From the modern point of view, infinity comes not only in various flavors (some of which Kant seems to have been aware of), but various sizes. So when Kant talks about conceptions as being too small or too large for the understanding, he seems to leave us room, nowadays, to ask whether it is not so much that infinity in general is always "too large," but that specific levels of this would outstrip our intuition's resources?

Moreover, if we confine ourselves to the horns of the size trilemma modulo Cantor's absolute infinity, we would have:

  1. Absolute finitude.
  2. Relative finitude (sometimes called the parafinite).
  3. Relative infinity.
  4. Absolute infinity.

... and the indefinite is assimilated to (2) and (3) variously (or, "worse," there is an indefinite distinction between (2) and (3) themselves). Another formulation of this quadrilemma might be:

  1. Finite in relation to itself and other things.
  2. Finite in relation to itself but infinite relative to other things.
  3. Infinite in relation to itself but finite relative to other things.
  4. Infinite in relation to itself and other things.

How does our modern understanding of different conceptions of finitude and infinity play into the representation of the antinomies and their possible solutions?

  • 2
    I think your "our modern understanding" will confuse many users because most associate it with Cantorian actual transfinite. Parafinite, just as Kant's mereological conception, are decidedly non-Cantorian, so you are talking about rather niche non-Cantorian modern understanding. Routledge gives a useful summary of Kant's understanding of human finitude and purely regulative use of the infinite, which does have some affinity to the parafinite.
    – Conifold
    Commented Oct 11, 2023 at 23:02
  • Behold and contemplate is there really any difference between your 'Finite in relation to itself' and your 'Infinite in relation to itself'... Commented Oct 12, 2023 at 6:30
  • @DoubleKnot I came up with those definitions before I was more knowledgeable about quantifiers, so later I considered, "Finite in relation to everything, infinite in relation to nothing," etc., except then either (A) the middle concepts would both be, "Finite in relation to some things and infinite in relation to some things," so they'd collapse into one again, or (B) there'd be concepts like, "Finite in relation to everything and infinite in relation to some things," and a fifth, "Finite and infinite relative to everything." But I worry about ending up with fourfold schemes in my theories... Commented Oct 12, 2023 at 7:07
  • 1
    Think about the transfer principle in Robinson's NSA, why all true 1st-order formulas can be transferred from all the standard finite (internal) reals to their infinite extension as internal sets of hyperreals by truth-preserved elementary embedding based on certain ultrafilter construction? I guess your so called 'in relation to itself' can be formalized as all 1st-order formulas of a σ structure, then transfer principle implies the above two 'in relation to itself' σ structures are almost model-theoretically elementarily equivalent except cardinality in the sense of Skolem's paradox... Commented Oct 14, 2023 at 5:34


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