Your concept of "infinitesimal" such that you can never get to 1 by multiplying it by a number no matter how large is not a part of "standard" mathematics; [and is not] particularly related to logarithms and orders of magnitude.
I'm pretty sure you meant to say finite number, here.
Are we talking about the same concept of orders of magnitude? (It might help to consider the notion that both torture and dust-speckings are distant from the same approximate "zero-magnitude" event which is the 1 anti-utilon.)
Your whole use of "really small numbers" and "really large numbers" in this thread (notably in the discussion with paper-machine) is inconsistent with the ways those concept are usually used in maths.
For any value of (finite) "really large number" n there is an equivalent "really small number" that can be expressed as 1/n. The notion of the logarithmic quantification of pain bears relevance to our discussion because of the fact that we have declared the dust-specking the "smallest possible unit of suffering". This essentially renders it nearly infinitessimal, and as such subject to the nature of infinitessimal values which are essentially the exact inverse of "mind-fuckingly large" numbers.
It is furthermore worth noting that there is, since we're on the topic of numbers and quantification, a sort of verbal slight-of-hand going on here: the 'priming' effect of associating 3^^^3 'dust-speckings' with a "mere" 50 'years of torture'. I have repeatedly been asked, "If 3^^^3 isn't enough, how about 3^^^^3 or 3^^^^^3?" -- or questions to that effect. When I note that this is priveleging the hypothesis and attempt to invert it by asking what number of years of torture would be sufficient to overwhelm 3^^^3 dust-speckings in terms of disutility, a universal response is given, which I will quote exactly:
" "
This, I feel, is very telling, in terms of my current point regarding that "slight-of-hand". Unlike units of measurement are being used here. I'll demonstrate by switching from measurement of pain to measurement of distance (note that I am NOT stating these are equivalent values; I'm demonstrating the principle I reference, here, and do NOT assert it to be a correct analogy of the torture-vs-specking answers.)
"Which is the longer distance? 50 lightcone-diameters or 3^^^3 nanometers?"
For those not familiar with the topic, Torture vs. Dustspecks asks the question: "Would you prefer that one person be horribly tortured for fifty years without hope or rest, or that 3^^^3 people get dust specks in their eyes?"
Most of the discussion that I have noted on the topic takes one of two assumptions in deriving their answer to that question: I think of one as the 'linear additive' answer, which says that torture is the proper choice for the utilitarian consequentialist, because a single person can only suffer so much over a fifty year window, as compared to the incomprehensible number of individuals who suffer only minutely; the other I think of as the 'logarithmically additive' answer, which inverts the answer on the grounds that forms of suffering are not equal, and cannot be added as simple 'units'.
What I have never yet seen is something akin to the notion expressed in Ursula K LeGuin's The Ones Who Walk Away From Omelas.If you haven't read it, I won't spoil it for you.
I believe that any metric of consequence which takes into account only suffering when making the choice of "torture" vs. "dust specks" misses the point. There are consequences to such a choice that extend beyond the suffering inflicted; moral responsibility, standards of behavior that either choice makes acceptable, and so on. Any solution to the question which ignores these elements in making its decision might be useful in revealing one's views about the nature of cumulative suffering, but beyond that are of no value in making practical decisions -- they cannot be, as 'consequence' extends beyond the mere instantiation of a given choice -- the exact pain inflicted by either scenario -- into the kind of society that such a choice would result in.
While I myself tend towards the 'logarithmic' than the 'linear' additive view of suffering, even if I stipulate the linear additive view, I still cannot agree with the conclusion of torture over the dust speck, for the same reason why I do not condone torture even in the "ticking time bomb" scenario: I cannot accept the culture/society that would permit such a torture to exist. To arbitrarily select out one individual for maximal suffering in order to spare others a negligible amount would require a legal or moral framework that accepted such choices, and this violates the principle of individual self-determination -- a principle I have seen Less Wrong's community spend a great deal of time trying to consider how to incorporate into Friendliness solutions for AGI. We as a society already implement something similar to this, economically: we accept taxing everyone, even according to a graduated scheme. What we do not accept is enslaving 20% of the population to provide for the needs of the State.
If there is a flaw in my reasoning here, please enlighten me.