I assume most people have heard of Pascal's Wager, in case you have not: https://en.wikipedia.org/wiki/Pascal%27s_Wager
By the Stanford encyclopedia of philosphy:
"We have a decision under risk, with probabilities assigned to the ways the world could be, and utilities assigned to the outcomes. In particular, we represent the infinite utility associated with salvation as ‘∞’. We assume that the real line is extended to include the element ‘∞’, and that the basic arithmetical operations are extended as follows: For all real numbers r: ∞+r=∞. For all real numbers r: ∞×r=∞ if r>0."
Doesn't this contradict one of the axioms of the expected utility theory? In particular the archimedean property states:
Let l, l’ and l’’ be three lotteries in L such that $l \succ l’ \succ l’’$. Then there are $p,q \in(0,1)$ such that: $pl + (1-p)l’’ \succ l’ \succ ql + (1-q)l’’$
and it basically means that there are no infinitely preferred or infinitely despised lotteries.
But in the Pascal's Wager, $l$ = salvation = $ + \infty$, $l'$ = nothing (god doesn't exist), $l''$ = damnation = $ - \infty$,
but then there does not exist a probability $p > 0$ such that any mixed combination of damnation and salvation is strictly preferred than "god doesn't exist".
What do you think?