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Unformatted text preview: 1 GODEL’S LEGACY IN MATHEMATICAL
PHILOSOPHY
by
Harvey M. Friedman
Ohio State University
LC ’06 Gödel Legacy Panel
Nijmegen, Netherlands
Presented August 2, 2006
Revised August 9, 2006
NOTE: This is an edited version of my 20 minute lecture at the Gödel
legacy panel of the LC ‘06. Gödel's definitive results and his essays leave us with a
rich legacy of philosophical programs that promise to be
subject to mathematical treatment. After surveying some of
these, we focus attention on the program of circumventing
his demonstrated impossibility of a consistency proof for
mathematics by means of extramathematical concepts.
This program has seen substantial progress through our new
Concept Calculus. 1. INCOMPLETENESS THEOREMS.
Gödel’s First Incompleteness theorem asserts that any
consistent formal system obeying rather weak conditions,
such as PA = Peano Arithmetic, is incomplete.
Gödel’s Second Incompleteness theorem asserts that any
consistent formal system obeying rather weak conditions,
equipped with a “standard” formalized proof predicate with
which to formulate its own consistency, cannot prove its
own consistency.
This suggests a number of obviously important programs,
which are, to various degrees, implicit and explicit in
Gödel’s writings, and certainly well known to him.
1a. What mathematics can be formalized in a way that is not
subject to the First Incompleteness theorem, so that one
has completeness?
Perhaps the most well known examples of real depth are the
complete axiomatizations of the ordered group of integers 2
(Presburger) and the ordered field of reals (Tarski) and
Euclidean geometry (Tarski).
I think that there is room for new dramatic results where
complete axiomatizations of natural significant portions of
mathematics are given. However, many of these will likely
involve rather imaginative delineations that are not
simply the full first order theory of some fundamental
structures, as was the case with Presburger and Tarski.
1b. Can we give a simple yet fully rigorous and very
general formulation of Gödel’s Second Incompleteness
theorem?
The difficulty surrounds the notion of a “standard” proof
predicate. This seems to be relevant to one interpretation
of what Ludwig Wittgenstein seemed to be complaining about.
I recently discussed some perhaps new and clearer forms of
Gödel’s Second recently on the FOM and the FMPC, which uses
only interpretability and not consistency. See
http://www.cs.nyu.edu/pipermail/fom/2006May/010529.html
http://www.cs.nyu.edu/pipermail/fom/2006May/010532.html
http://www.cs.uky.edu/fmpc/archive.dir/0605005.html
1c. To what extent can we formalize mathematics in such a
way that we avoid Gödel’s Second entirely, and have a
system with a consistency proof in a weak fragment of PA?
In Strict Reverse Mathematics, we show how little one needs
about finite sequences of integers in order to gain logical
strength. This can probably be pushed much farther. See
‘Strict reverse mathematics', January 31, 2005, 24 pages,
draft.
'The inevitability of logical strength', May 31, 2005, 13
pages, draft. in:
http://www.math.ohiostate.edu/%7Efriedman/manuscripts.html
1d. An often cited consequence of Gödel’s Second is that
“mathematics cannot establish its own consistency”. Can
this be gotten around in an interesting way by proving the
consistency of mathematics using principles outside 3
mathematics? Perhaps even principles of ordinary
commonsense reasoning?
My recent Concept Calculus deals with a large variety of
such systems of such principles. I will discuss this
shortly. See
Concept Calculus, at
http://www.math.ohiostate.edu/%7Efriedman/manuscripts.html 2. CONSTRUCTIBLE SETS.
In his 1938 announcement in PNAS of the consistency of AxC
and GCH, Gödel wrote
“The proposition [V = L] added as a new axiom seems to give
a natural completion of the axioms of set theory, in so far
as it determines the vague notion of an arbitrary infinite
set in a definite way. In this connection it is important
that the consistency proof for [V = L] does not break down
if stronger axioms of infinity (e.g., the existence of
inaccessible numbers) are adjoined to T. Hence the
consistency of [V = L] seems to be absolute in some sense,
although it is not possible in the present state of affairs
to give a precise meaning to this phrase.”
Does V = L really have a special status? Modern set
theorists say no, citing inner model theory. L does rely on
the concept of ordinal, which is not made definite, in
sharp contrast to the definiteness of the L construction at
each stage.
Perhaps relevant is an old result of mine which shows that
any set theory for which the ordinals determine the sets
(in a rather strong sense involving arbitrary models), must
prove V = L. See
Categoricity with Respect to Ordinals, Higher Set Theory,
Springer Lecture Notes, Vol. 669, (1978), pp. 1720. 3. REALISM AND RUSSELL’S PARADOX.
Gödel distinguishes between concepts such as “truth,
concept, being, class, etc.” and the iterative concept of
set. For the former, he credits Russell for having shown
that our intuitions are contradictory. For the latter, he 4
denies that there was ever any hint of a paradox.
SUGGESTED PROBLEM: Develop a natural and powerful theory of
classes that is not just a dressed up theory of the
iterative concept of set.
Some time ago, I wrote a paper
A Cumulative Hierarchy of Predicates, Zeitschrift für
Mathematische Logik und Grundlagen der Mathematik, Bd. 21,
(1975), pp. 309314.
which made a modest start in this direction.
Gödel sharply criticizes Russell’s “vicious circle
principle” which asserts that
no totality can contain members definable
only in terms of this totality.
He says that this is correct only if the entities whose
totality is in question are “constructed by ourselves”.
Also,
“If, however, it is a question of objects that exist
independently of our constructions, there is nothing in the
least absurd in the existence of totalities containing
members, which can be described ... only by reference to
the totality”.
Another important quote:
“For how can one hope to solve mathematical problems by
mere analysis of the concepts occurring, if our analysis so
far does not even suffice to set up the axioms?”
SUGGESTED PROBLEM: Are our usual axiom systems uniquely
characterized by minimal adequacy conditions together with
simplicity? Perhaps we can “set them up” this way. It
appears that, e.g., the axioms of ZFC have a special
syntactic simplicity to them that is not matched by various
proposed additions to ZFC.
As a modest step in this direction, I wrote
Three quantifier sentences, Fundamenta Mathematicae, 177 5
(2003), 213240.
Also see
Primitive Independence Results, Journal of Mathematical
Logic, Volume 3, Number 1, May 2003, 6783. 4. CONCEPT CALCULUS.
The 42 page abstract
Concept Calculus, http://www.math.ohiostate.edu/%7Efriedman/manuscripts.html
has table of contents:
1. BETTER THAN.
1.1. Better Than, Much Better Than.
1.2. Better Than, Real.
1.3. Better Than, Real, Conceivable.
2. VARYING QUANTITIES.
2.1. Single Varying Quantity.
2.2. Two Varying Quantities, Three Separate Scales.
2.3. Varying Bit.
2.4. Persistently Varying Bit.
2.5. Naive Time.
3. BINARY RELATIONS.
3.1. Binary Relation, Single Scale.
3.2. Binary Relation, Two Separate Scales.
4. MULTIPLE AGENTS, TWO STATES.
5. POINT MASSES.
5.1. Discrete Point Masses in One Dimension.
5.2. Discrete Point Masses with End Expansion.
5.3. Discrete Point Masses with Inner Expansion.
5.4. Point Masses with Inner Expansion.
5.5. Discrete Point Masses with Inner Expansion
Revisited.
6. TOWARDS THE MEREOLOGICAL.
We will use 1.1 as a sample:
"better than" (>), and “much better than” (>>).
BASIC. ÿx > x. x >> y Æ x > y. x >> y Ÿ y > z Æ x >> z. x
> y Ÿ y >> z Æ x >> z. x >> y Æ ($z)(x >> z > y). ($z)(z
>> x,y). 6
MINIMAL. There is nothing that is better than all minimal
things.
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.
HORIZON. Let y > x be given, as well as a true statement
about x, using “better than”, and “much better than x”. The
corresponding statement about x, using “better than”, and
“much better than y” is also true.
THEOREM. Basic + Minimal + Existence + Horizon is
mutually interpretable with ZFC. This is provable in EFA. ...
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