Comments (by well-wishers)
02
& Notes (by B. S. Anand)
(Random thoughts and extracts from
on-going correspondence on issues of current interest in mathematics,
mathematical logic, philosophy and related subjects)
22nd
April 2001
Comment :
[Jailton da Costa Ferreira<jailton@cnen.gov.br>] My objection to
Cantor's argument is not presented in "Notes - 13 April 2001".
Note : Cantor’s work
needs critical evaluation.
It is very
likely that I have not quite followed your reasoning as intended due to the
obvious language barrier.
For instance,
the remarks in your e-mail reproduced below are not at all clear to me; perhaps
you can amplify on these whenever you have the time. I have not understood what
you mean by 'completion', 'conform' and 'argumentation'.
Meanwhile, the
impression I have of the direction of your arguments is that you are unable to
duplicate Cantor's Platonist reasoning constructively. This, of course, is
impossible, and forms the basis for Intuitionist objections to creation by
definition.
However, I do
agree with you that both Cantor's and Gödel's works need more critical
evaluation, clarification, and possibly a complete re-interpretation. One topic
that may interest you in this respect is the 'anomalous' construction of
the Cantor set in real analysis, which today is presented more as a 'fractal'.
21st
April 2001
Comment : [Stephen R. Fennell <srf22@cam.ac.uk>] Your query on Gödel's Q-function in line 8.1 of page 188 (if I remember correctly) of On formally undecidable propositions of Principia Mathematica and related systems I is most interesting: it is one of the functions my revised version refers to where Gödel has used the same kind of function-apparatus that I use. (My contention, you may remember, is that if you have enough 'kit' to do Gödel's proof, you also have enough to do Fennell's inconsistency proof.) My revised version pretty much spells out the whole 'setup' of that function explicitly, so if there is genuinely a problem there, it will certainly be of interest to me. (If my own remarks are any use to your analysis, on the other hand, then all to the good.) I have enclosed as an attachment the current revised version for you; I'm afraid it is still in German (which is the language I normally write in) as I haven't yet had time to translate the changes.
Note : The legitimacy for
Gödel's Theorem VI
lies in his Theorem V.
As I cannot
follow your German adequately, and am largely (very largely) filling in
the blanks, my understanding of what you are specifically stating is likely to
be considerably off the mark. However, I do have a sense of your overall
direction and intent, in which you seem to be right on track. Specifically,
your premise that Gödel's reasoning itself should yield a formal 'Liar' type of
inconsistency is, I believe, the correct one.
The key to his
reasoning though, I have found, does not lie in the arguments he uses in his Theorem
VI where he constructs his specific undecidable sentence. Almost all the
writings on the subject that I have come across - outside of text-books - focus
entirely on this theorem.
This theorem,
however, follows almost trivially from his thesis in his Theorem V that
every primitive recursive function and relation can be formally represented,
and the latter Gödel-numbered by a natural number that is independent of the
variables contained in the formal representation of the function or relation.
I have found
that constructing inconsistencies of the kind you are considering - a path that
I believe is the correct one - will not insulate you adequately from serious
criticism so long as the reasoning in Theorem V sustains.
This, of course,
is today obvious to me in hindsight.
Since Gödel's
formal system is clearly a constructive and intuitionistically unobjectionable
one due to the very obvious 'finitist' nature of its construction, we cannot,
almost by definition, have an inconsistency in it without taking the stand that
all mechanistic reasoning is theoretically inconsistent. No serious
mathematician or scientist would consider this possibility even remotely,
although philosophers and meta-physicists might consider it worth a discussion.
Absurd as it
sounds today (it didn't seem all that absurd then!), I have, quite
seriously, proposed that Gödel's formal system has a 3-valued logic, with what
I termed 'contrary' sentences, in some of my earlier notes, which I
intend putting up on my web-site later this year.
Coming back to
your revised paper, can you re-phrase the reasoning so that the main Theorem
is obviously a logical expression about sentences or sentential forms? As it
stands, both 'r' in your earlier version, and 'R' in the current one, are
defined as Gödel-numbers. Hence both 'r & ~r' and 'R & ~R' will not be accepted by a critic
as valid or meaningful expressions in any reasoning.
The reason is
obvious if you consider the interpretation of the expressions in any model.
They have no interpretation that would make sense to a critical evaluator
without additional qualification. The situation will change, though, if you
replace 'r' and 'R' with the actual formulas that they represent.
This, of course, is precisely what Gödel has done with his 'Bew', 'B', 'Sb' and 'z' functions.
Like you, my
thesis is that Gödel's reasoning does not fully explore - at least it does not
convey the sense of having explored - the consequences of its applications to
formula forms other than the ones he does consider. Like you, I believe that
this area of investigation will yield significant new directions and results
for truly assessing the value of Gödel's revolutionary concepts.
20th
April 2001
Comment :
[Stephen R. Fennell <srf22@cam.ac.uk>] I have made some
minor (but probably necessary) changes to the Antinomies paper: the wf-hood of
the first line has been queried, so there is now a little prelim-proof of its
expressibility in P attached to the start of the proof proper, and a very
slight simplification of the form of Line 1. I haven't reput the changes on the
e-print arXive yet, but should do so in the next day or two. The main body of
the proof and its contentions remain the same however. If any of your
correspondents is interested in such details they would do best to wait a
couple of days for the reput. My correspondence with the journal is still
ongoing on the subject.
Note : What is the
Gödel-number of Gödels undecidable sentence?
I am attaching a
short note on the problem I too faced with the first numbered line of your
argument, reproduced below. (Though you appear to have already attended to
it, I felt the similarity with the problem I face with Gödel's proof may help
you. I found your use of a straightforwardly inconsistent expression - if my
interpretation of your symbolism is right - quite novel and intriguing.)
s = Gn (Sb [n zn]
® Neg Sb
[n zn]) (1)
I take this to mean that ‘s’ is defined as the Gödel-number of the formal expression ‘N(n) → ~N(n)’, where ‘n’ is a variable representing the natural number that is the Gödel-number of the undefined (since ‘n’ is a variable) formal primitive recursive expression (class expression) ‘N(z)’.
Since ‘N(z)’ is an undefined relation (class expression), how do we specify its primitive recursiveness (if we are appealing to Gödel’s Theorem V), and how do we determine the Gödel-number ‘s’ constructively and in an intuitionistically unobjectionable way?
My problem with Gödel’s proof is somewhat similar.
Despite Gödel’s thesis in Theorem V that every primitive recursive relation is expressible in his formal system, and so it can be Gödel -numbered, I am unable to determine, or even determine a procedure that leads me to, the Gödel -number ‘q’ that he states as being that of the open primitive recursive formula, Q(x, y), which he defines at the start of his proof at para 188 eqn. no. 8.1 of On formally undecidable propositions of Principia Mathematica and related systems I.
17th
April 2001
Comment : [Harry Fisher <harry@harrylfisher.fsnet.co.uk>] So, taking this inconsistency to be established, I ask: Of all the involvements, which can be identified as causative? The system has been designed to be completely foolproof - how has this intention been betrayed?
Note : Should we seek
consistency in how we communicate, not in the perception we seek to
communicate?
As you aptly emphasise, the significance does not lie in
demonstrating an inconsistency in Gödel's formal system, but tracing its origin
and the causes for, and implications of, its oversight.
What I stress in my papers is that Gödel's undecidable sentences are 'undecidable' not due to any lack of 'decidability' criteria in an adequate formalisation of Peano's Arithmetic, but due to lack of sufficient definition whilst constructing an adequate formalisation of Peano's Arithmetic.
Whether such a formalisation is at all possible is perhaps the question that will determine the survival of strong AI, and I have not seriously given thought to this. At this point of time, however, I doubt whether such a formalisation can be achieved.
The points you raise are interesting.
In www.alixcomsi.com/Intuitionism.htm, I briefly indicate that such issues pre-suppose - and our understanding of them is thereby considerably influenced and determined by - a personal, philosophical answer to the question of what we understand by 'mathematics'.
I hold that we answer this question implicitly in every context, but not consistently.
My personal view is to treat mathematics as a language, and to treat the ‘Law of the Excluded Middle’, the ‘axiom of choice’ and other ‘axioms of infinity’ as meta-specifications on how we are to construct and extend the language, as needed, to describe some meaningful or significant ‘content’ with as high a degree of correlation between that which is sought to be described, and our manner of describing it.
The degree of correlation would be reflected in the degree of correlation between that which is sought to be described by the language by two different perceptions.
That each such perception (which I take to cover each instance of a perception even by the same perceiver) is a unique one-off that can only be subjective is, to me, a reflection not only of the richness of that which is sought to be described (and which, I take as a premise, can never be completely captured within the language), but also of the richness of the perception potential of the perceiver at the point of perception.
Thus that which we perceive is, in a crude sense, a construct of our perception. As our perception fluctuates, changes and evolves, so, correspondingly, does our reality.
The question of what is reality I thus treat as a meta-physical query, lying in the domain whose value lies less in its attempts to marshal facts into abstractions, and more in its ability to provide the data of raw experiences and phenomena for a meaningful scientific and philosophical enquiry into the inter-relationship between such experiences and phenomena.
This contrasts with my understanding of the Platonist doctrine where this question seems to be one of philosophy; and where our perception of reality is, in a sense, determined by our ability to perceive that which there is in an objective universe that we only discover, whether an abstract mathematical one or a physical one. The focus, thus, is on determining, with ever-greater comprehension, the real nature of such an objective universe, common to all perceivers, and one which subsists even without any perception.
I am not sure how this will relate to the issues raised by you, but I am advocating that a paradigm shift in focus towards deciding and defining what we consider consistent, and then specifying such consistency into our processes for construction and extension of the language, rather than on seeking an assurance of consistency for the language we define, may be more illuminating.
So, perhaps, the significant question to address is : Should we concentrate on being consistent in how we communicate, rather than in seeking consistency in either our perception, or in that which is sought to be perceived?
(Updated : 4/20/01 11:56:38 PM by re@alixcomsi.com)
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