◄ Index 
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Main essay
Why we must heed Wittgenstein’s “notorious paragraph”
Bhupinder
Singh Anand[1]
(A .pdf file of this
essay before the current update is available at http://arxiv.org/abs/math/0306024)
We argue that, although
Wittgenstein’s reservations on Gödel's interpretation of his own formal
reasoning are, indeed, of historical importance, the uneasiness that
academicians and philosophers continue to sense, and express, over standard
interpretations of Gödel's formal reasoning - even seventy years after the
publication of his seminal 1931 paper - is of much greater significance, and
relevance, to us today; it should be seen as indicating specific points of
possible ambiguity in foundational mathematical concepts that need to be
addressed on philosophical grounds, rather than dismissed on technicalities.
1. Introduction
2.1 Mathematics
as a set of formal languages
2.2 An
interpretation must be an effectively decidable translation
2.3 Is the
converse necessarily true?
2.4 Tarskian
truth under the standard interpretation
3. What is PA?
3.1 Is
PA an interpretation of its standard interpretation?
3.2 Defining
effective satisfiability and truth
4. Standard
interpretations of Gödel's reasoning
4.1 How
definitive is the standard interpretation of Gödel's reasoning?
4.2 We
must conclude that PA is omega-inconsistent
4.3 Rosser’s
argument may be invalid
5. When does a formal assertion “mean”
what it represents?
5.1 When does a formal assertion
“mean” what it represents?
5.2 Formal
expressibility and representability
5.3 When may
we assert that A(x1, ..., xn) “means” R(x1,
..., xn)?
5.4 What
is a mathematical object?
5.5 Introduction
of new symbols by definition
6. Why
P cannot be interpreted as “P is not provable in PA”
6.1
A
recursive number-theoretic function that is not a mathematical object
6.2 No
PA-formula P can interpret as “P is not provable in PA”
6.3 No
PA-formula can interpret as “PA is consistent”
7. Conclusion
1. Introduction
In
their paper “A
note on Wittgenstein's ‘notorious paragraph’ about the Gödel Theorem”[2] [FP00],
I imagine someone asking my advice; he says: “I have
constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism,
and by means of certain definitions and transformations it can be so
interpreted that it says: ‘P is not provable in Russell’s system.’ Must I not
say that this proposition on the one hand is true, and on the other hand is
unprovable? For suppose it were false; then it is true that it is provable. And
that surely cannot be! And if it is proved, then it is proved that it is not
provable. Thus it can only be true, but unprovable.”
Just as we ask, “‘Provable’ in what system?,” so we
must also ask, “‘True’ in what system?” “True in Russell’s system” means, as
was said, proved in Russell’s system, and “false in Russell’s system” means the
opposite has been proved in Russell’s system.--Now what does your “suppose it
is false” mean? In the Russell sense it means, “suppose the opposite is proved
in Russell’s system”; if that is your assumption you will now presumably give
up the interpretation that it is unprovable. And by “this interpretation” I
understand the translation into this English sentence. --If you assume that the
proposition is provable in Russell’s system, that means it is true in the
Russell sense, and the interpretation “P is not provable” again has to be given
up. If you assume that the proposition is true in the Russell sense, the same thing
follows. Further: if the proposition is supposed to be false in some other than
the Russell sense, then it does not contradict this for it to be proved in
Russell’s system. (What is called “losing” in chess may constitute winning in
another game.)
Floyd
and Putnam then argue that this paragraph contains a “philosophical claim of
great interest which”:
... is simply
this: if one assumes (and, a fortiori if one actually finds out) that ~P is
provable in Russell’s system one should ... give up the “translation” of P by
the English sentence “P is not provable”.
Now,
although Wittgenstein’s reservations on Gödel's interpretation of his own
formal reasoning are, indeed, of historical importance, the uneasiness that
academicians and philosophers such as Floyd and Putnam continue to sense, and
express, over standard (text-book) interpretations of Gödel's formal reasoning
- even seventy years after the publication of his seminal 1931 paper [Go31a] -
is of much greater significance, and relevance, to us today.
In a
set of related essays[4], we argue that standard interpretations of
foundational concepts of classical mathematical theory may be implicitly
influenced by, and built upon, some of Gödel's debatable interpretations of his
own formal reasoning[5]; hence such interpretations can be argued as being,
prima facie, either ambiguous, or non-constructive, or both.
We now
argue further that Wittgenstein’s reservations, and Floyd and Putnam’s
uneasiness, can - and arguably must, as we advocate in this essay - be seen as
indicating specific points of such ambiguity that need to be addressed on
philosophical grounds, rather than dismissed on technicalities.
2. What is mathematics?
2.1 Mathematics as a set of formal languages
Without
attempting to address the issue in its broader dimensions, we take
Wittgenstein’s remarks as implicitly indicating that mathematics is to be
considered simply as a set of precise, symbolic, languages.
Any
language of such a set, say Peano Arithmetic PA[6] (or Russell and Whitehead’s Principia Mathematica,
PM, or ZFC[7]), is intended to express - in a finite, unambiguous,
and communicable manner - relations between concepts[8] that are external to the language PA (or to PM, or to
ZFC). Each such language is, thus, in a Platonic sense, two-valued[9], if we assume that a relation[10] either holds or does not hold externally (relative to
the language).
Further,
a selected, finite, number of primitive formal assertions[11] about a finite set of selected primitive relations
of, say, a language L are defined as axiomatically L-provable; all other
assertions about relations that can be effectively[12] defined in terms of the primitive relations are
termed as L-provable if, and only if, there is a finite sequence of assertions
of L, each of which is either a primitive assertion, or which can effectively
be determined in a finite number of steps as an immediate consequence of any
two assertions preceding it in the sequence by a finite set of rules of
consequence.
2.2 An
interpretation must be an effectively decidable translation
An
effective interpretation[13] of a language L into another language, say PM (or PA,
or ZFC, or English, etc.), is essentially the specification of an effective
method by which any assertion of L is translated unambiguously into a unique
assertion of PM (or PA, or ZFC, or English, etc.). Clearly, if a relation is
provable in L, then it should be effectively decidable in any interpretation of
L - since a finite proof sequence of L would, prima facie, translate as a
finite proof sequence in the interpretation.
2.3 Is the converse necessarily true?
The
question arises: If an assertion is decidable in an interpretation M of L, then
does such decidability translate into an effective method of decidability in L?
Obviously,
such a question can only be addressed unambiguously if there is an effective
method for determining whether an assertion is decidable in M. If there is no
such effective method, then we are faced with the following thesis that is
implicit in, and central to, Wittgenstein’s “notorious” remark:
Thesis: If
there is no effective method for the unambiguous decidability of the assertions
of a mathematical language L under an interpretation M, then L can only be
considered a mathematical language of intuitive expression, but not a
mathematical language of effective, and unambiguous, communication.
What
this means is that, in the absence of an effective method of decidability in a
mathematical language M that is a model of, say, a mathematical language such
as PA, any correlation of “soundness” between a PA-assertion and an assertion
in M is essentially arguable; so it is meaningless to ask whether, in general,
an assertion of PA is decidable under interpretation in M or not (the question
of whether the assertion is decidable in PA or not is, then, an issue of
secondary consequence).
2.4 Tarskian truth under the standard interpretation
The
philosophical dimensions of this thesis emerge if we take M as the standard,
arithmetical, interpretation of PA, where [Me64]:
(a) the set of
non-negative integers is the domain,
(b) the
integer 0 is the interpretation of the symbol “0” of PA,
(c) the
successor operation (addition of 1) is the interpretation of the “ ' ” function
(i.e. of f11),
(d) ordinary
addition and multiplication are the interpretations of “+” and “.“,
(e) the
interpretation of the predicate letter “=” is the equality relation.
Now,
post-Gödel, the standard interpretation of classical theory seems to be that:
(f) PA can, indeed, be interpreted in M;
(g) assertions
in M are decidable by Tarski’s definitions of satisfiability and truth (cf. [Me64], p49-53);
(h) Tarskian
truth and satisfiability are not always effectively verifiable in M[14].
However,
the question, implicit in Wittgenstein’s argument regarding the possibility of
a semantic contradiction in Gödel's reasoning, then arises:
How can we assert that a PA-assertion (whether such an
assertion is PA-provable or not) is true under interpretation in M, so long as
such truth remains effectively unverifiable in M?
Since
the issue is not resolved unambiguously by Gödel in his 1931 paper (nor,
apparently, by subsequent standard interpretations of his formal reasoning and
conclusions), Wittgenstein’s remark can be taken to argue that, although we may
validly draw various conclusions from Gödel’s formal reasoning and conclusions,
the existence of a true or false assertion of M cannot be amongst them.
3. What is PA?
3.1 Is
PA an interpretation of its standard interpretation?
A
related philosophical issue is, then, the question:
Is PA an interpretation of its standard interpretation
M?
In
other words, since PA is intended to formalise our intuitive arithmetic M as
expressed by Dedekind in his formulation of the Peano Postulates, do we accept
that PA must be an interpretation of M?[15]
(The standard response to this question seems to lie
at the heart of Wittgenstein’s reservations, and to be the cause of the
uneasiness felt by subsequent philosophers who question the standard
interpretations of classical mathematical theory.)
Now, a
negative answer would imply that PA cannot be taken as a faithful formalisation
of our intuitive, Dedekind, arithmetic; so, either (as standard interpretations
of Gödel's reasoning and conclusions implicitly imply) such arithmetic is not
formalisable in principle, or there is some interpretation of M that formalises
M more faithfully.
The
former is, intuitively, an unappealing, Platonic, and implicitly self-limiting,
admission; the latter, an unacceptable reflection on the competence of
mathematicians to adequately select an appropriate set of primitive, axiomatic,
assertions for PA as may be needed for PA to be an effective, and unambiguous,
mathematical language of precise communication.
An
affirmative answer, on the other hand, whilst validating PA as a formalisation
of our intuitive, Dedekind arithmetic, would further imply that, since an
assertion would then be effectively decidable in PA only if it were effectively
decidable in M, there must be some effective method of defining Tarskian
satisfiability and truth in M.
3.2 Defining effective satisfiability and truth
Although
Wittgenstein does not appear to have attempted such a definition - possibly as
it may have seemed to involve technicalities beyond the scope of his
reflections - we note in §5.1
of Anand [An02c] that such an effective
method is, indeed, made available to us by, curiously, a constructive
interpretation of Gödel's reasoning and conclusions; an interpretation that is,
ironically, more in sympathy with Wittgenstein’s constructive approach than
Gödel’s Platonic one.
3.3 Undecidability
in PA
Now,
the thesis of a constructive interpretation of Gödel's reasoning and
conclusions ([An02c], §5.1)
is that we may not interpret, for instance, the meta-assertion “PA proves: [(Ax)F(x)][16]” as the non-verifiable, Tarskian meta-assertion:
F(x) is satisfied by any given element x of the domain of M[17].
We must
interpret it, instead, as the verifiable meta-assertion:
There is a uniform effective method (algorithm/Turing
machine) such that, given any element x of the domain of M, it will effectively decide that F(x) is satisfied in M.
It
follows that both the meta-assertions, “PA does not prove [(Ax)F(x)]”, and “PA proves [~(Ax)F(x)]”, then
interpret constructively as the meta-assertion:
There is no uniform effective method (algorithm/Turing
machine) such that, given any element x of M, it will effectively decide that F(x) is satisfied in M.
Consequently,
a constructive interpretation of Gödel's reasoning and conclusions implies that
there can be no undecidable propositions in PA; in other words, that PA is
syntactically complete!
4. Standard interpretations of Gödel's
reasoning
4.1 How
definitive is the standard interpretation of Gödel's reasoning?
However,
we are, then, immediately faced with the question (cf., also, [An03h]): Since the standard interpretation of
Gödel's reasoning and conclusions asserts that PA is syntactically incomplete,
how definitive is the standard interpretation?
Now, in
Theorem VI of his 1931 paper, Gödel essentially argues that his “undecidable”
proposition, [(Ax)R(x)], is such that (cf. [An02b],
§1.3-§1.6):
If [(Ax)R(x)] is
P-provable, then [~R(n)] is
P-provable for some numeral [n].
Now, by
standard logical axioms, we have that:
If [~R(n)] is P-provable for some numeral [n], then [~(Ax)R(x)] is
P-provable.
It thus
follows that Gödel has essentially argued that:
If [(Ax)R(x)] is P-provable, then [~(Ax)R(x)] is
P-provable.
Clearly,
it should now follow, by the standard Deduction Theorem of first order logic,
that:
[(Ax)R(x) => ~(Ax)R(x)] is P-provable,
and so:
[~(Ax)R(x)] is
P-provable.
4.2 We
must conclude that PA is omega-inconsistent
However,
at this point, standard interpretations of Gödel’s reasoning appeal to his
explicit assumption that PA is omega-consistent in order to conclude that the
PA-provability of [~(Ax)R(x)] cannot be inferred from the above meta-argument.
Now,
unless the omega-consistency of PA has some deeper, intuitive significance
philosophically, this is not a reasonable inference; since the Deduction
Theorem is a fundamental theorem of classical logic, we must, using Occam’s
razor, conclude from Gödel's reasoning, firstly, that [~(Ax)R(x)] is PA-provable, and, secondly, that PA is omega-inconsistent
(and so Gödel's Theorem VI holds vacuously [An02b]).
We note
that Wittgenstein’s remarks indicate that, prima facie, there appear no intuitively
significant philosophical grounds for treating the omega-consistency of PA as a
primitive concept; prima facie, there are thus no reasonable grounds for
allowing it to over-ride an application of the Deduction Theorem (cf. [An03h]).
In
sharp contrast, we note that the omega-inconsistency of PA is natural, and
intuitively unobjectionable, under a constructive interpretation of the concept
of “PA proves: [(Ax)F(x)]” as described earlier. Under such interpretation,
an omega-inconsistent PA does not imply that PA, or any of its interpretations,
are either inconsistent, or unnaturally consistent (cf. [An02b], Appendix 2); it simply
implies that there are arithmetical relations that cannot be verified uniformly
by a common algorithm over the domain of their interpretation.
Moreover,
as we argue in Anand
([An03c], Appendix 1),
this interpretation implies that PA can express relations that are
deterministic, yet essentially unpredictable; such a language would have
significance for the expression of natural phenomena that are best described in
quantum mechanical terms.
The
above suggests that it may be the absence of an adequately technical
counter-argument that leaves Wittgenstein’s viewpoint - and that of others who
have shared his reservations - vulnerable to the arguments advanced by standard
interpretations of Gödel's reasoning and conclusion; these implicitly imply
that any interpretation of Gödel's reasoning and conclusion must be accepted as
essentially counter-intuitive on the basis of purely technical considerations.
4.3 Rosser’s argument may be invalid
Prima
facie, the standard interpretation of Gödel’s reasoning and conclusions seems
strengthened by Rosser’s argument that Gödelian undecidability can be
established in a simply consistent language without assuming omega-consistency.
However, as shown in Anand ([An02a], §7.4(b)(x)(3)),
if a standard (text-book) exposition of Rosser’s proof, such as in Mendelson ([Me64], p145, Proposition 3.32), can be
accepted as definitive, then Rosser’s argument is invalid.
5.
When does a formal assertion “mean” what it represents?
An
important philosophical issue - which does not seem to have been adequately
addressed by standard interpretations of classical theory as yet - is implicit
in the key thesis of Floyd and Putnam’s paper, and is reflected in
Wittgenstein’s remark:
If you assume that the proposition is provable in
Russell’s system, that means it is true in the Russell sense, and the
interpretation “P is not provable” ... has to be given up.
5.1 When
does a formal assertion “mean” what it represents?
We may
state this issue explicitly as:
(*) When does
a formal assertion “mean” what it represents?
Now,
if, as argued earlier, we accept that PA formalises our intuitive arithmetic M,
and that M is the standard interpretation of PA, it follows that every
well-formed formula of PA interprets as a well-defined arithmetical expression,
and every well-defined arithmetical expression can be represented as a
PA-formula.
The
question then arises:
When is an arbitrary number-theoretic function or
relation representable in PA?
5.2 Formal
expressibility and representability
Now,
the classical PA-expressibility and representability of number-theoretic
functions and relations is addressed by the following three definitions (cf. [Me64], p117-118):
(a) A number-theoretic relation R(x1, ..., xn) is said to be expressible in PA if, and only
if, there is a well-formed formula [A(x1, ..., xn)] of PA with n free variables such that, for any natural numbers k1, ..., kn:
(i) if R(k1, ..., kn) is true, then PA proves: [A(k1, ..., kn)],
(ii) if R(k1, ..., kn) is false, then PA proves: [~A(k1, ..., kn)],
(b) A number-theoretic function f(x1, ..., xn) is said to be representable in PA if, and
only if, there is a well-formed formula [A(x1, ..., xn, y)] of PA,
with the free variables x1, ..., xn, y, such
that, for any natural numbers k1, ..., kn, l:
(i) if f(k1, ..., kn) = l, then PA proves: [A(k1, ..., kn, l)],
(ii) PA
proves: [(E!l)A(k1, ..., kn, l)],
(c) A
number-theoretic function f(x1, ..., xn) is said to be strongly representable in PA
if, and only if, there is a well-formed formula [A(x1, ..., xn, y)] of PA,
with the free variables x1, ..., xn, y, such
that, for any natural numbers k1, ..., kn, l:
(i) if f(k1, ..., kn) = l, then PA proves: [A(k1, ..., kn, l)],
(ii) PA
proves: [(E!l)A(x1, ..., xn, y)],
5.3 When may we assert that A(x1, ..., xn) “means” R(x1, ..., xn)?
We can,
thus, re-phrase our earlier question (*) as:
If a number-theoretic relation R(x1, ..., xn) is expressible by a PA-formula [A(x1, ..., xn)], when may we assert that, under the standard
interpretation, A(x1, ..., xn) “means” R(x1, ..., xn)?
Now we note
that, if R(x1, ..., xn) is arithmetical, then one of its PA-representation
is [R(x1, ..., xn)][18], whose standard interpretation is R(x1, ..., xn). Hence every arithmetical relation is the standard
interpretation of some PA-formula that expresses R(x1, ..., xn) in PA, and we can adapt this to give a formal
definition of the term “means”:
Definition 1:
If a number-theoretic relation R(x1, ..., xn) is expressible by a PA-formula [A(x1, ..., xn)], then we say that, under the standard
interpretation, A(x1, ..., xn) means R(x1, ..., xn) if, and only if, R(x1, ..., xn) is the standard interpretation of some PA-formula
that expresses R(x1, ..., xn) in PA.
The
question (*) can now be expressed precisely as:
(**) When is a
number-theoretic relation the standard interpretation of some PA-formula that
expresses it in PA?
Now, by
definition, the number-theoretic relation R(x1, ..., xn), and the arithmetic relation A(x1, ..., xn), can be effectively shown as equivalent for any
given set of natural number values for the free variables contained in them.
However,
for R(x1, ..., xn) to mean A(x1, ..., xn), we must have, in addition, that R(x1, ..., xn) can be effectively transformed into an arithmetical
expression, so that it can be the standard interpretation of some PA-formula
that expresses it in PA.
5.4 What
is a mathematical object?
Clearly,
this implies, firstly, that we must be able to add [R] as a primitive, n-ary, relation letter to PA, along with suitable
axioms, without inviting inconsistency; and, secondly, that R(x1, ..., xn) must define a unique mathematical object, where we
define (cf. [An02c], §1.2):
(i) Primitive
mathematical object: A primitive
mathematical object is any symbol for an individual constant, predicate letter,
or a function letter, which is defined as a primitive symbol of a formal
mathematical language.
(ii) Formal
mathematical object: A formal
mathematical object is any symbol for an individual constant, predicate letter,
or a function letter that is either a primitive mathematical object, or that
can be introduced through definition into a formal mathematical language without
inviting inconsistency.
(iii) Mathematical
object: A mathematical object is
any symbol that is either a primitive mathematical object, or a formal
mathematical object.
Now,
the logical and mathematical antinomies show that we cannot unrestrictedly
assume that every arithmetical relation defines a mathematical object[19]. However, if we can introduce the n-ary relation letter [R] into PA as above, without
inviting inconsistency, then we can, reasonably, assert that the relation R(x1, ..., xn) does, indeed, define a mathematical object in a
constructive and intuitionistically unobjectionable way, and that A(x1, ..., xn) does mean R(x1, ..., xn).
5.5 Introduction of new symbols by definition
Now,
the question:
When may we introduce an additional function or
relation letter into PA without inviting inconsistency?
is
addressed classically by the following proposition (cf. [Me64], Proposition 2.29, p82):
Proposition: Let K be a first-order theory with equality. Assume
that K proves: [(E!u)A(u, y1, ..., yn). Let K' be the first-order theory with equality
obtained by adding to K a new function letter f of n arguments, and the proper axiom [A(f(y1, ..., yn), y1, ..., yn)], as well as all instances of the axioms of a first
order theory involving f. Then there is an effective transformation
mapping each well-formed formula B of K' into a well-formed formula B'
of K such that:
(a) if f
does not occur in [B], then [B'] is [B];
(b) [(~B)']
is [~(B')];
(c) [(B
=> C)'] is [B' => C'];
(d) [((Ax)B)'] is [(Ax)B'];
(e) K'
proves: [B <=> B'];
(f) if K'
proves: B, then K proves: B'.
Hence, if B does not contain f, and K' proves:
B, then K proves: B.
It
follows that a number-theoretic function f (or relation R, if we
treat R as a Boolean function) may be taken to define a mathematical
object if it is strongly representable in PA; we may then introduce the
function letter [f] into PA without risking inconsistency, and we would
then have that f(x1, ..., xn) is represented in PA by the formula [f(x1, ..., xn)], whose standard interpretation is f(x1, ..., xn).
6.
Why P cannot be interpreted as “P is not provable in PA”
6.1 A recursive
number-theoretic function that is not a mathematical object
Apart
from the trivial resolution of the mathematical and logical paradoxes (footnote 18), the significance of the above
definitions is that Gödel's primitive recursive substitution function, Sb(x v|y) (cf. [Go31a],
definition 31, p20), may also not be a mathematical object!
The significance of these definitions is seen in Meta-hypothesis 1. We postulate, there,
the existence of an asymmetrical
recursive number-theoretic relation[20] - one that is intuitively decidable constructively,
but which cannot be introduced through definition as a formal mathematical
object into any formal system of Peano Arithmetic without inviting
inconsistency; nor, ipso facto, into any Axiomatic Set Theory that admits
models[21] (cf. [Me64], p192-3)
of such Arithmetic. Hence, it would not be a formal mathematical object, and
the range of its characteristic function[22] would not be a recursively enumerable set[23]!
This would be an astonishing result[24]! Firstly, recursive number-theoretic functions and
relations are classically accepted as the basic building blocks for defining
constructive, and intuitionistically unobjectionable, mathematical objects[25]. Secondly, and in vivid contrast, the relative
consistency, and independence, of the Continuum Hypothesis would imply, prima
facie, that we may also treat Cantor’s non-constructive cardinal, aleph1,
as a valid formal mathematical object[26]; thus, we may introduce axiomatic definitions - and
an individual constant symbol - for it into any Axiomatic Set Theory without
inviting inconsistency!
In
other words, Meta-hypothesis
1 in Anand
[An02c] argues that there is no PA-formula
that can mean Sb(x v|y) under
interpretation.
6.2 No
PA-formula P can interpret as “P is not provable in PA”
Now we
note that Gödel’s number-theoretic relation Bew(x) - which holds if, and only if, x is the Gödel-number of some PA-provable formula P -
is defined ([Go31a], definition 46, p22) in
terms of his primitive recursive number-theoretical relation Sb(x v|y). It follows that, if the latter is not a
mathematical object, then, neither is the former.
In
other words, Meta-hypothesis
1 in Anand
[An02c] suggests that there is no PA-formula
that can mean Bew(x) under
interpretation; ipso facto, there would be no PA-formula that can interpret as
an arithmetic assertion that is equivalent to the assertion “P is not provable
in PA”. As we remark in Anand
([An02c], §II(3)),
such an assertion would only, then, be an arguable convention, not a logical
inference.
6.3 No
PA-formula can interpret as “PA is consistent”
It also
follows that if the concept of “PA-provability”, as defined by Gödel, cannot be
expressed in PA, then Gödel’s Theorem XI [Go31a],
which asserts that the consistency of PA cannot be established within a
consistent PA, must hold vacuously. This argument is discussed in detail in Anand ([An02c], §II),
where we, further, prove Theorem
2 independently of Meta-hypothesis
1, to the effect that no PA-formula can mean “PA is consistent” under the
standard interpretation.
Thus, Theorem
2 can, debatably, be taken to address, and resolve, the issue:
If we assume that the number-theoretic sentence (Ex)(Form(x) & ~Bew(x)) ([Go31a], p36, footnote 63), abbreviated Wid(PA),
defines the assertion “PA is consistent” in a constructive and
intuitionistically unobjectionable way[27], then can we consistently assume further that Wid(PA)
is equivalent to the standard interpretation of some PA-formula [Con(PA)]?
... the latter assumption is one of the implicit
meta-theses that appear to underlie Gödel’s proof of, and the conclusions he
draws from, his Theorem XI ([Go31a], p36).
Clearly, if Meta-hypothesis
1 is valid, then, by Meta-lemma 2,
such an assumption may be invalid[28]. Thus, in that case, there may be no P-formula,
Con(P), whose standard interpretation is the number-theoretic assertion (Ex)(Form(x) & ~Bew(x)) - which is defined by Gödel as equivalent to “P is consistent”.
7. Conclusion
We
conclude that standard interpretations of Gödel’s reasoning and conclusions are
not definitive. Wittgenstein’s reservations on Gödel's interpretation of his
own formal reasoning, as reflected in his “notorious” paragraph, and the
uneasiness that academicians and philosophers such as Floyd and Putnam continue
to sense, and express, over standard interpretations of Gödel's formal
reasoning, must be respected as significant indicators of possible ambiguities
that may be rooted in implicit assumptions underlying standard interpretations
of classical foundational concepts.
[An02a] Anand, B. S.
2002. Reviewing
Gödel’s and Rosser’s meta-reasoning of undecidability. (Web essay)
<Preprint: http://alixcomsi.com/Constructivity_consider.htm>
[An02b] Anand, B. S.
2002. Omega-inconsistency in
Gödel’s formal system: a constructive proof of the Entscheidungsproblem.
(Web essay)
<Preprint: http://alixcomsi.com/CTG_00.htm>
[An02c] Anand, B. S.
2002. Some
consequences of a recursive number-theoretic relation that is not the standard
interpretation of any of its formal representations. (Web essay)
<Preprint:
http://alixcomsi.com/CTG_06_Consequences.htm>
[An02d] Anand, B. S.
2002. Is a
deterministic universe logically consistent with a probabilistic Quantum
Theory? (Web essay)
<Preprint:
http://alixcomsi.com/CTG_08_Quantum_consistency1.htm>
[An03a] Anand, B. S.
2003. Is
there a duality in the classical acceptance of non-constructive, foundational,
concepts as axiomatic? (Web essay)
<Preprint:
http://alixcomsi.com/CTG_06_Consequences_Bringsjord.htm>
[An03b] Anand,
B. S. 2003. Three beliefs
that lend illusory legitimacy to Cantor’s diagonal argument? (Web
essay)
<Preprint:
http://alixcomsi.com/Three_beliefs.htm>
[An03c] Anand, B. S.
2003. Is there a
“loophole” in Gödel’s interpretation of his formal reasoning and its
consequences? (Web essay)
<Preprint:
http://alixcomsi.com/Is_there_a_loophole.htm>
[An03d] Anand, B. S.
2003. Can Turing
machines capture everything we can compute? (Web essay)
<Preprint:
http://alixcomsi.com/Can_Turing_machines.htm>
[An03e] Anand, B. S.
2003. The
formal roots of Platonism (Web essay)
<Preprint:
http://alixcomsi.com/The_formal_roots_of_Platonism.htm>
[An03f] Anand, B. S. 2003. Can we express every transfinite
concept constructively? (Web
essay)
<Preprint: http://alixcomsi.com/Can_we_express_every_transfinite.htm>
[An03g] Anand,
B. S. 2003. Is the Halting probability a
Dedekind real number? (Web
essay)
<Preprint: http://alixcomsi.com/Is_the_Halting_probability.htm>
[An03h] Anand, B. S. 2003. How
definitive is the standard interpretation of Gödel’s Incompleteness Theorem? (Web essay)
<Preprint: http://alixcomsi.com/How_definitive_is_the_standard.htm>
[Br93] Bringsjord, S. 1993. The Narrational Case
Against Church's Thesis. Easter APA meetings,
<Web
page: http://www.rpi.edu/~brings/SELPAP/CT/ct/ct.html>
[Bu01] Budnik, Paul P. Jr. 2001. What is and what will be -
Integrating spirituality and science. Mountain Math Software,
<Preprint:
http://www.mtnmath.com/whatth/whatth.html>
[Ca01] Calude,
Cristian S., Calude, Elena, and Marcus, Solomon. 2001. Passages of Proof.
Workshop, Annual Conference of the Australasian Association of Philosophy (
<PDF
file: http://arXiv.org/abs/math.
HO/ 0305213>
[Da95]
<PDF
file: http://citeseer.nj.nec.com/davis90is.html>
[FP00] Floyd,
Juliet., Putnam, Hilary. 2000. A Note on Wittgenstein's ‘Notorious
Paragraph’ about the Gödel Theorem. The Journal of Philosophy 45,
11: 624-632.
<PDF
file: http://staff.washington.edu/dalexand/Putnam%20Readings/Notorious.pdf>
[Go31a] Gödel,
Kurt. 1931. On formally undecidable propositions of Principia Mathematica
and related
[Go31b] Gödel, Kurt.
1931. On formally
undecidable propositions of Principia Mathematica and related systems I. Translated by B. Meltzer.
<Web
page: http://home.ddc.net/ygg/etext/godel/index.htm>
[Ha47] Hardy, G.H.
1947, 9th ed. Pure Mathematics.
[Ho00] Hodges, A.
2000. Uncomputability
in the work of Alan Turing and Roger Penrose.
<Unpublished
lecture: http://www.turing.org.uk/philosophy/lecture1.html>
[Ka59] Kalmár, L.
1959. An Argument Against the Plausibility of Church's Thesis. In
Heyting, A. (ed.) Constructivity in Mathematics. North-Holland,
[Kl36]
[La51] Landau,
E.G.H. 1951. Foundations of Analysis. Chelsea Publishing Co.,
[Me64] Mendelson,
Elliott. 1964. Introduction to Mathematical Logic. Van Norstrand,
[Me90] Mendelson, E.
1990. Second Thoughts About Church's Thesis and Mathematical Proofs.
Journal of Philosophy 87.5.
[Pe90] Penrose, R.
1990 (Vintage edition). The Emperor's
New Mind: Concerning Computers, Minds and the Laws of Physics.
[Pe94] Penrose, R.
1994. Shadows of the Mind: A Search for the Missing Science of Consciousness.
[Po01] Podnieks,
Karlis. 2001. Around Goedel's Theorem.
<e-Textbook:
http://www.ltn.lv/~podnieks/gt.html>
[RH01]
<
Web page: http://psy.ucsd.edu/chip/ramapubs.html>
[Ru53] Rudin,
Walter. 1953. Principles of Mathematical Analysis.
[Ti61] Titchmarsh,
E. C. 1961. The Theory of Functions.
[Tu36] Turing,
Alan. 1936. On
computable numbers, with an application to the Entscheidungsproblem.
Proceedings of the
<Web
version: http://www.abelard.org/turpap2/tp2-ie.asp#index>
[Wi78] Wittgenstein,
Ludwig. (1978 edition). Remarks on the Foundations of Mathematics. MIT Press,
Acknowledgement: The idea
of relating the uneasiness expressed by Wittgenstein, Floyd, Putnam and others
- over standard interpretations of classical theory - to a constructive
interpretation was conceived after viewing the Apr-May 2003 discussion in the
Foundations of Mathematics Forum, FOM, over Floyd and Putnam’s paper [FP00].
Author’s e-mail:
anandb@vsnl.com
(Updated: Thursday 3rd June 2004 7:03:03
PM IST by re@alixcomsi.com)
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