>On the other hand, sometimes painful issues do arise. The Whitehead >problem, whether or not Ext^1(A, Z)=0 for A an abelian group entails >that A is free, was a shocker to algebraists, you may recall.
And yet (perhaps tellingly, perhaps not) Osofsky's result wasn't--as far as I've ever been able to see--considered to be a shocker, or even more than a mild "ho hum", among topologists (at least, geometric topologists), though Whitehead (J.H.C.) brought up the problem for topological reasons in a topological context. As a matter of fact and practice, (geometric) topologists never (or hardly ever) need to deal with spaces so "big" that that independence result makes any difference to them. Similar comments apply, less strongly, to similar independence results in point-set topology (in my opinion; some people who style themselves "point-set topologists" look--to the extent that there are still any of them on the hoof--like straight out "logicians with an interest in pathology" to most other topologists).
>We would >like to think what seem like basic algebraic questions of that nature >are independent of set theory considerations--and usually, they are. >But not this time, and in fact the whole question is bound up with the >continuum hypothesis. > >Cabal-type intutions, that the continuum hypothesis is false and >Martin's axiom true, lead to one answer, whereas the minimalist >hypothesis that all sets are constructible leads to another. You get >the exact same division over Suslin lines; there are two very different >flavors of set theory extensions which these reflect.