In 1936 Tarski proved a fundamental theorem of logic: the undefinability of truth.   Roughly speaking, this says there's no consistent way to extend arithmetic so that it talks about 'truth' for statements about arithmetic.  Why not?  Because if we could, we could cook up a statement that says "I am not true."  This would lead to a contradiction, the Liar Paradox: if this sentence is true then it's not, and if it's not then it is.

This is why the concept of 'truth' plays a limited role in most modern work on logic... surprising as that might seem to novices!

However, suppose we relax a bit and allow probability theory into our study of arithmetic.  Could there be a consistent way to say, within arithmetic, that a statement about arithmetic has a certain probability of being true? 

We can't let ourselves say a statement has a 100% probability of being true, or a 0% probability of being true, or we'll get in trouble with the undefinability of truth.  But suppose we only let ourselves say that a statement has some probability greater than X and less then Y of being true, where 0 < X < Y < 1.  Is that okay?

Yes it is, according to this draft of a paper by Paul Christiano, Eliezer Yudkowsky, Marcello Herresho and Mihaly Barasz!   

But there's a catch, or two.  First there are many self-consistent ways to assess the probability of truth of arithmetic statements.  This suggests that the probability is somewhat 'subjective' .  But that's fine if you think probabilities are inherently subjective.  

A bit more problematic is this: their proof that there exists a self-consistent way to assess probabilities is not constructive.  In other words, you can't use it to actually get your hands on a consistent assessment.

Fans of game theory will be amused to hear why: the proof uses Kakutani's fixed point theorem!  This is the result that John Nash used to prove games have equilibrium solutions, where nobody can improve their expected payoff by changing their strategy.  And this result is not constructive.

If you don't know Tarski's original result, try this:
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