monads and Turing machines
dennis.hamilton at acm.org
dennis.hamilton at acm.org
Mon Feb 27 20:45:30 EST 2023
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
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