  We consider the Jeffreys-Lindley paradox from an objective Bayesian
perspective by attempting to find priors representing complete indifference to
sample size in the problem. This means that we ensure that the prior for the
unknown mean and the prior predictive for the $t$-statistic are independent of
the sample size. If successful, this would lead to Bayesian model comparison
that was independent of sample size and ameliorate the paradox. Unfortunately,
it leads to an improper scale-invariant prior for the unknown mean. We show,
however, that a truncated scale-invariant prior delays the dependence on sample
size, which could be practically significant. Lastly, we shed light on the
paradox by relating it to the fact that the scale-invariant prior is improper.
