[SEQ RERUN] 0 And 1 Are Not Probabilities

Successful use would count as evidence for the laws of probabilities providing "good" values right? So if we use these laws quite a bit and they always work, we might have P(Laws of Probability do what we think they do) = .99999 We could discount our output using this. We could also be more constructive and discount based on the complexity of the derivation using the principle "long proofs are less likely to be correct" in the following way: Each derivation can be done in terms of combinations of various sub-derivations so we could get probability bounds for new, longer derivations from our priors over other derivations from which it is assembled. (derivations being the general form of the computation rather than the value specific one).

ETA: Wait, were you sort of diagonalizing on Bayes Theorem because we need to use that to update P(Bayes Theorem)? If so I might have misread you.

the probability that some elementary event in the entire sample space will occur is 1

I believe that a part of the post's point is that the entire sample space is hard to find in most real-life cases. From the post:

However, in the real world, when you roll a die, it doesn't literally have infinite certainty of coming up some number between 1 and 6. The die might land on its edge; or get struck by a meteor; or the Dark Lords of the Matrix might reach in and write "37" on one side.

EDIT: Another example, this time from the Martin Gardner's excellent book, Mathematical Games :

The hotel's cocktail lounge before the dinner hour was noisy with prestidigitators. At the bar I ran into my old friend "Bet a Nickel" Nick, a blackjack dealer from Las Vegas who likes to keep up with the latest in card magic. The nickname derives from his habit of perpetually making five-cent bets on peculiar propositions. Everybody knows his bets have "catches" to them, but who cares about a nickel? It was worth five cents just to find out what he was up to. "Any new bar bets, Nick?" I asked. "Particularly bets with probability angles?"

Nick slapped a dime on the counter beside his glass of beer. "If I hold this dime several inches above the top of the bar and drop it, chances are one-half it falls heads, one-half it falls tails, right ?"

"Right," I said.

"Betcha a nickel," said Nick, "it lands on its edge and stays there."

"O.K.," I said.

Nick dunked the dime in his beer, placed it against the side of his glass and let it go. It slid down the straight side, landed on its edge and stayed on its edge, held to the glass by the beer's adhesion. I handed Nick a nickel. Everybody laughed.

Nick tore a paper match out of a folder, marked one side of the match with a pencil. "If I drop this match, chances are fifty-fifty it falls marked side up, right?" I nodded. "Betcha a nickel," he went on, "that it falls on its edge, like the dime."

"It's a bet," I said.

Nick dropped the match. But before doing so, he bent it into the shape of a V. Of course it fell on its edge and I lost another nickel.

Jaynes didn't like Kolmogorov's axioms, and I expect Eliezer would agree. I remember he mentioned somewhere in the sequences that he thought probability could be axiomatized without reference to probabilities of 0 or 1, but it wouldn't have much practical use to do so.