Mathematics with the potential or actual infinite or without emptiness

[dennis.hamilton at acm.org]

-----Original Message-----
From: FOM <fom-bounces at cs.nyu.edu> On Behalf Of I.V. Serov
Sent: Saturday, February 11, 2023 08:52
To: fom at cs.nyu.edu
Subject: Mathematics with the potential or actual infinite or without 
emptiness

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Dennis Hamilton writes
(https://cs.nyu.edu/pipermail/fom/2023-February/023742.html):

Now it is certainly a fair point that obtaining zero as the limit as n goes to 
infinity of 2^-n is the limit of an infinite sequence.

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Antonio Drago replies (in
https://cs.nyu.edu/pipermail/fom/2023-February/023768.html):

Not true. Any approximation does not equate the zero, always a distance 
remains; or evenly: a segment, representing an approximation, is defined by 
two extreme points; it cannot be reduced to one point only; this is an old 
criticism to epsilon-delta technique .

. illusion that leads to believe that the approximations at last crash to 
zero. It is just an appeal to actual infinity that justifies this crashing as 
effective. Really, it is true in a world of mathematical ideas only according 
to the "realist [Platonic] attitude". The entire undergraduated teaching of 
mathematics is based on the actual infinity.

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Does zero exist at all, does it not?

Potentially it does not exist as it can never be reached as in the
example of 2^-n.
Limits are always potential unless they are actually achieved.

[orcmid]  [ ... ]

 Serov

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I clearly stepped on quicksand here, and I need to jump for an overhanging 
branch.

I will concede that the sequence 2^-n provide an inexhaustible set of 
progressive values that gets us as close to 0 as we might desire.  To speak of 
a limit seems to slip into actual infinities and I will forswear that.

I will also be content to say the same for 1-10^-n providing an inexhaustible 
series of values that can be used to get as close to 1 as we desire.  It's 
basically 0.999...9 with an inexhaustible supply of 9's.

I have no quarrel whatsoever with regard to 0 (and 1) as mathematical entities 
and I see no difficulty with the mathematical standing of the empty set (and 
the null string) as such.  There is no metaphysical claim and I see no need 
for concern in that respect.   There is significant pragmatic value.

 - Dennis