A simultaneous collision that produces paradoxical indeterminism (involving N0 hypothetical particles in a classical three-dimensional Euclidean space) is described in Section 2. By showing that a similar paradox occurs with long-range forces between hypothetical particles, in Section 3, the underlying cause is seen to be that collections of such objects are assumed to have no intrinsic ordering. The resolution of allowing only finite numbers of particles is defended (as being the least ad hoc) by looking at both -sequences (in the (...) context of a very basic supertask, in Section 4) and *-sequences (reversed -sequences, in the form of paradoxical results from the recent literature). Introduction The simultaneous collision The paradox in other contexts The basic problem is N0 things The recent literature Generating N0 things. (shrink)

The metaphysical concept of continuity is important, not least because physical continua are not known to be impossible. While it is standard to model them with a mathematical continuum based upon set-theoretical intuitions, this essay considers, as a contribution to the debate about the adequacy of those intuitions, the neglected intuition that dividing the length of a line by the length of an individual point should yield the line’s cardinality. The algebraic properties of that cardinal number are derived pre-theoretically from (...) the obvious properties of a line of points, whence it becomes clear that such a number would cohere surprisingly well with our elementary number systems. (shrink)

Mawson recently argued that since a temporal God can’t know what we’ll freely choose, so he’s not completely omniscient and hence not omnipotent, whence his beneficence is a matter of luck. However, even (transfinite) arithmetic is inde-finitely extensible and only an everlasting, changeable God could learn forever. Furthermore an epistemically perfect being would hardly, I argue, be completely certain that there were no other perfect beings, because such negative empirical be-liefs could hardly be fully justified. So if God could learn, (...) then heavenly souls would probably ask to be born into a world this far from heaven (causally and epistemically) because that would probably help God to learn more about such matters. And since an omnipotent God’s perfect goodness is most likely to lead to human suffering and divine hiddenness if omnipotence includes the power to change, so it probably does. (shrink)