I'm wondering if there are examples of statements that have been proven whose consistency proofs came before the proofs of the statements themselves.


More informally, I'm wondering how promising in general is the approach of attempting a consistency proof for a statement when faced with a statement that seems true but difficult to prove.


Background:


If a statement is provable from a set of axioms, then that statement is obviously consistent (assuming the set of axioms is consistent). So provability is stronger than consistency. This might lead one to think that constructing a consistency proof for a statement should be strictly easier than constructing a proof.


Yet consistency proofs (at least oft-cited ones, for example those by Godel and Cohen about the Continuum Hypothesis) seem to require a high level of sophistication (though this might be a byproduct of the fact that consistency proofs like these are for the special class of statements that cannot be proven).


For statements that can be proven then, are there cases where their consistency proofs are easier or came before the proofs themselves?


**Update:**


Thanks a lot, everyone, for the great answers so far. The number and existence of these examples is interesting to me, as well as the fact that they all rely on the same technique of first proving something using an additional axiom (an approach first suggested by Michael Greinecker). That hadn't occurred to me. I wonder if there are other approaches.



Are there examples of statements that have been proven whose consistency proofs came before their proofs?
Here are my favorite examples of statements whose consistency was established and cherished before their proof.


**1.** The [Keisler-Shelah isomorphism theorem](http://arxiv.org/PS_cache/math/pdf/9204/9204203v1.pdf) stating that two elementarily equivalent structures have isomorphic ultrapowers (proved using in $ZFC+GCH$ by Keisler in 1964, and in $ZFC$ by Shelah in 1971).


**2.** Several algebraic results (normal forms, divison algorithms, etc) concerning the "left-distributive algebras of one generator" were first established by Richard Laver by **assuming (very) large cardinals** (known as **(I3)** in the literature). Later Patrick Dehornoy eliminated the large cardinal assumption in these results by giving a representation in the braid group. 


By the way, a left distributive algebra is a set with one binary operation \* satisfying the left distributive law
$a\*(b\*c)=(a\*b)\*(a\*c)$; the operation of conjugation in a group is an example. Here is an [old paper of Laver](http://arxiv.org/PS_cache/math/pdf/9204/9204203v1.pdf) about this topic; there is by now a large literature on the subject.


**3.** In some cases, the consistency proof of a statement $S$ can be combined with some absoluteness argument to yield a proof of $S$ (this was hinted at in the second example cited by Steprans). There are all sorts of absoluteness theorems in logic; the standard tool of the trade is the [Shoenfield absoluteness theorem](http://en.wikipedia.org/wiki/Absoluteness#Shoenfield.27s_absoluteness_theorem) that shows that all sorts of consistency results can be translated into $ZFC$-proofs.


