1

My question is rather about terminology.

I know that $F_\sigma$ set is a countable union of closed sets, an $F_{\sigma\delta}$ is a set that can be expressed as a countable intersection of $F_\sigma$-sets etc. etc.

I have seen these notions in various papers several times and know them under "Borel hierarchy". But then, when I simply search Borel hierarchy on Wikipedia, I get $\Pi$s, $\Sigma$s, and $\Delta$s, but no mention of $F´s$ or $G´s$.

How are these notions connected? What am I missing? Thank you.

Tereza Tizkova
  • 2,238

1 Answers1

3

We have two different notational systems for the same thing: $F_\sigma$ is synonymous with $\Sigma^0_2$, $G_\delta$ is synonymous with $\Pi^0_2$, etc. For most purposes, once we get above the third level of the Borel hierarchy the $\Sigma/\Pi$ notation is much more convenient (this is especially true once we get into the infinite levels!), and so sources like the wiki page which present the whole hierarchy itself as opposed to focusing on a small initial segment typically use that notation.

(That said, there is one situation where the $F/G$ notation is nicer: when we drop the axiom of choice. For example, in $\mathsf{ZF}$ it is not true that $F_\sigma=F_{\sigma\sigma}$. But this is a rather technical aspect which should be ignored at first.)

Noah Schweber
  • 245,398
  • 1
    I've always been very leery of trying to avoid AC, and your last remarks remind me of a 30 July 2001 sci.math thread I began -- Every set is F_sigma-delta-sigma in the Feferman-Levy model?, which has more useful comments by Abhijit Dasgupta in this thread and this thread. – Dave L. Renfro Mar 28 '22 at 16:20
  • 2
    @DaveL.Renfro Indeed, see also the introduction to this paper by Arnie Miller, where it's observed that in the Feferman-Levy model every set of reals is both $F_{\sigma\sigma}$ and $G_{\delta\delta}$, despite the non-collapse of the usual Borel hierarchy being provable (when phrased appropriately) in $\mathsf{ZF}$ alone. – Noah Schweber Mar 28 '22 at 16:37
  • @NoahSchweber Thank you! Just to be sure - so $F_\sigma$ and $G_\delta$are called Borel sets of \textbf{order one}? I got confused that they are the same as $\Sigma_2^0$ etc., with number 2, not 1, right? So $\Sigma_1^0$ then means just open sets, while $\Sigma_2^0$ is their countable intersection? – Tereza Tizkova Mar 28 '22 at 18:42
  • 1
    @TerezaTizkova I've never heard the phrase "Borel set of order $1$" (or "order anything" for that matter). I would say that $F_\sigma$ and $G_\delta$ sets are Borel sets of rank $2$ (not $1$). – Noah Schweber Mar 28 '22 at 18:47
  • @NoahSchweber Ah, alright. So then rank 1 would be just $F$ and $G$ or this notation is not used? And rank 1 would mean just open and closed subsets? – Tereza Tizkova Mar 28 '22 at 18:50
  • 1
    @TerezaTizkova I've never seen $F$ or $G$ used by themselves; I don't think there is a standard $F$/$G$-style notation for just open or closed sets. But in the $\Pi/\Sigma$-style, "closed" is $\Pi^0_1$ and "open" is $\Sigma^0_1$. – Noah Schweber Mar 28 '22 at 18:53
  • 1
    I've seen the word "order" used in older literature (before 1960s), although google searches (regular and books and scholar) show much less usage than I expected. Sometimes $G_0$ and $F_0$ are used for open and closed sets (e.g. some of the books in my MSE answer cited earlier along with Bingman's thesis cited here). However, (continued) – Dave L. Renfro Mar 28 '22 at 19:26
  • @DaveL.Renfro Huh, didn't know that, thanks! – Noah Schweber Mar 28 '22 at 19:26
  • since "order" is so over-used in math and "rank" has a more restricted array of meanings, I think "rank" is much better. "Rank" is, of course, not just used in set theory, but the various uses of "rank" tend have much less "mathematical variance" than the various uses of "order". (order of a group, asymptotic growth order, partial and total and well orders, order of a singularity in complex analysis, $\ldots$ the list seems endless, with almost no cohesion among the various meanings) – Dave L. Renfro Mar 28 '22 at 19:30