Infinity minus an infinity yields logically impossible scenarios. Notably, one can take away identical quantities from identical quantities and arrive at contradictory remainders.1 This is relevant because if an infinity could be instantiated in the real world, then so too could these contradictions. (But in fact, contradictions can’t be instantiated, and so modus tollens infinities can’t either.)
Infinities yield metaphysically impossible scenarios (absurdities)
•… Actual infinities are mathematically legitimate (in set theory).1 [See below]
So? Couldn’t it simply be that…
•… These only show certain kinds of infinity are impossible.2
A part of the whole contains less than the whole (i.e. In reality, if M’ is a submultitude of M, then intuitively there are more things in M than M’). This is relevant because if infinities are possibly instantiated in the real world, then this intuitive proposition about the world is actually false.
“The notion of an 'actual infinity' is logically consistent/possible within Axiomatized Set Theory.”
But, so what? A concept's being logically possible (free of formal contradictions) doesn't entail that it is actually/metaphysically possible.1
Any interval contains an infinity of subintervals (e.g. a meter and minute can both be divided in half an infinity of times).1, 2 This is relevant because if there are an infinity of sub-intervals inside any interval, then an actual infinity of subintervals must exist.
By way of response, however, intervals can only potentially infinitely be divided (i.e., divided, then divided again, then again, with no end), so it is not an actual (completed) infinity. That is to say, the interval is not comprised of an infinity of point-parts or divisions; there is not infinitieth cut. Instead, the cuts depend on the interval (i.e. the interval is logically prior to any potential infinity of divisions/points that we continually impose on it with our conceptual dividings).
God is an infinite being. This is relevant because if God is an infinite being, then God's existence entails the existence of an actual infinity.
But wait, when we say God is infinite, we are ascribing a qualitative attribute to God (e.g. God is maximally good and powerful). We are not ascribing a quantitative infinity.
The extent of God's knowledge would be infinite (given God exists and is omniscient). So if actual infinities are impossible so is God.1
But so what if the extent is infinite? Couldn't it be that God's knowledge is metaphysically "simple", akin to a map, which isn't inherently propositional in form, but which can nevertheless be _represented _propositonally (e.g. city A is 3 miles from city B;one can get there via route xyz, etc.).2