Timeline for Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?
Current License: CC BY-SA 3.0
14 events
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| Apr 6, 2016 at 4:22 | comment | added | Danielle Ulrich | Let us continue this discussion in chat. | |
| Apr 6, 2016 at 4:21 | comment | added | Danielle Ulrich | I don't follow. As an example, consider $H = \mathcal{P}_{\aleph_1}(\mathbb{R})$, the set of countable subsets of $\mathbb{R}$. There are a variety of nice Borel equivalence relations $E$ one can put on $\mathbb{R}$ to represent this, and they are all isomorphic. But what will be the Borel quotient space structure on $H$? | |
| Apr 6, 2016 at 4:14 | comment | added | Burak | In most of the examples that inspired this proposed thesis, we start with a collection of objects coding the structures we have and then put a standard Borel structure on them. Here, we start with LO and collapse LO-WO into a single class. In this case, the collection coding countable well orders is WO or $LO/E_{\omega_1}$ (whichever you like). What is the standard Borel structure on these collections? I don't see a reasonable way of putting one on the latter quotient space and the former space being a proper analytic set creates a problem for simply using the restriction. | |
| Apr 6, 2016 at 4:09 | comment | added | Burak | Well, you can declare $A \subseteq X/E$ to be Borel if $\cup A$ is Borel in $X$ for a standard Borel space $X$. However, this will not give you a standard Borel structure unless your relation is smooth. You can see the backwards direction of Proposition 6.3 here (where countability is not used). I will elaborate on what kind of examples I was looking for in the next comment. | |
| Apr 6, 2016 at 4:02 | comment | added | Danielle Ulrich | What is a quotient Borel space structure? Can you elaborate on what you want to happen? (Is this related to saying "$E_{\omega_1}$ is not a Borel equivalence relation?") | |
| Apr 6, 2016 at 3:56 | comment | added | Burak | By the way, even though I was pleased learning this fact, I am not sure that $(LO,E_{\omega_1})$ is a "coding" in the sense that was meant in the proposed principle. The reason is that in this coding, we are representing countable well-orders as the quotient space $LO/E_{\omega_1}$, which is not standard Borel in the quotient Borel space structure. So I don't see any natural way of putting a standard Borel structure of $LO/E_{\omega_1}$. (This applies to the other coding as well.) | |
| Apr 6, 2016 at 1:20 | comment | added | Danielle Ulrich | I agree that $\Delta^1_2$ is huge... I guess in my mind the question of what reductions should be permitted is context-sensitive, with Borel being the most restrictive option (used in particular when the equivalence relations are Borel), and with provably $\Delta^1_2$ (and related things, like $HC$-reduction) being the least restrictive option. $\Delta^1_2$ is too big to be useful in $ZFC$--consistently (if $V=L$ say), there are $\Delta^1_2$ reductions from $(\mathbb{R}, E_{\omega_1})$ to $(\mathbb{R}, =)$, which is bad. | |
| Apr 6, 2016 at 0:17 | comment | added | Joel David Hamkins | In my opinion, $\Delta^1_2$ is a far more powerful class than Borel. After all, the least transitive model of ZFC is $\Delta^1_2$ definable (if it exists), and so one has much more powerful methods in that class. Basically, $\Delta^1_2$ is huge. | |
| Apr 5, 2016 at 23:30 | history | edited | Danielle Ulrich | CC BY-SA 3.0 |
addendum strengthened
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| Apr 5, 2016 at 23:18 | comment | added | Danielle Ulrich | I edited the post to add that there is a $\Delta^1_2$ reduction | |
| Apr 5, 2016 at 23:16 | history | edited | Danielle Ulrich | CC BY-SA 3.0 |
Added addendum
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| Apr 5, 2016 at 23:12 | comment | added | Burak | This is quite nice. Perhaps, one should add that Gao points out that there exists a $\Delta^1_2$ reduction from $F_{\omega_1}$ to $E_{\omega_1}$ (which is described in $\S 9.2$). It seems that $F_{\omega_1}$ codes countable linear orders in an "unnatural" way that obtaining the well order back in the other coding requires a non-Borel reduction. (Of course, one might argue that this coding is "natural" enough so that we should not believe in the proposed thesis.) | |
| Apr 5, 2016 at 23:05 | comment | added | Joel David Hamkins | This is very nice. Since as you noted your relations are not Borel, one naturally wonders whether there are such examples with Borel relations. | |
| Apr 5, 2016 at 21:52 | history | answered | Danielle Ulrich | CC BY-SA 3.0 |