresses the compactness thing, here. Hope it helps. Also, if you are not expecting new new answer, you can accept mine. – Pedro Sánchez Terraf Jul 06 '20 at 18:26
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Thank you for more information. This article seems very interesting and readable! – minerva Jul 08 '20 at 09:43