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.)
If infinities existed, they would entail the existence of metaphysically impossible scenarios (absurdities)
This is relevant, because if an infinity could be instantiated in the real world, then so too could these impossible scenarios. (But in fact, these scenarios can’t be instantiated, and so modus tollens infinities can’t either.)
No, after all…
• …actual infinities are mathematically legitimate (in set theory).1 [See below]
• …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 deeply intuitive (almost commonsensical) proposition about the world has to be false. (I.e. belief in the possible existence of an infinity would require significant evidential justification.)
“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)—especially within an ad hoc system of rules designed to prevent contradiction, like subtraction's being disallowed in set theory—doesn't entail that the concept 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.
• …the interval is not comprised of an infinity of point-parts or divisions (i.e. there is not infinitieth cut. Instead, the cuts depend on the pre-existing interval.)1
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”. (That is to say, couldn't it represent the world the way a map does, which is not inherently propositional in form, but which can nevertheless be described with a potential infinity of propositions [e.g. city A is 3 miles from city B; one can get there via route 1 or 2, or 3, etc., ad infinitum]).2