Introducing Hyperset Theory
http://edisk.fandm.edu/tony.chemero/pap ... tonomy.pdf
Toward the end of his book Life Itself, Robert Rosen
(1991; see also Rosen 2000) claimed that the defining
feature of living systems is that they are “closed to efficient
cause”. The idea of closure to efficient cause is wrapped
up with a series of other concepts (metabolism-repair
systems, computatibility, impredicativity, complexity) in a
somewhat obscure way. (See Letelier, Soto-Andrade,
Abarzúa, Cornish-Bowden and Cárdenas 2006 for a recent
attempt at clarification.) The first purpose of this paper is
to set out exactly how these concepts are related. The
upshot of this is that closure to efficient causation is a
variety of complexity, where complex systems are systems
whose models contain impredicativities, and, therefore, are
not computable. Because of this complexity, the
mathematical and computational tools of the mainstream
cognitive sciences, derived from computability theory and
designed for simple systems, are inappropriate as tools for
analyzing systems that are closed to efficient causation
(e.g., metabolism-repair systems). So, the second, more
important. purpose of this paper is to introduce non-well
founded set theory or hyperset theory (Aczel 1988,
Barwise and Etchemendy 1987, Barwise and Moss 1996,
Kercel 2003), and show that it is useful tool to make sense
of the kind of complexity that is characteristic of living
systems.
During the late 19th and early 20th century work that
codified and established set theory, it became evident that
certain paradoxes threatened the foundations of set theory.
The most famous of these paradoxes was first discussed by
Bertrand Russell, and is now known as Russell’s Paradox
(1903). Imagine the barber who shaves all and only those
who don’t shave themselves. Who shaves the barber? It
turns out that if the barber shaves himself, he does not
shave himself, and vice versa. Put in terms of sets, the
paradox is as follows. Call all the sets that do not contain
themselves as members normal sets; those that do contain
themselves are abnormal sets. Consider S the set of all
normal sets. Is S normal? If so, S is a member of the set of
all normal sets. But since S is the set of all normal sets, S
would then be abnormal. If S is abnormal, then S is a
member of itself. But if S is a member of itself (the set of
all normal sets), then S must be normal. So: S is a member
of itself just in case it is not a member of itself, or,
equivalently, S is normal just in case it is abnormal. This
means that the sentence “S is a normal set” is
simultaneously true and false. (We ask that readers
convince themselves of this before continuing.)
This was taken to be a serious problem with the theory of
sets, which was intended to be deducible from logic. In
response to this problem, Russell introduced the vicious
circle principle, which outlawed any sentence which was
circular. This was later formalized as the theory of types,
most of the details of which we do not need to be
concerned with, but whose main effect was to make it
illegal for sets to be members of themselves. According to
the theory of types, that is, there are no abnormal sets, and
there is also no set of all sets. Paradox solved. Poincaré’s
(1906) banishment of impredicative definitions from
mathematics, which may be more familiar to some readers,
serves a similar purpose. Roughly, a predicative definition
applies to members of some domain so that its application
is not altered by addition of new members to the domain;
an impredicative definition, on the other hand, picks out
different members of a domain should new individuals be
added to the domain. The definition that picks out “the
individuals in Ms. Riley’s class whose first names begins
with P” is predicative, but the definition that picks out “the
4th tallest individual in Ms. Riley’s class” is impredicative.
A new student’s arrival does not change the first letter of
Peter’s name, but it may make him 5th tallest. Essentially,
impredicative definitions pick out individuals or properties
whose falling under that definition depend on other
members of a set. Another way to put this is that
impredicative definitions pick out individuals in a way that
is context-dependent. “Abnormal set” and “the barber who
shaves all and only those who do not shave themselves” are
defined impredicatively, so outlawing impredicativities
makes defining these types of set impossible. Again,
paradox solved.
Non-well-founded sets were banished from set theory in
the interest of the logical soundness of the theory, and with
the explicit goal of reducing mathematics to set theory, and
set theory to logic. Re-introducing non-well-founded sets
into our set theory makes such reduction impossible. But
this loss (if it really is a loss) is offset by the increased
ability to model real-world phenomena. Work by Gupta
(1981), Barwise and Etchemendy (1987), Gupta and
Belnap (1993) and Barwise and Moss (1996) makes very
clear that many concepts and real-world systems are
circular or otherwise not-well-founded, hence illegal
according standard set theory. The most well-known
example is the concept ‘truth’, and the predicate ‘is true’.
Gupta (1981) argues that the semantic paradoxes involving
truth, such as the liar (“This very sentence is false”) and the
truth-teller (“This very sentence is true”) show that the
predicate “is true” can only be defined circularly, and
develops a logical technique (the revision theory) for
dealing with circular concepts. Later, Barwise and
Etchemendy (1987) use hypersets to understand the truth
predicate. Another example from Quine (1963) is “the
most average Yale student”.[/b]
