Ben Wallis’s counter-apologetics blog

William Lane Craig uses Hilbert's hotel in an attempt to illustrate the impossibility of an existing actually infinite multitude, but I find several serious gaps in his arguments which, in my judgment, prevent them from having any force. His approach takes two forms: First, he claims that certain logical contradictions follow from the existence of actual infinites; second, his intuition tells him that an existing infinite multitude is absurd. In response, I want to resolve the alleged contradictions, and argue that intuition is an unreliable guide to the possibility or impossibility of an existing actual infinite.

Philosopher and mathematician Alexander Pruss in 2008 , and again in 2009 , discussed on his blog the following paradox: Consider an infinite collection of grim reapers indexed by the positive integers n=1,2,..., where the nth grim reaper is scheduled to kill Fred at 11:00am + 1/n minutes, and where Fred's life is otherwise safe during that period. By hypothesis, he dies from the hand of a grim reaper, say the kth grim reaper. However the (k+1)th grim reaper visited Fred before the kth grim reaper, which means that Fred must already be dead by the time the kth grim reaper visits him. This is a contradiction, and we conclude that there is a logical error in the construction of this unusual and hypothetical situation.

In a blog post dated 2010 Mar 17, mathematician and philosopher Alexander Pruss expressed some interesting ideas regarding probability and countably infinite samples in an attempt to show an absurdity which would, in his judgment, lend support to the claim that the existence of what he and others call "actual" infinite collections of physical objects is impossible. [ 1 ] He suggests a set of hypotheses which, taken together, appear to violate our intuition regarding probability. While I do not believe this constitutes evidence against actual infinities, I find the argument interesting in another way which I shall discuss here. In particular, I maintain that we ought not assign probability values under certain conditions whereby the probability measurements in question are insufficiently interpreted. Paul Castell calls this position abstention , [ 2 ] and Pruss's ideas yield an opportunity to give an example of how we might find reason for adopting it.