Mathematics with the potential or actual infinite or without emptiness

[Patrik Eklund]

In set theory, the empty set is axiomatized. No other set is, except for 
the powerset, but the powerset is a constructor of sets from sets, and 
in the end boils down to the axiomatized existence of the empty set.

When we move to using set theory as a metalanguage for developing other 
languages, like logic languages, we depart from strict set theory in the 
sense of being generated from the empty set, and we start to use 
"notation" of all kind. Indices come into play, and we use indexed 
notations for sets where the indices themselves are sets.

And natural numbers come into play. Where do they actually come from? At 
the very beginning, they are just there out of the blue, so that they at 
least can be used in indexing.

When all this happens, we come to a situation where we have several 
"languages" that are intertwined and we have lost the sense of which 
comes first, and which one is derived room what. This then means that 
"from-to" of languages is removed, and this enables to go back and forth 
between them. I have called this a breach of the principle of 
'lativity'.

This is what happens e.g. in Gödel 1931. There is PM, there is Peano, 
and there is set theory. And when the "from-to" is removed, everything 
is very "untyped", and the notion of "proposition" is all too vague, 
Gödel is able to apply the Liar and Richard's techniques (himself 
admitting it) to develop what during a century has been called a 
"Theorem", which in fact is nothing but a Paradox.

I would tend to find this these discussions on infiniteness also rather 
senseless without more precise foundations.

In general, FOM has never allowed debates on foundations that would 
question the establishment. My view was always that we need to go back 
to where Hilbert stopped (because he was too old to continue himself), 
and dismiss, or at least seriously question, results that are based on 
tricks enabled by the Liar and Richard.

Best,

Patrik



On 2023-02-12 03:02, dennis.hamilton at acm.org wrote:
> -----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
> 
> ***
> ***
> 
> 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.
> 
> ***
> ***
> 
> 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.
> 
> ***
> ***
> 
> 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
> 
>  [orcmid]  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
> - - - -
>  - - - - - - - - -
> 
> 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