Actual and Potential infinite
Sam Sanders
sasander at me.com
Tue Feb 7 15:20:10 EST 2023
Dear Joe, dear FOM,
In the discussion of “potential vs actual” infinite, I imagine most would say
that Cantor space or the set of real numbers count as an actual infinite set.
> ACA_0 doesn’t require actual infinity in my book, because any arithmetical consequence of it can be proven from ZF without using the axiom of infinity, right?
However, ACA_0 and MUCH stronger (second- and third-order) systems are consistent with
“there is an injection (or bijection) from R to N” (*)
or
“there is a sequence of finite sets (X_n)_{n in N} such that R = U_n X_n,” (**)
as proved in [1]. This includes (standard) systems at the level of Z_2.
[1] Dag Normann and Sam Sanders, On the uncountability of R, JSL, https://arxiv.org/abs/2007.07560
Does it then follow that Cantor space / the reals is a potential infinity?
Best,
Sam
PS:
Regarding (*), one can list the first n reals for any n in N, but NOT put them all together in a sequence (of course).
This is all very “potential infinity”.
PPS:
For those who have trouble imagining third-order objects, one can replace “injection” in (*) by “injection in a nice function class from real analysis”.
PPPS:
Strangely, (*) does not boast many equivalences (in Kohlebach’s higher-order rm), while (**) does.
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