A quick note for those who might be interested.

I wrote this recently – “The notion of the infinite is contingent on notions of finitude. If notions of finitude are empty, so are notions of the infinite.”

That was within a Buddhist context concerning emptiness.

Classically speaking, the notion of the finite and the notion of the infinite are complicit conceptions.

Finite means, defining; definite; finished. Infinite means, not finite.

So of course both words have each a range of meanings which are not necessarily strictly synonymous, but those meanings are connected and have a logic.

Obviously, finite and infinite are inverses of each other, but it’s probably important to keep track of whatever specific inversions are in play with respect to specific notions of the finite, and their corollary, specific inversions of the infinite.

How this is useful, is where notions of the infinite do indeed rest on specific characterisations. They are infinite with respect to some specific quality, substance, tendency, extent, et cetera. Those specificities are finite structures, that is to say, defined or ‘finished’ structures, structures of the finite which are then opened up to various modes of infinity procedures or infinitisations, whether it be endless recursion of, or just endless extent of, whatever specific, finite structuring, is being addressed.

The important thing to remember is the specific complicity between the two concepts.

I don’t think finite and infinite are inverses. Take for example Cantor’s diagonal slash argument. The construct that is finite (the listing) does not include the listing that is constructed on the diagonal. These are not inverses of a common structure.

Look at it this way – whatever the principle indicating or configuring the specific uncountability might be, that is the finite or defining structure of the infinity characterised by it.

If you’re suggesting that there might be infinities which are not susceptible to any kind of principled reduction in finite or definitive terms, I would agree with you, but such infinities would transcend any specific definition, except the definition of being indefinable.

If it’s indefinable, then can it be defined as infinite? On what basis does the definition of being infinite apply? The possibility that there are infinite progressions or sequences beyond any cardinality?

The context you are assuming is the mathematical or set theory conception of infinity. That might not be the only way to model infinity.

Etymologically speaking, the infinite means ‘not finite’, so there is inversion, there. The finite is a seeming localisation of infinite possibility, but in being a localisation it is susceptible to reflecting the infinite possibility that it localises, as it is contextually considered. A doorway to that infinite possibility.