Actual and Potential infinite

[Sam Sanders]

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.