You are making a mistake in reasoning if you don't change your belief through that evidence. Your belief should change by orders of magnitude. A change from 10^{-18} to 10^{-15} is a strong change.

Where do those numbers come from?

I agree with this comment, but I want to point out that there may be a problem with equating the natural language concept "strength of evidence" with the likelihood ratio.

You can compare two probabilities on either an additive or multiplicative scale. When applying a likelihood ratio of 1000, your prior changes by a multiplicative factor of 1000 (this actually applies to odds rather than probabilities, but for low probability events, the two approximate each other). However, on an additive scale, a change from 10^{-18} to 10^{-15} is really just a change of less than 10^{-15} , which is negligible.

The multiplicative scale is great for several reasons: The likelihood ratio is suggested by Bayes' theorem, it is easy to reason with, it does not depend on the priors, several likelihood ratios can easily be applied sequentially, and it is suitable for comparing the strength of different pieces of evidence for the same hypothesis.

The additive scale does not have those nice properties, but it may still correspond more closely to the natural language concept of "strength of evidence"

Yes, that is probably clear to most of us here. But, in reality, I and most likely also you discount probabilities that are very small, instead of calculating them out and changing our actions (we'll profess 'this is very unlikely' instead of 'this is not true', but what actually happens is the same thing). There's a huge amount of probability 10^{-18} deities out there, we just shrug and assume they don't exist unless enough strong (or 'good', I still don't see the difference there) evidence comes up to alter that probability enough so that it is in the realm of probabilities worth actually spending time and effort thinking about.

This hypothetical skeptic, if pressed, would most likely concede that sure, it is /possible/ that Zeus exists. He'd even probably concede that it is more likely that Zeus exists than that a completely random other god with no myths about them exists. But he'd say that is fruitless nitpicking, because both of them are overwhelmingly unlikely to exist and the fact that they still might exist does not change our actions in any way. If you wish to argue this point, then that is fine, but if we agree here then there's no argument, just a conflict of language.

I'm trying to say that where you would say "Probability for X is very low", most people who have not learned the terminology here would normally say "X is false", even if they would concede that "X is possible but very unlikely" if pressed on it.

In the Bayesian worldview you always have probabilities for your beliefs.

That is empirically false.

Maybe you meant "To be a proper Bayesian, one should have probabilities for one's beliefs"?