Scott Garrabrant comments on Asymptotic Logical Uncertainty: Uniform Coherence

[Scott Garrabrant]

It does not imply belief in all true ${\Pi }_1$ sentences. If $f\left(x\right)$ is true for all $x$, the probability of $f\left(x\right)$ will go to 1 as $x$ goes to infinity, but the probability of $\left(\neg \neg {)}^n\forall xf\right(x)$ may stay bounded away from 1.

[abramdemski]

Ahh, right. Silly mistake.

So the difference in this example is that a coherent prior with approximation $P_t\left(n\right)$ can have $li{m}_t{P}_t\left({\varphi }_{s_n}\right)\to 1$ for all $n$, but not have $li{m}_t{P}_t\left({\varphi }_{s_t}\right)\to 1$; uniformly coherent must have this. To be precise, if we set $P_t\left(\varphi \right):=M(┌(\neg \neg {)}^t\varphi ┐)$ with $M$ uniformly coherent, we must have this.

[Scott Garrabrant]

Correct.