Background and Motivation




Question, Loosely Stated







  • $\begingroup$ I posted a question to look for Whitehead-like problems, but despite some upvotes no answers were posted. mathoverflow.net/questions/101212/… $\endgroup$ – Asaf Karagila Jan 4 '13 at 19:53
  • $\begingroup$ Your question, also interesting, looks for open problems that were proven in some definition of "countable case" and not necessarily independent or implied by stronger set theoretic axioms, so I am hoping that the different criteria and wording might prompt some answers. $\endgroup$ – Jason Polak Jan 4 '13 at 20:04
  • $\begingroup$ Jason, while you are technically correct, I have realized that just like the axiom of choice ensures that the passing from finite to countable is relatively smooth; combinatorial principles (which hold in $V=L$) ensure similar transitions for higher cardinals. Our questions are indeed different, but they do share a common background (in my opinion). $\endgroup$ – Asaf Karagila Jan 4 '13 at 20:49









  • $\begingroup$ (I meant to say $\mathrm{Ext}^1(A,\mathbb{Z}) = 0$) $\endgroup$ – Jason Polak Jan 4 '13 at 20:44
  • $\begingroup$ Jason, yes, probably the right question here is simply to ask for the best examples of algebraic statements independent of ZFC. We don't have so many examples. You can peruse mathoverflow.net/questions/1924/…. $\endgroup$ – Joel David Hamkins Jan 4 '13 at 20:48

One can find examples of algebraic statements that are independent of ZFC + $V = L$ by considering "absolute" versions of standard algebraic propositions. This happens for example when one seeks large strongly non-isomorphic (i.e. non-$L_{\infty \omega}$-equivalent) or absolutely rigid families of structures, absolutely indecomposable abelian groups and modules, or absolute $E$-rings of arbitrarily big cardinality. Roughly, up to the first $\omega$-Erdos cardinal $\kappa(\omega)$, in general such families exist, but above $\kappa(\omega)$, they do not.

The beauty of $\kappa(\omega)$ is that if it exists, then it exists in $L$, so that the existence of absolutely rigid systems in every cardinal is independent of ZFC + $V = L$.

In loose terms, the absolute version of a property $P$ is obtained by requiring that any structure $M$ retains $P$ in any generic extension. This can be recast as in algebraic terms using back-and-forth systems or infinitary equivalence.

Questions about the existence of this type of strongly non-isomorphic structures were first asked by Nadel and pursued more recently by Eklof, Fuchs, Goebel, Herden and Shelah.


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