monads and Turing machines
José Manuel Rodríguez Caballero
josephcmac at gmail.com
Sun Feb 26 03:01:56 EST 2023
Dear FOM members,
Using a Turing machine, it is easy to define finite and countable sets. To
define an uncountable set, one can imagine the implementation of a virtual
amoeba having a binary register inside. This register represents a rational
number between zero and one written in binary notation. I use this
particular microorganism to follow Hugh Everett's amoeba metaphor on the
many-worlds interpretation of quantum mechanics [1]. I will use the
mechanism that produces true randomness for most observers in a
deterministic system. This procedure was motivated by the question of how
can randomness arise in a deterministic universe [3].
Initially, there is only one amoeba and its register is empty, representing
the zero. During the n-th iteration, the Turing machine produces a copy of
all the existing amoebas, the only difference being that the registers of
the new amoebas are increased by 1 over 2 to the power n. The register can
be enlarged if more bits are necessary. The computational process is as
follows (I represent the registers between the parentheses):
initial condition:
( )
after the first iteration
( ) (1)
after the second iteration
( ) (1) (01) (11)
after the third iteration
( ) (1) (01) (11) (001) (101) (011) (111)
etc.
Inspired by Leibniz's monads [2], we can add the hypothesis that each
amoeba also contains a copy of the Turing machine running the entire
simulation and that the simulation also runs inside each of them. 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. Indeed, at any
cluster point of the simulation, one can assign a cluster point of the
simulation inside the corresponding amoeba in the limit. But there is no
need to stop here: we can consider that this computation also defines the
set of functions from the closed interval between 0 and 1 to the set of
functions from the closed interval between 0 and 1 to itself. And so on.
Some questions could be: Any suggestions for formalizing this framework?
References to similar constructions? How far can we go in the hierarchy of
infinities using this procedure? Critique of this thread of thought.
Kind regards,
Jose M.
[1] Everett, Hugh. The Everett interpretation of quantum mechanics:
Collected works 1955-1980 with commentary. Princeton University Press, 2012.
[2] Leibniz, Gottfried Wilhelm. *La monadologie*. Presses Électroniques de
France, 2013.
[3] Wolfram, Stephen. "Origins of randomness in physical systems." Physical
Review Letters 55.5 (1985): 449.
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