Significance of Omega
The finite is essential to this solution in order to produce finite integrals. In a bounded universe, actually must be finite, although this is not usually an argument used by those proposing the SIA rebuttal. However, other proponents of indefinite survival of human (and posthuman) intelligence have postulated a finite endpoint, as the (extremely high) “Omega”.
Specifying any finite upper limit, was not a part of Dieks's argument, and critics of the SIA have argued that an infinite upper bound on N creates an Improper integral (or summation) in the bayesian inference on N, which is a challenge to the logic of the critique. (For example Eastmond, and Bostrom, who argues that if the SIA cannot rule out an infinite number of potential humans, it is fatally flawed.)
The unbounded vague prior is scale invariant, in that the mean is arbitrary. Therefore no finite value can be selected with more than a 50% chance of being above N (the marginal distribution of N). Olum's critique depends on such a limit existing; without this his critique is technically not applicable. Therefore it must be cautioned that the simplification here (to bound N's distribution at ) omits a significant hurdle to the credibility of the Self-Indication Assumption Doomsday argument rebuttal.
Read more about this topic: Self-Indication Assumption Doomsday Argument Rebuttal
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