3 Removed outdated info from "update" section after new responses.

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(Michael Greinecker, Tanmay Inamdar, Joel David Hamkins, and Juris Steprans), 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)Michael Greinecker). That hadn't occurred to me. I wonder if there are other approaches.

I've selected Joel's answer because to me it is the most well-written and easiest to understand though all of the answers are enlightening.

2 added 585 characters in body

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 (Michael Greinecker, Tanmay Inamdar, Joel David Hamkins, and Juris Steprans), 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). That hadn't occurred to me. I wonder if there are other approaches.

I've selected Joel's answer because to me it is the most well-written and easiest to understand though all of the answers are enlightening.

1

# Are there examples of statements that have been proven whose consistency proofs came before their proofs?

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?