RE: ​Mathematics with the potential infinite - some inexhaustible?

[dennis.hamilton at acm.org]

From: FOM <fom-bounces at cs.nyu.edu> On Behalf Of Vaughan Pratt
Sent: Tuesday, January 31, 2023 17:55
To: fom at cs.nyu.edu
Subject: Re: ​Mathematics with the potential infinite
 
[orcmid] [ … ]
 
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.  What is bothering me is the idea that because the sequence is infinite, it is therefore not something we can experience.
 
[orcmid] I should have read all of this more closely.  It strikes me that there is a case of inexhaustibility that fits with the notion of potential infinite without having to assume an infinite entirety as an actuality.    The case of series that are convergent at the limit might be the boundary of it.
 
Some series are easier than that.  The Peano numbers produced by succession seems an appropriate case, and the usual mathematical induction is a kind of inexhaustibility condition, might one say?
 
A SIMPLE CASE IN ARITHMETIC
 
The lengthy example below can be summarized as follows.  In 2’s complement binary arithmetic, a carry into an inexhaustible supply of 1’s is an inexhaustible supply of 0’s, and this seems particularly fine and might be a simplest-possible argument facing into a potential infinity.
 
DETAILED ILLUSTRATION
 
A simpler case is at hand in something I am doing.  Consider a *numeral* expression that is written using binary notation, in form
 
[u_0 u_1 …, u_n : b] where b is the “bumper” and it is either 0 or 1.  The bumper is an inexhaustible supply of the particular binary digit (bit).
 
It is convenient (in my usage) that these numerals are written in order from lowest-order bit to higher orders, with the bumper supplying as many more higher-order bits as might be required for an arithmetic operation.  So, with this little-endian reading,
 
[:0] is a numeral for Z number 0.
[1:0] is a number for Z number 1.
[01:0] is a number for Z number 2
[11:0] 3
[001:0] 4, 
Etc.
These expressions are interpreted as 2’s complement numerals, so
[:1] is a numeral for Z number -1
[0:1] is a numeral for Z number -2
[10:1] is a numeral for Z number -3,
Etc.
 
There are canonical forms of these numerals that can be taken as finite strings uniquely identifying numbers in Z.  Simply drop  out any values to the left of the bumper that are the same as the bumper value.  
 
In 2’s-complement arithmetic the numeral for the negative of a represented Z number is by taking the 1’s complement (flipping all the bits) and adding one.  We can treat the case adding one as introducing a carry at the left.
 
The 1’s complement of [:0] is [:1] and the carry into :1 can be seen to produce 0:1 with a carry into the bumper, ad nauseum.  The claimed canonical form is [:0], satisfying the condition of Z 0 being its own algebraic complement (additive inverse).
 
The reasoning is that we do not (and cannot) continue the carry into :1 indefinitely.  I claim we can safely conclude that such carrying produces an inexhaustible supply of 0’s as far as we ever need them and hence :0.  
 
I suppose this is a bit like Zeno’s case, in that the  gobbling carry into :1 cannot be caught 😊.   This strikes me as an argument to within potential infinity, but there need be no presumption of actual infinity.  Is there agreement about that?
 
This is easier than a rational series for which there is convergence “at the limit,” yes?  I think it is more like those expansions that can be taken as far as we like for successively “better” results in terms of nearness to something, but the simplest case here.
  
 
 
 
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