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Unformatted text preview: 1 CONCEPT CALCULUS
by
Harvey M. Friedman
Mathematical Methods in Philosophy
Banff Research Center
University Professor
Ohio State University
February 21, 2007
1. THE GENERAL NATURE OF CONCEPT CALCULUS.
Concept Calculus is a new mathematical/philosophical
program of wide scope. The development of Concept Calculus
began in Summer, 2006.
Concept Calculus promises to connect mathematics,
philosophy, and commonsense thinking in a radically new
way.
Advances in Concept Calculus are made through rigorous
mathematical findings, and promise to be of immediate and
growing interest to philosophers.
Developments in Concept Calculus generally consist of the
following.
a. An identification of a few related concepts from
commonsense thinking. In the various developments, the
choice of these concepts will vary greatly. In fact, all
concepts from ordinary language are prime targets.
b. Formulation of a variety of fundamental principles
involving these concepts. These various principles may have
various degrees of plausibility, and may even be
incompatible with each other. There may be no agreement
among philosophers as to just which principles to accept.
Concept Calculus is concerned only with logical structure.
c. Formulation of a variety of systems of such fundamental
principles in b. These systems generally combine several
such fundamental principles in some attractive way.
d. An analysis of the “interpretation power” of these
resulting systems. 2
e. In particular, for each of the resulting systems, a
determination of whether they interpret mathematics – as
formalized by ZFC.
A system T having interpretation power at least that of
mathematics (ZFC) has special significance. This means that
if T is without contradiction then mathematics (ZFC) is
without contradiction.
I.e., we have a consistency proof for mathematics (ZFC)
relative to that of T.
Furthermore, relative consistency proofs arising this way
are generally very finitary.
2. INTERPRETATION POWER.
The interest of Concept Calculus rests considerably on the
significance of interpretation power.
Interpretability between formal systems was first precisely
defined by Tarski.
We define this for systems S,T in first order predicate
calculus with equality. S,T may have completely different
symbols.
An interpretation of S in T consists of
i. A one place relation defined in T which is meant to
carve out the domain of objects that S is referring to,
from the point of view of T.
ii. A definition of the constants, relations, and functions
in the language of S by formulas in the language of T,
whose free variables are restricted to the domain of
objects that S is referring to (in the sense of i).
iii. It is required that every axiom of S, when translated
into the language of T by means of i,ii, becomes a theorem
of T.
In ii, we usually allow that the equality relation in S
need not be interpreted as equality – but rather as an
equivalence relation. 3
We give some examples.
S consists of the axioms for linear order, together with
“there is a least element”.
i. ÿ(x < x).
ii. (x < y Ÿ y < z) Æ x < z.
iii. x < y ⁄ y < x ⁄ x = y.
iv. ($x)("y)(x < y ⁄ x = y).
T consists of the axioms for linear order, together with
“there is a greatest element”.
i. ÿ(x < x).
ii. (x < y Ÿ y < z) Æ x < z.
iii. x < y ⁄ y < x ⁄ x = y.
iv. ($x)("y)(y < x ⁄ x = y).
Note that S,T are theories in first order predicate
calculus with equality, in the same language: just the
binary relation symbol <.
CLAIM: S is interpretable in T and T is interpretable in S.
They are mutually interpretable.
Here is the obvious interpretation of S in T. In T, take
the objects of S to be everything (according to T).
Define x < y of S to be y < x in T.
Interpretation of the axioms of S formally yields
i’. ÿ(x < x).
ii’. (y < x Ÿ z < y) Æ z < x.
iii’. y < x ⁄ x < y ⁄ x = y.
iv’. ($x)("y)(y < x ⁄ x = y).
These are obviously theorems of T.
Here is a more sophisticated example. PA = Peano Arithmetic
is the first order theory with equality, using 0,S,+,•. The
axioms: successor axioms, defining equations for +,•, and
scheme of induction for all formulas in this language.
Now consider “finite set theory”. This is a bit ambiguous:
could mean either 4
ZFC without the axiom of infinity; i.e., or ZFC\I; or
ZFC with the axiom of infinity replaced by its negation;
i.e., ZFC\I + ÿI.
THEOREM (well known). PA, ZFC\I, ZFC\I + ÿI are mutually
interpretable.
PA in ZFC\I: nonnegative integers become finite von Neumann
ordinals. Induction in PA gets translated to a consequence
of foundation and separation.
ZFC\I + ÿI in PA: Sets of ZFC\I + ÿI, are coded by the
natural numbers in PA – in an admittedly ad hoc manner.
The various axioms of ZFC\I + ÿI get translated into
theorems of PA.
3. STARTLING FACT ABOUT INTERPRETATION POWER.
We begin with the observation that for any two S,T, if T is
inconsistent (proves a sentence and its negation) then S is
interpretable in T.
A very fundamental fact about interpretation power is that
there is no greatest interpretation power – short of
inconsistency.
THEOREM 3.1. (In ordinary predicate calculus with
equality). Let S be a consistent recursively axiomatized
theory in a finite language. $ a consistent finitely
axiomatized conservative extension T of S which is not
interpretable in S.
THEOREM 3.2. Let S1,...,Sk be consistent recursively
axiomatized theories. There exists a consistent finitely
axiomatized theory T, where
i) each Si is interpretable in T;
ii) T is not interpretable in any of the Si.
These can be proved using Gödel’s second incompleteness
theorem.
COMPARABILITY. Let S,T be recursively axiomatized theories.
Then S is interpretable in T or T is interpretable in S? 5
Now there are plenty of quite interesting and natural
examples of incomparability for finitely axiomatized
theories that are rather weak. To avoid trivialities, we
give an example of incomparability where there are only
infinite models.
S is the theory of discrete linear orderings without
endpoints.
T is the theory of dense linear orderings without
endpoints.
We get plenty of incomparability arbitrarily high up:
THEOREM 3.3. Let S be a consistent recursively axiomatized
theory. There exist consistent finitely axiomatized
theories T1,T2, both in a single binary relation symbol,
such that
i) S is provable in T1,T2;
ii) T1 is not interpretable in T2;
iii) T2 is not interpretable in T1.
BUT, are there examples of incomparability between natural
theories that are metamathematically strong?
Metamathematically strong can be rephrased as “subject to
the Gödel phenomena”, and formally as “EFA is
interpretable”. Here EFA = exponential function arithmetic.
Natural theories are all recursively axiomatized, and in
fact they are axiomatized by finitely many schemes. This
includes finitely axiomatized theories.
STARTLING OBSERVATION. Any two natural theories
to interpret EFA, are known (with small numbers
exceptions) to have: S is interpretable in T or
interpretable in S. The exceptions are believed
have comparability. S,T, known
of
T is
to also Because of this observation, there has emerged a rather
large linearly ordered table of “interpretation powers”
represented by natural formal systems. Generally, several
natural formal systems may occupy the same position.
We call this growing table, the Interpretation Hierarchy.
4. RESTATEMENT OF CONCEPT CALCULUS. 6 We restate what we said at the beginning, altered to take
into account the Interpretation Hierarchy.
Concept Calculus develops with
a. An identification of a few related concepts from
commonsense thinking. In the various developments, the
choice of these concepts will vary greatly. In fact, all
concepts from ordinary language are targets.
b. Formulation of a variety of fundamental principles
involving these concepts. These various principles may have
various degrees of plausibility, and may even be
incompatible with each other. There may be no agreement
among philosophers as to just which principles to accept.
Concept Calculus is concerned only with logical structure.
c. Formulation of a variety of systems of such fundamental
principles in b. These systems generally combine several
such fundamental principles in some attractive way.
d. An identification of the position in the Interpretation
Hierarchy of these resulting systems.
e. In particular, if the position is at or higher than that
of ZFC, then we will generally have a finitary consistency
proof of ZFC relative to the consistency of the system.
5. SOME DEVELOPMENTS IN CONCEPT CALCULUS.
We begin with the notions: better than (>), and much better
than (>>). These are binary relations. This is an example
of what we call concept amplification.
One can also view > and >> mereologically, as
x > y iff y is a “proper part of x”. x >> y iff y is a “small proper part of x”.
BASIC. > is a linear ordering. x >> y Æ x > y. x >> y Ÿ y >
z Æ x >> z. x > y Ÿ y >> z Æ x >> z. ($x)(x >> y,z). If x
>> y then x >> some z minimally > y.
MINIMAL. There is nothing that is better than all minimal
things. 7 EXISTENCE. Let x be a thing better than a given range of
things. There is something that is better than the given
range of things and the things that they are better than,
and nothing else. Here we use L(>,>>) to present the range
of things.
Existence is like fusion. Here the “range of things” is
given by a first order formula in >,>>, with parameters
allowed.
AMPLIFICATION. Let y > x be given, as well as a true
statement about x, using the binary relation > and the
unary relation >> x. The corresponding statement about x,
using > and >> y, is also true.
THEOREM 5.1. Basic + Minimal + Existence + Amplification is
mutually interpretable with ZFC. This is provable in EFA.
AMPLIFICATION (binary). Let y > x be given, as well as a
true statement about x, using the binary relations > and z
>> w >> x. The corresponding statement about x, using > and
z >> w >> y, is also true.
AMPLIFIED LIMIT. There is something that is better than
something, and also much better than everything it is
better than.
Leads to much higher places in Interpretation Hierarchy
than ZFC:
THEOREM 5.2. Basic + Minimal + Existence + Amplification
(binary) interprets ZFC + “for all x Õ w, x# exists” and is
interpretable in ZFC + “there exists a measurable
cardinal”.
THEOREM 5.3. Add Amplified Limit: well above measurable
cardinals. Below ‘conentrating’ measurable cardinals, in
interpretation power.
I also considered a lot of theories based on time, and some
on time and space. The simplest one to present involves a
single varying quantity – where the time and quantity scale
are the same.
The language has >,>>,F, where >,>> are binary relations,
and F is a one place function. 8 F(x) is the value of the varying quantity at time x.
When thinking of time, >,>> is later than and much later
than. When thinking of quantity, >,>> is greater than and
much greater than.
BASIC. > is a linear ordering. x >> y Æ x > y. x >> y > z
Æ x >> z. x > y >> z Æ x >> z. ($x)(x >> y,z). If x >> y
then x >> some z minimally > y.
ARBITRARY BOUNDED RANGES. Every bounded range of values is
the range of values over some bounded interval. Here we use
L(>,>>,F) to present the bounded range of values.
AMPLIFICATION. Let y > x
statement about x, using
unary relation >> x. The
using > and >> y is also be given, as well as a true
the binary relation > and the
corresponding statement about x,
true. This also lands at ZFC in the Interpretation Hierarchy. We
can strengthen as before:
AMPLIFICATION (binary). Let y > x be given, as well as a
true statement about x, using the binary relations > and z
>> w >> x. The corresponding statement about x, using > and
z >> w >> y, is also true.
AMPLIFIED LIMIT. There is something that is greater than
something, and also much greater than everything it is
greater than.
As before, these latter two principles push the
interpretation power well into the large cardinal
hierarchy.
There are versions with continuously ordered >, using
equidistance of intervals.
PRINCIPLE OF PLENITUDE
From Wikipedia, Plenitude Principle.
The principle of plenitude asserts that everything that can
happen will happen.
The historian of ideas Arthur Lovejoy was the first to 9
discuss this philosophically important Principle explicitly, tracing it back to Aristotle, who said that no
possibilities which remain eternally possible will go
unrealized, then forward to Kant, via the following
sequence of adherents:
Augustine of Hippo brought the Principle from NeoPlatonic
thought into early Christian Theology.
St Anselm 's ontological arguments for God's existence used
the Principle's implication that nature will become as
complete as it possibly can be, to argue that existence is
a 'perfection' in the sense of a completeness or fullness.
Thomas Aquinas's belief in God's plenitude conflicted with
his belief that God had the power not to create everything
that could be created. He chose to constrain and ultimately
reject the Principle.
Giordano Bruno's insistence on an infinity of worlds was
not based on the theories of Copernicus, or on observation,
but on the Principle applied to God. His death may then be
attributed to his conviction of its truth.
Leibniz believed that the best of all possible worlds would
actualize every genuine possibility, and argued in
Théodicée that this best of all possible worlds will
contain all possibilities, with our finite experience of
eternity giving no reason to dispute nature's perfection.
Kant believed in the Principle but not in its empirical
verification, even in principle.
The Infinite monkey theorem and Kolmogorov's zeroone law
of contemporary mathematics echo the Principle. It can also
be seen as receiving belated support from certain radical
directions in contemporary physics, specifically the manyworlds interpretation of quantum mechanics and the
cornucopian speculations of Frank Tipler on the ultimate
fate of the universe.
See Concept Calculus, Preprints, #53,
http://www.math.ohiostate.edu/%7Efriedman/manuscripts.html ...
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