monads and Turing machines

[dennis.hamilton at acm.org]

Thank you for your informative response, José.  I have two supplementary 
concerns.

From: José Manuel Rodríguez Caballero
Sent: Sunday, February 26, 2023 14:11
Subject: Re: monads and Turing machines

> Below, I answer questions from Dennis Hamilton. I have listed the questions 
> in order of difficulty rather than in order of appearance.

[orcmid] [ . ]

> Q2:
> > In fact, this scheme doesn't even enumerate all of the rationals in the
interval [0,1], considering incommensurate prime denominators.  For example,
1/3 and 1/5 are not captured in this sequence.  There are repairs for this,
but they don't extend the cardinality beyond the denumerable.

> The numbers 1/3 and 1/5 are defined as cluster points. Indeed, the closed 
> interval of real numbers between 0 and 1 can be defined as the set of 
> cluster points associated with this computation.

I don't understand how that addresses my concern that the rationals are 
denumerable and for the interval [0, 1] there are binary schemes that capture 
all of them exactly.  It does not bear on ideas about convergence at all, a 
matter that does attend attempts to represent irrational and transcendental 
numbers by rational approximations taken to some finite degree of proximity.

> Q3:
> > One can defend the thesis that this computation defines the set of 
> > functions
> > of the closed interval between 0 and 1 to itself, which has a cardinality
> > strictly greater than the cardinality of the set of real numbers.

> > It is not clear to me that this claim is defensible.  First, 
> > denumerability
has not been escaped, so it is unclear how a non-denumerable set of functions
have been represented.

> Consider a function f from the closed interval of real numbers between 0 and 
> 1 to itself. For any element x in the closed interval between 0 and 1, we 
> can approximate this number as much as we want by a sequence of amoebas 
> generated by this computation. Now, inside each of these amoebas 
> approximating x, there is a sequence of amoebas approximating f(x) as much 
> as we want. Thus, such a function f is realized as a limiting process. Of 
> course, in most cases, the selection of such sequences is a non-computable 
> process.

I am reminded that not even the set of all functions over N onto N is 
countable, and, if one accepts the set-theoretic notion of that set, the 
consequence is that there are non-computable functions.  So we are already 
constrained under a ceiling of denumerability (as to computation) and we are 
not going to get any higher cardinality of computable functions by 
entertaining approximations of functions over R onto R.

So to suddenly consider functions, f, on the closed interval [0,1] of real 
numbers and the cardinality of those seems to be unapproachable, if we're 
talking about Turing Amoebas.

I don't see how to make any leap to the contrary.

 - Dennis