Hierarchical Introspective Logics

This research was initially stimulated by the announcement in

June 1993 of a proof of the FLT conjecture ("Fermat's Last Theorem").

Once again a mathematical question that might conceivably have been

unanswerable was being answered by the efforts of humans to find an

acceptable proof of a standing conjecture.

Earlier the famous 4 Color Conjecture had yielded to an attack

aided by computers.

So the question that arose in my mind was that of whether or not

there actually exists any mathematically interesting true mathematical

assertion (such as, for example, perhaps the Riemann Hypothesis) which

cannot be proved and which must ultimately be accepted as a new axiom

if mathematics is to proceed in terms of assertions and proofs (Satz;

Beweis).

I was of course aware of Goedel's establishment of the existence

of assertions concerning the natural integers that are not provable

modulo the proof rules of specific given formal systems and also of

the general results on undecidability which also imply the existence

of such assertions that are true but not provable in the context of

a given formal system.

The specific sort of formula that is produced by the construction

of Goedel, or by Rosser's variation on Goedel's construction, is true

but not provable within the formal system on the basis of which it was

constructed. However an acceptable mathematical proof, in verbal form,

was given by Goedel of the truth of his constructed formula provided

that it could be assumed that the formal system at issue was ideally

consistent.

Thus the Goedel formula or assertion is not really of itself

mysterious; it is provably true, on a common-sense basis, and it shows

therefore that the formal system studied must have been inadequate to

achieve the ideal goal of completeness. (It can be remarked that

logical systems of a more limited application, specifically systems

not adequate for the proof of general propositions of "number theory",

for example the "calculus of sentences", can be complete. For systems

with a narrower scope of applicability completeness does not introduce

paradoxes.)

If a Goedel or Goedel-Rosser assertion is added to one of these

incomplete systems as an axiom then one obtains a new logical system

with similar characteristics to the original. It is thus again

incomplete and again a Goedel assertion can be found within it, a

formula that is true but that is not provable by the formal machinery

of the system.

But better than adding the Goedel or the Goedel-Rosser assertion

to the initial system as an axiom one can instead add to it an axiom

of consistency. That is one can add an axiom stating that the initial

system was formally consistent. This does not also say that the new

system including the added axiom is itself consistent.

The axiom asserting formal consistency could be expressed with

the use of a scheme of Goedel numbering for the formulae expressible

in the initial system and also for proofs possible in that system.

Then it simply asserts that no such proof can result in a formal proof

of falsehood (which is represented by the second truth symbol of the

pair of T and F).

It is known that adding such an axiom of formal consistency to the

initial system makes it then possible, using the added axiom, to prove

either the Goedel assertion or the Rosser assertion for that system

(the initial system).

Extended Systems

This sort of supplementary axiom, which serves naturally to deal

with the incompleteness of a logical system, this supplementation

produces a natural extension of the original system. This type of

extension process was studied in the mid- and later 30's by Turing

in a project that became his Ph.D. thesis at Princeton.

One thing that is notable is that when an axiom is added that

asserts formal consistency of an initial system that this added axiom

is not itself included in relation to the consistency issue; so the

extended system formed by the addition of that axiom is not itself

asserted to be consistent by the added axiom. And thus, for the

newly formed extended system to be itself axiomatically stated to

be consistent requires the introduction of a second added axiom in

a parallel proceeding.

Turing considered this process in his paper (the thesis) "Systems

of logic based on ordinals" published in 1939 in Proc. Lond. Math.

Soc. But Turing did not construct his systems, his towers of

extension, on a basis of "ideal ordinals", corresponding to the

mathematician's traditional way of thinking of mathematical objects,

but rather he worked in terms of arrays of ordinals which were indexed

somehow by ordinary integers and described by a table of ordering by

which the order relation of any two of the indexed entities was

specified. And this order table was required to be specified by

recursive functions.

When I began to think about the same general theme, that of

axiomatically extending an initially specified logical system so as

to "correct" or fill in for the incompleteness of the type discovered

by Goedel, it occurred to me that Turing's concept of how ordinals

could be used may have been intrinsically too limiting and that

it would be desirable to explore what might be done with a less

restrictive concept of ordinal numbers than that of the recursively

presented sets of them used by Turing.

There are many logical "paradoxes" that can arise if we speak

loosely about ordinals and their definability. On the other hand I do

think that Turing's concept of usable ordinals was too restrictive to

allow the related extension scheme to proceed to really interesting

results relating to incompleteness.

Another Form of Incompleteness

The discovery of incompleteness by Goedel was supplemented in

the 30's by the work of Turing, Church, and others who showed that

there was "undecidability" with regard to the totality of assertions

possible within a formal system of the same general category as those

for which Goedel's discovery applied. This showed that it was not

possible, systematically, to decide which assertions were true and

which false. And consequently to that there could not exist proofs

for all of the true assertions since that would make possible the

systematic decision process. Thus a true but unprovable assertion

MUST EXIST although this route, via "undecidability", did not produce

any specific example.

But much more recently a paper was published by Paris and

Harrington showing that a popular form of Peano arithmetic called PA-

(upper suffix -) was incomplete because of the lack of a sufficiently

strong "axiom of infinity". This result was very interesting since it

showed that incompleteness could have this sort of a connection with

"knowledge of infinity" and with the possibilities for infinite

arguments such as proofs by induction.

Of course there are many long known paradoxes, or potential

logical traps, that arise from talking about infinity. One of these

can be illustrated by imagining two mathematical logicians named

Adams and Bentley. So Bentley says "Let beta be the limit ordinal

(number) which is the least ordinal that is larger than any ordinal

alpha definable in the formal language of the system of Adams. This

illustrates the paradoxes that inherently arise because we seek

to "talk about" infinite concepts but we have only finite resources

to use for the actual purpose of communication in words. Any book

written by humans has a text that contains only a finite amount

of information, a certain number of "bytes" as it were. And this

illustrative paradox relates to the classical "Burali-Forti paradox"

of logic and set theory.

Dangers of Paradoxes, Personal Caution

I am speaking about a research project that is not fully complete

since I have not yet written up and submitted for publication any

paper or papers describing the work. Also the details of what axioms

to use and how to select the basic set theory underlying the

hierarchical extension to be constructed are not fully crystallized.

I have also a great fear of possible error in studying topics in

this area. It is not rare, historically, for systems to be proposed

that are either inconsistent or that have unexpected weaknesses. So

I feel that I must be cautious and proceed without rushing to a goal.

And this psychology of fear has also inhibited me from consulting

other persons expert in logic before I could feel that I had gotten

my own ideas into good shape.

The Overview Concept

It is notable that Goedel's construction of an unprovable, yet

true, proposition in a generic sort of formal system is accompanied

also by a proof, under reasonable hypotheses, of the truth of that

"unprovable" proposition. How is this achieved? Well, of course, the

proof does NOT occur WITHIN the formal system for which the Goedel

proposition was constructed, rather it is given verbally in the normal

style of argument for published mathematical papers.

But it can be observed that if another logical system were

constructed with the specific goal of studying the first system

and the proofs possible within it and if this "overview" system

also included axioms to the effect that the machinery of procedure

of the first system is truth-preserving and that the axioms of

the first system are true then it would become provable, in this

overview system, that the first system is consistent and also

"Omega-consistent". And from these proofs would follow, in the

overview system, the proof of the truth of a Goedel assertion for

the first system.

These observations relate to what Turing studied in his paper

"Systems of logic based on ordinals". In hindsight one could say that

Turing's concept of extension was (italicize) THE UNIQUE AND NATURAL

CONCEPT so far as the range of the finite ordinal numbers is

concerned.

This results in an ascending ladder of systems in which each successor

is more complete than any of its predecessors but this ladder ascends

only through the finite ordinal numbers, 0th, 1st, 2nd, 3rd, 4th,

5th, .... nth, ... and does not reach any transfinite levels.

Of course Turing did actually concern himself with transfinite

ordinal levels but he did not have a way of uniquely describing them.

(This was the "invariance" problem in the terminology of his paper.)

Although there seems to be essentially no confusion about the

naming or proper description of finite ordinal levels, beyond that

there are serious difficulties that are classically known. Thus

mathematically there should be a non-enumerable totality of ordinal

numbers each of which has only an enumerable set of predecessor

ordinals. Then how many can we expect to properly name or describe

in the words of any precise mathematical language? And what can be

said about the description: "the least ordinal not thus properly

describable"? Relating to this is the mathematical property of the

ordinals that any subset of them has a least member. This shows

by illustration how difficult it is to form an ideal concept of

definable ordinals or of canonical names for ordinals.

Turing's hierarchy of levels in any one of his systems was based

on association of the levels with ordinals which were themselves

associated with natural integers through a recursive description of

a subset of all ordinals. Unfortunately this specification did not

have uniqueness, or "invariance" as he called it, a property that

he knew was most desirable.

In my own thinking, after a long period of study and after

encountering the problem that although one could talk about "ideal"

(or mathematical) ordinals one would need names for them in order to

be able to refer to them in formulae of a formal system. Ultimately

I came to the idea that instead of associating the levels of a system

with ordinal numbers they could instead be associated with definitions

of ordinals. This is not the same as indexing the levels by ordinals.

Two quite different definitions might easily define the same

mathematical ordinal yet also it could be very difficult to determine

the precise comparative relationship of two different definitions,

each of them stating the definition of a specific mathematical

ordinal.

And it is also easily observed that any specified language would

enable the formation of only an enumerable array of finite formulae

to serve as definitions for ordinals while mathematically the number

or cardinality of the ordinals is unlimited. (And as was mentioned

above, even the set of ordinals where each has only enumerably many

predecessors, this set should itself be of higher cardinality and

non-enumerable.)

An Hierarchy of Levels

So we evolved the idea that the formal system serving to extend a

given basic system (or "ground level") could be structured by making

use of special functions enabling references to levels. In effect,

if we used references indicating a higher level then it would become

permissible to "overview" the formulae and procedures of a lower

level.

And the basic idea, at least at the beginning of the construction

of an hierarchy of extension levels, is quite like Turing's concept

in that the first extension level, the first level above the ground

level, incorporates the possibility to prove as a theorem the Goedel

assertion for the logical system of the "ground level".

The first higher level, corresponding in our approach to any

acceptable definition of the first ordinal number, would be such that

the Goedel assertion from the ground level would now become provable

if that Goedel assertion had been simply added as an axiom usable on

the higher level. Or instead it would be effective to introduce as an

axiom an assertion of formal consistency of the ground level logic.

This type of assertion can be seen to imply the assertion of Goedel's

type. Formal consistency is the assertion to the effect that "on the

ground level nothing false can be proved."

The verbal argument for the truth of Goedel's formula depends

in effect on an overview of the specific formal system in which

it is stated and on the basis of the axioms, etc. of which it was

constructed. Also this argument depends on the assumption that that

formal system is actually perfectly consistent. Our idea of extension

and of "hierarchical introspective logics" is that there is to be

considered an hierarchy of levels of logic such that higher levels

have an "overview" of the proceedings and results obtainable on

lower levels. Thus the totality is "introspective" because it is looking

inward to study itself but this self-study is not unrestricted. And

effectively a higher level is supposed to be able to see the truth of

a Goedel assertion deriving from a lower level by a process analogous

to the verbal argument originally known for the truth of the Goedel

assertion in a formal system although there is no proof of that

assertion in that original formal system.

A logical system cannot effectively state its own consistency;

this relates to the incompleteness originally found by Goedel. But

one logical system CAN easily state the formal consistency of another

system. We use this idea in creating our hierarchy of levels so that

on a higher level it is possible to, in effect, assume the validity

and the consistency of proceedings possible on lower levels.

Axioms and Special Functions

(This portion of the talk should make use of transparencies

presenting the axioms or axiom schemata and the special functions

introduced.)

Our design for the extension hierarchy has the feature that

certain special logical functions are introduced as components of

formulae on all levels of the hierarchy except the ground level,

the ground level being simply the formal logic that we are extending.

The initial or ground level system should be adequate to be of the

sort studied by Goedel and to be able to deal with "number theory",

"functional calculus", and "set theory". And we prefer the viewpoint

of Zermelo, that sets can be based on ur-elements which are not

themselves sets. Also we like to think of ordinal numbers as being

basically a special type of mathematical objects such that they

satisfy appropriate axiomatic properties. Thus the ordinals, as

mathematical entities, are analogous to the quaternions. Of course

this doesn't really matter, it's a question of taste.

We assume, for convenience, that a specific "canonical"

"Goedel numbering" has been chosen, both for formulae (wff's) and

for "strings" of formulae that are presented as proofs. This is with

regard to the extended system, not just the "ground level". Then

each wff or string of wff's has its index which is simply one of the

natural integers. And the special functions that we introduce are

to be understood to use these index integers as arguments. We favor

the convention that the formula or string that is indexed may be

substituted for the indexing integer in an instance of any of these

special functions.

(Give oral verbal description with display of transparencies for

axioms and special functions. Explain that ORDDEF( "delta" ) is to

be affirmed or true ONLY when "delta" is a definition in a standard

form using only notation/symbols appropriate for the ground level.

And explain also that such a definition, to be approved, must be

recognizable as PROVABLY the def. of an unique ordinal by the logic

of the GROUND LEVEL. And remark that the function ORDDEF(~~) is

NOT recursive but is definable from a set of recursive functions relating

to provability and form.)

Regarding axioms, we are not sure that we have the final set of

main axioms and also we see some overlapping or redundancy of

axioms dealing with the KT(~~,~~) or "known true" function. It is also

a technical need that certain axioms should be present that simply

assert that the special functions are what they are supposed to be.

Inductive and General Incompleteness

The incompleteness found and exhibited in the paper of Paris and

Harrington has a relation to proofs by induction and the possibility

of achieving desired results by that means given favorable "axioms

of infinity". Parallelwise, for our system of extension, which builds

upon a given initial system or ground level, the extension process

naturally ends as the range of definable ordinals becomes exhausted.

So an extension system of our type is incomplete, which is fitting

in relation to general undecidability results, but the extension

process can be easily revived and renewed provided that new "axiomatic

ordinals" are introduced. That is, it is always possible to add an

axiom or axioms specifically directed to the theme of the existence

of ordinal numbers so that it becomes possible to define a "new"

ordinal level, with the validity of the "orddef" associated with this

level dependent on the new axiom or axioms, and such that the new

definable ordinal is larger than all of those that were previously

usable. And then the extension process is renewed and in the

re-extended system the proof of formal consistency of the system

as it was before becomes possible because the former range of the

system is all subject to overview from the perspective of levels

depending on the new axiom or axioms.

The interesting possibility is that it now becomes plausible or

conceivable that the incompleteness phenomenon will not cause us to

need to add anything like the Riemann Hypothesis or the Goldbach

Conjecture as an axiom (because of the impossibility of otherwise

establishing their truth) but rather that appropriate set theoretic

axioms relevant to the existence of ordinal numbers may be added

as needed.

Cultural Evolution

If it had been possible to achieve logical completeness with a

system like Principia Mathematica then all of mathematical research

to the extent of its not depending on tasteful definitions and

inspired inventions of topics, and thus perhaps all "number theory",

in the sense of the study of questions of classical elementary form,

could be worked out straightforwardly by a robot operating in an

isolation chamber simply on the basis of the rules.

But the history of human progress in science and mathematics

reveals that observation of the phenomena of Nature has always

played a large ro^le. Mathematics itself can be viewed as having

evolved from the "language of precise quantitative communication"

and many precise logical concepts have become included with the

quantitative concepts. The Sumerian scribes wrote down records for

the quantities of grain received or sent from their central granaries.

So mathematical logic itself looks like a language that is

naturally capable of evolution like also mathematics as a language

and as an encyclopedia. But we feel that a "translatability" property

should hold true here. Thus, for example, a relatively modern proof

in geometry by Pascal should be translatable into a form, written in

Greek, that Euclid would have found acceptable. Or the proof of the

FLT by Wiles should be translatable into a version in French or Latin

such that, with its introductory material developing the theory of

elliptic curves and modular functions, it could have been studied and

accepted by Fermat, if he could devote enough time to the effort.

In this context of the theme of cultural evolution, as applied

specifically to mathematical logic, it seems possible that a system of

logic of the future could be translated into a form corresponding to a

system of the present time with the addition of a few axioms that give

what is needed to give the potentialities of the future system. Our

concept of an "hierarchical introspective logic" suggests that for

"number theory" at least that there might be no need to adopt any new

axioms other than essentially axioms of infinity that make it possible

to assert the existence of ordinals that otherwise could not be

thought of as definitely existing.

Already we have in set theory a variety of popular potential

axioms. It is not yet generally assumed, but the "Axiom of Choice"

is an extremely popular option in set theory. And if the adoption

of conceivable or popular axioms went further then the GCH or

"Generalized Continuum Hypothesis" could be adopted.

It seems plausible or at least conceivable that knowledge

actually gained from the study of Nature, plus cultural evolution,

would in time lead to decisions, positive or negative, about the

adoption of axioms relating to set theory that seem to us now as

quite optional in merits.