Timeline for Periodic point-free maps and free ultrafilters.
Current License: CC BY-SA 3.0
7 events
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| Mar 12, 2013 at 17:05 | comment | added | Clinton Conley | Asaf, I confess that my primary motivation for looking at such examples comes not from the perspective of choiceless math but rather from the attempt to understand descriptive set-theoretic complexity among orbit equivalence relations. Nevertheless, they are occasionally useful as sanity checks for what is provable from ZFDC. In any case, I think your example is very nice and has a more classical set-theoretic flavor. | |
| Mar 11, 2013 at 19:26 | comment | added | Asaf Karagila♦ | @Clinton, thank you for this example. I am trying to expand my "counterexample sources" to models of ZF+DC+BP, but amorphous (and $\kappa$-amorphous) sets are still my first go to solution. They are just so lovely because they terrifyingly violate so many choice principles at once... | |
| Mar 11, 2013 at 18:36 | comment | added | Clinton Conley | So for example one recovers a version of Asaf's example by looking at the complementation map on $2^\omega / E_0$ (or equivalently $\mathcal{P}(\omega)/\mathrm{FIN}$). For an example with no finite orbits, one can look at for instance the free part $X$ of the shift action of the free abelian group $\mathbb{Z}^2$ on generators $a, b$ acting on $2^{\mathbb{Z}^2}$. The map $[x]_A \mapsto [b\cdot x]_A$ works on $X/A$, where $A$ is the orbit equivalence relation of $\langle a \rangle$. | |
| Mar 11, 2013 at 18:35 | comment | added | Clinton Conley | I'm not sure this is of interest to anybody, but if you're willing to work in a model of ZF+DC+BP then many examples are furnished by a little bit of ergodic theory. The point is that if $E$ is any generically ergodic meager equivalence relation on Polish $X$, the collection of subsets of $X/E$ which have comeager lift to $X$ forms a ($\sigma$-complete!) ultrafilter. This ultrafilter is of course preserved by maps on the quotient induced by homeomorphisms of $X$, since they leave category on $X$ unchanged. [cont.] | |
| Mar 11, 2013 at 2:24 | comment | added | Joel David Hamkins | Great ! | |
| Mar 11, 2013 at 2:08 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
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| Mar 11, 2013 at 1:59 | history | answered | Asaf Karagila♦ | CC BY-SA 3.0 |