# What role has Whitehead's Conjecture played in the thinking on the plurality of set theories?

I am curious about the history of the Whitehead's Conjecture, as this was the first natural mathematical statement, in the sense that mathematicians were actually interested in the answer, that was shown to be independent of ZFC. That is, in Gödel's language, it is undecidable. Before then, all statements shown to be undecidable were artificial, in the sense that no mathematician was actually interested in the statement itself. They were merely examples showing that such statements actually existed.

John Constatine Henry Whitehead, to give him his full name, and who also happens to be the nephew of Alfred North Whitehead who collaborated with Bertrand Russell on their Principia Mathematica, was lead to this question by his investigation of the Cousain problems in complex analysis/geometry. The linked article is clear enough for those with some mathematical maturity. It's expressed in the language of homological algebra but it's explained what this means in terms in a more prosaic manner. To rephrase:

An abelian group G, is called a Whitehead group when every surjective morphism, f, to it from any other abelian group splits. That is, it has a right inverse.

Now, it turns out that every free group is a Whitehead group. J. C. H. Whitehead asked whether the converse is true. That is:

Is every Whitehead group also a free group?

Ekloff, in an article in The American Mathematical Monthly (vol.83, no.10, Dec.'76) points out that this has a natural interpretation in terms of topological groups by taking advantage of Pontryagin duality: that is, is every path connected abelian group, a power of the circle group, that is a product of a number of copies of the circle group.

Stein actually showed that Whiteheads conjecture follows when the group is countable. But otherwise progress was slow until Saharon Shelah in '74, completely unexpectedly showed it was undecidable from ZFC. In fact more, he showed it was undecidable if one also assumes GCH, the Generalised Continuum Hypothesis.

Equally interestingly, he also showed that if one assumes that V=L, that the von Neumann universe is equivalent to Gödel's constructible universe then it does happen to be true!

Q. Have any mathematicians postulated that V=L is a natural axiom to add to ZFC as it resolves Whiteheads question in a natural way? More, is it also evidence that one ought to consider a plurality of set theories in the same way that we have a plurality of number systems (of many kinds: groups, modules etc).

• Many mathematicians, starting with Cantor, were interested in the continuum hypothesis before that, which was proved independent of ZFC earlier. Quine suggested adopting V=L, Devlin and Fraenkel were sympathetic, but most mathematicians find it "unnatural" as it goes against the grain of maximizing ontology in mathematics, not minimizing it, see Maddy, Naturalism in Mathematics, pp. 73-81. Set theory and algebraic systems are not of a kind, the latter are not intended to be "categorical". Nov 16 '20 at 6:26
• @Conifold: That was my initial thought as well. The continuum hypothesis arose in a large number of mathematical pursuits related to real, complex, and functional analysis during the first few decades of the 1900s. In fact, Sierpiński even wrote a book about results equivalent to the continuum hypothesis -- Hypothèse du Continu (1934). I imagine it also arose in some algebra pursuits as well (Ulam invariants in group theory, existence of certain types of field extensions, etc.), but probably only for reducing possibilities. Nov 16 '20 at 11:51
• Keith Devlin does exactly this in his book The Axiom of Constructibility. A Guide for the Mathematician (1977). I actually have a copy of this, purchased back when university bookstores (well, at least the bookstore where I was an undergraduate) received all the Springer Lecture Notes in Mathematics series books as they were published. The Whitehead problem is discussed on pp. 44-58. Nov 16 '20 at 12:01
• The continuum question is a question purely in set theory --- So is the axiom of choice, but like the axiom of choice, there are plenty of results outside of set theory that were originally proved using the continuum hypothesis and which were later shown to be false without this assumption, such as if a bounded and non-negative function $f(x,y)$ is such that both the $x$-integral of $f$ followed by the $y$-integral of $f$ exists and also the reverse-order iterated integral exists, do the two iterated integrals have to be equal? (continued) Nov 17 '20 at 13:56
• What the continuum hypothesis does is allow you to use transfinite induction over $\mathfrak c$-length transfinite sequences in such a way that at each limit ordinal stage there are only countably many preceding steps, which allows for sequential-based and other countability tools (e.g. countable dense sets, series of positive terms having a finite sum, etc.) to be applied. For instance, you can well order the reals (or the Borel sets, or the set of continuous functions, etc.) so that each element (real, set, function, etc.) has only countably many predecessors in that order. Nov 17 '20 at 13:57