Excerpt from
Googol room essays: one © 2003 by Rolf Mifflin
The grail machine: Two
ZF+: A set theory for describing the mind
by Rolf Mifflin
Abstract: The complete description of the mind, as well as of the universe that allows the mind, requires a formal language beyond any in common use today ('03). Here I present a modified version of Zermelo-Fraenkel set formalism sufficient to make statements completely equivalent to our thoughts and, so, symbolically present the systems of artificial minds I argue are constructible to modern science. This modified set theory is called ZF+.
Table of contents:
Fig. 1: Illustration of the logical Future and Past
5: Axioms of ZF+ and redefinition of the
reals
B: Axioms of ZF and ZFC abandoned by ZF+
Having introduced an extension to formal logic called the
temporal propositions in The grail
machine: One, I will here extend these propositions into the structure of
modern set theory. This discussion will
be useful both to the general reader and the student of logic. The subject matter is technical but the
presentation will be in plain English.
Those few technical moments or details can be easily glossed as
unimportant to the non-mathematical reader, and the treatment of ordinary ZF
will be entirely peripheral to essential arguments of extended logic.
The discrete structure of thoughts lends itself most
readily to set theory. For discussions
of the physical universe, group theory is the more natural language of
discourse. Group theory’s adaptation to
temporal propositions will be a more subtle matter than that presented here.
Set theory, as it was assembled in the 20th century, is
incomplete for discussions of the mind.
To make statements emulating thought, set theory must be expanded in
some fashion. The route I choose for
that expansion is the adoption of the temporal propositions, although there may
be others.
First, I will present this specific adoption as a
modification to the basic symbols of logic.
Then I will expand these changes into Zermelo-Fraenkel formalism,
adapting the axioms of ZF to better reflect the utility of these
modifications. This will produce a
theory called ZF+. This new system
involves very little change to ZF, only a few minor additions and a few
redefinitions to make use of those additions; the name reflects this
subordinancy.
For the time being, I simply use the name of the previous
theory modified with a + sign. Whether
other thinkers should be included in the name of a theory is an important
consideration. My own thoughts were
developed in an attempt to unify the writings of Ludwig Wittgenstein and John
Stuart Mill, great thinkers, on the one hand, of deduction and, on the other,
of induction. As said, this theory is
still mainly the work of Zermelo and Fraenkel.
It is sometimes said that we should also remember Skolem, who along with
Fraenkel helped to adapt Zermelo’s work, so a more properly deferential name
for this theory might be ZFS+, reminding us of the logicians who went before.
The basis of symbolic logic are well stated by the three
Aristotelian Laws, the three pre-assumptions on which mathematical logic is
based. One makes a statement, an
assertion, and that statement occurs in one of two truth-states. It is either True or False. The laws fix these states:
(i) Law of Identity - A True proposition is True.
(ii) Law of Contradiction - A proposition can not be both
True and False.
(iii) Law of the Excluded Middle - A proposition must be
True or False.
This divides the universe of possible discourse into statements that assert the truth of a proposition, j, and statements that negate the truth of a proposition, ~j. This forms the basis of atomic logical theory.
In order to expand logic into the temporal realm, these laws are modified by adding a third truth-state that mixes the other two. These three truth-states are organized by two modes of interpretation:
(iv) A proposition
is always interpreted in one, and only one, of two modes: the Unresolved mode
or the Resolved mode.
(v) In the
Unresolved, or Future, mode, all
propositions exists in one, and only one, of three allowed truth-states: True,
False, or Unresolved.
a. In the Unresolved truth-state, a proposition is seen as existing in two parallel logic structures: one structure where it is True and one structure where it is False. An Unresolved proposition simultaneously violates both the Law of Contradiction and the Law of the Excluded Middle.
b.
True and False are contraries.
c.
Unresolved is contrary to itself.
(vi) In the
Resolved, or Past, mode, all
propositions exists in one, and only one, of two allowed truth-states: True or
False. All three Aristotelian laws are
obeyed by all propositions in the Resolved mode.
a. Propositions
True in the Unresolved mode are True in the Resolved.
b. Propositions False in the Unresolved mode are False in the Resolved.
c.
Propositions that are Unresolved become either True or False in
the Resolved mode. The proposition is
in one, and only one, of these two states.
One can not choose which truth-state the proposition adopts. The two truth-states are not equally likely,
they are both possible. Assertions in
logic are about existence, not probability.
Propositions are not said to adopt the same truth-state every time that
same Proposition is Unresolved. As the
actual truth-state of a Resolution event can not be specifically identified, to
say that it is different under different Resolutions is meaningless.
d. True and False
are still contrary, as in ordinary logic.
On the basis of
this structure, a complete symbolics of the mind can be built.
Four kinds of assertion are now possible:
((vii) j Asserts that j is True.
(viii) ~j Asserts that j is False. (~j negates j.)
(ix) Ŧj Asserts that j is Unresolved. (Ŧj Unasserts j.)
(x) ŧj Asserts that j is Resolved. (ŧj Resolves j.)
From Unassertion, ZF+ generalizes to other Unresolved operators. For example, the operator ŦÎ:
( (x) i) u ŦÎ v Asserts as an Unresolved truth-state does,
both that uÎv and that uÏv in a two-part parallel structure.
Likewise, other operators with a preceding Ŧ divide the truth-value into two structures, one a Truth assertion and the other a Truth negation. Resolution transforms these operators into either simply themselves or their negation; choosing one outcome over the other is not allowed, both are possible and both are singular after Resolution. Resolution puts the operator into one state; which particular state is unpredictable and undecidable. These Ŧ and their analogues ŧ may at times carry indexes to link different instances of Unresolution together. For example:
(xii) aŦÎv Ù bŦÎv Asserts four different parallel logic structures.
(xiii) aŦ1Îv Ù bŦ1Îv Asserts only two parallel structures.
(xiv) ŧ(aŦÎv Ù bŦÎv) Resolves both Ŧs,
asserting one structure.
(xv) ŧ1(aŦ1Îv
Ù
bŦÎv) Resolves the Ŧ1,
but not the Ŧ, asserting two.
5:
Axioms of ZF+ and the redefinition of the reals
ZF+
incorporates eight of the Axioms of ZF.
I will reiterate these eight in Appendix A, so as not to clutter this
discussion. The theory abandons one
axiom on formal grounds, the Power Set Axiom (PSA). It also abandons the Axiom of Choice (AOC) as a practical matter. The existence of Unresolved truth-states,
the basic logic operators, the extended logic operators implied by Unresolved
truth-states, and the eight axioms of Appendix A are the components of ZF+.
PSA is abandoned in order to mirror the physical world
through which ZF+-based systems will be constructed. There is no expressible content that can be accessed by power
sets that can not be presented through ordinary discrete symbols. The PSA itself is a demonstration that
finite symbols can be used to represent grades of non-finite objects. Versions of ZF+ will discuss power sets in
the exactly the same way that the mind discusses them, by defining for these
appropriate discrete symbols that reflect their relationships, mapping the
repetitive cycling of the power sets onto an indexed reference to simple ZF+
sets.
But this raises the question how, without the Power Set Axiom, does the theory gain access to the real numbers?
ZF+ still accepts the Axiom of Infinity:
(xvi) $ω (ω¹ø Ù "u (uÎω
®
u È
{u}Î
ω)) ))
From this follows the ZF+ definition of the real numbers:
(xvii) $r "u
(uÎω
®
u ŦuÎ r)
The set of all real numbers is now a purely Unresolved set and the individual real numbers are instances of that set's Resolution. Individual real numbers cannot be chosen out of this definition, commensurate with the Resolution of any Unresolved truth-state. Real numbers can still be constructed by ordinary means. The definition of p still stands, for instance. But there are still a number of inaccessible real numbers equal to the cardinality of the reals. Sets larger than the real numbers can not be constructed in ZF+ without the adoption of extended definitions.
To choose specific elements from a family of countable sets
is trivial, so the AOC is not necessary for Resolved sets. But ZF+ does allow the reals as an
Unresolved set, so it might allow a method for choosing an elements from that
set. When choosing an element from an
Unresolved set, that chosen element comes with the same inherent Unresolution
as that possessed by the original set.
Even with the Axiom of Choice it is never possible to choose the
outcomes of the Resolution of Unresolved truth-states or Unresolved logic
operators. It is possible to
assert that any specific individual real number is a possible outcome of the
Resolution of the reals and arguing from there. The AOC is, so, a reasonable addition. To model physical devices, though, we will begin with a
completely constructible theory and move to add axioms for proving theorems
when the theory needs these kinds of capabilities. Again, the fact that AOC can
be expressed in discrete symbols makes it positable by ZF+.
The purpose behind ZF+ is to create a language that can
echo any statement of which the human mind is capable. ZF and ZFC are notably incomplete, but ZF+
avoids incompleteness by incorporating structures of exactly that unknowability
observed in nature.
It is perhaps reasonable to say that ZF+ is not
mathematics. It certainly violates the
philosophical principles from which 20th century thinkers built modern
mathematics. One of my contentions is
that mathematics and metaphysics are the same discipline. The first scrubs itself of all physical
connections, while the second focuses entirely on these connections, and this
makes their difference. Whereas
metaphysics, the study of those inescapable realities in which the physical
world is embedded, will speak of matter or space or time, mathematics will
speak of numbers and infinities and ordering.
If any of metaphysics lacks an analogue in mathematics, we can say
therefore say that mathematics is lacking.
State-reduction events in Quantum Mechanics have no analogue in
mathematics, but they have a clear image in ZF+. This theory does what no other formal system can. It incorporates time into its structure
without dismissing that metaphysical category and its three components, the
past, present, and future, as merely subjective phenomena. It is one of the first attempts to grapple
formally with time and the way it creates the mind.
Call this set-theory-analogue-language an instance of metaphysics,
mathematics, or physical-mathematics, its power always lies in its practical
application. It is my purpose in the
following series of essays to present the construction of the mind in symbols
and machinery and, as best I can, biology.
ZF+ will appear only in brief sidebars until various philosophical
concepts have been identified with their proper physical phenomena. Then the theory will become the way through
which to organize these concepts into larger systems.
From this dissection it will become apparent that ZF+
expands formal thought to the limits of human thought, that it can incorporate
everything we know or can come to know as human beings.
The first concept for dissection into terms usable by ZF+
will be the ideal of free will. The fact
that the outcome of an Unresolved truth-state can not be chosen would seem to
prohibit its use for building a system with what we commonly call free will,
but this is not the impediment that it might seem. Unresolved truth-states provide the Freedom for the such a
system, while the extended logical structure around these truth-states provides
the Will. A careful and logical
division of the common concept will provide the proper logical statement. As with many discoveries in logic, the
proper solution to what free will is depends on the proper realization of how
it is constructed from simpler components.
Appendix A: The eight axioms shared by ZF and ZF+.
(i) Extension Axiom
"x, y (x=y « "z (zÎx « zÎy)).
(ii) Empty Set Axiom
$ø "x (xÏø).
(iii) Foundation Axiom
"x (x¹ø ® $yÎx "zÎx (zÏy)).
(iv) Pairing Axiom
"x, y $z "u (uÎz « u=x Ú u=y).
(v) Union Axiom
"x $y "z (zÎy « $u (uÎx Ù zÎu)).
(vi) Separation Axiom Schema
"z $y "x (xÎy « xÎz Ù j(x)).
(vii) Replacement Axiom Schema
"a ("xÎa $!yj(x, y) ® $z "xÎa $yÎzj(x, y)).
(viii) Axiom of Infinity
$ω (ω¹ø Ù "u (uÎω ® u È {u}Î ω)).
Appendix B: The two axioms of ZF abandoned by ZF+.
(i) Power Set Axiom
"x $y "z (zÎy « z Í x).
(ii) Axiom of Choice
xn nonempty pair-wise disjoint sets
$z "n |z Ç xn|=1