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On the other hand, in summaries for a broad readership, the posteriors are the most important result to report. Now most readers don't have the expertise to bring their own priors, so you have to give them yours. And then do the calculation for them.
Good point. It would be irresponsible to publish a news item that "the Prime Minister's support for this bill is three times more likely if he is, in fact, a lizard alien than if he is a human" without noting that the prior probability for him being a lizard alien is pretty low.
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Mainstream status:
The presentation of the natural numbers is meant to be standard, including the (well-known and proven) idea that it requires second-order logic to pin them down. There's some further controversy about second-order logic which will be discussed in a later post.
I've seen some (old) arguments about the meaning of axiomatizing which did not resolve in the answer, "Because otherwise you can't talk about numbers as opposed to something else," so AFAIK it's theoretically possible that I'm the first to spell out that idea in exactly that way, but it's an obvious-enough idea and there's been enough debate by philosophically inclined mathematicians that I would be genuinely surprised to find this was the case.
On the other hand, I've surely never seen a general account of meaningfulness which puts logical pinpointing alongside causal link-tracing to delineate two different kinds of correspondence within correspondence theories of truth. To whatever extent any of this is a standard position, it's not nearly widely-known enough or explicitly taught in those terms to general mathematicians outside model theory and mathematical logic, just like the standard position on "proof". Nor does any of it appear in the S. E. P. entry on meaning.
I remember explaining the Axiom of Choice in this way to a fellow undergraduate on my integration theory course in late 2000. But of course it never occurred to me to write it down, so you only have my word for this :-)