[Fis] _ Concerns regarding a questioning of Goedel's theorem . . .

[Maxine Sheets-Johnstone]

To all concerned colleagues,

I appreciate the fact that discussions should be conversations about 
issues,
but this particular issue and in particular the critique cited in my 
posting
warrant extended exposition in order to show the reasoning upholding the 
critique.
I am thus quoting from specific articles, the first phenomenological, 
the second
analytic-logical--though they are obviously complementary as befits 
discussions
in phenomenology and the life sciences.

EXCERPT FROM:
SELF-REFERENCE AND GÖDEL'S THEOREM: A HUSSERLIAN ANALYSIS
Husserl Studies 19 (2003), pages 131-151.
Albert A. Johnstone

The aim of this article is to show that a Husserlian approach to the 
Liar paradoxes and to their closely related kin discloses the illusory 
nature of these difficulties. Phenomenological meaning analysis finds 
the ultimate source of mischief to be circular definition, implicit or 
explicit. Definitional circularity lies at the root both of the 
self-reference integral to the statements that generate Liar paradoxes, 
and of the particular instances of predicate criteria featured in the 
Grelling paradox as well as in the self-evaluating Gödel sentence 
crucial to Gödel's theorem. Since the statements thereby generated turn 
out on closer scrutiny to be vacuous and semantically nonsensical, their 
rejection from reasonable discourse is both warranted and imperative. 
Naturally enough, their exclusion dissolves the various problems created 
by their presence. . . .

VII: THE GOEDEL SENTENCE
Following a procedure invented by Gödel, one may assign numbers in some 
orderly way as names or class-numbers to each of the various classes of 
numbers (the prime numbers, the odd numbers, and so on). Some of these 
class-numbers will qualify for membership in the class they name; others 
will not. For instance, if the number 41 should happen to be the 
class-number that names the class of numbers that are divisible by 7, 
then since 41 does not have the property of being divisible by 7, the 
class-number 41 would not be a member of the class it names.
	Now, consider the class-number of the class of class-numbers that are 
members of the class they name. Does it have the defining property of 
the class it names? The question is unanswerable. Since the defining 
property of the class is that of being a class-number that is a member 
of the class it names, the necessary and sufficient condition for the 
class-number in question to be a member of the class it names turns out 
to be that it be a member of the class it names. In short, the number is 
a member if and only if it is a member. The criterion is 
circular--defined in terms of what was to be defined--and consequently 
not a criterion at all since it provides no way of determining whether 
or not the number is a member.
	The situation is obviously similar for the class-number of the 
complementary class of class-numbers--those that do not have the 
defining property of the class they name--since the criteria in the two 
cases are logically interdependent. The criterion of membership is 
likewise defined in circular fashion, and hence is vacuous. In addition, 
the criterion postulates an absurd analytic equivalence, that of the 
defining property with its negative. The question of whether the 
class-number is a member of the class it names is unanswerable, with the 
result that any proposed answer is neither true nor false. In addition, 
of course, any answer would generate paradox: the number has the 
requisite defining property if and only if it does not have it.
	As might be expected, the situation is not significantly different for 
the class-number of classes of which the definition involves semantic 
predicates. Consider, for instance, the class of class-numbers of which 
it is provable that they are members of the class they name. The 
question of whether the class-number of the class is a member of the 
class it numbers is undecidable. The possession by the class-number of 
the property requisite for membership is conditional upon the question 
of whether it provably possesses the property, with the result that the 
question can have no answer. Otherwise stated, the number has the 
defining property of the class it names if and only if it provably has 
that property. In these circumstances, the explanation of what it means 
for the class-number to have the property has to be circular in that it 
must define having the property in terms of having the property. The 
vacuity that results is hidden somewhat by the presence of the 
requirement of provability, but while provability might count as a 
necessary condition, in the present case it cannot be a sufficient one. 
In fact, its presence creates a semantically absurd situation: the 
analytic equivalence of having the property and provably having it. The 
statement of the possession of the property by the class-number in 
question is consequently both vacuous and semantically absurd, hence an 
undecidable pseudo-statement.
	The analytic equivalence of the number's having the property and 
provably having it has a further and quite interesting consequence. In 
principle, since the equivalence is analytic, it explains what it means 
to say that the class-number in question has the requisite property, 
that is, it explains what is being said by the statement that attributes 
the property to the number. What the statement is saying, according to 
the equivalence, is that it is provable that the number has the 
property, which is to say, it is saying of itself that it is provable. 
Thus, the statement is self-evaluating. It is not, strictly speaking, 
self-referential since it contains no designator, and so cannot refer to 
itself. However, it mirrors the self-referential statements of the sort 
discussed earlier in that it predicates a semantic property of itself 
(or at least purports to do so).
	In these circumstances, it is not overly surprising to find that a 
sentence having a vacuously defined semantic predicate of provability is 
ambiguous or leads a double life. It may be used to express either of 
two statements, a pseudo-statement that purports to evaluate itself as 
provable, or, a genuine statement that evaluates the pseudo-statement, 
which genuine statement is, of course, false since a pseudo-statement is 
in principle not provable. The two statements, genuine and pseudo, are 
not the same statement. The two have distinct truth-values, but the 
basic point is that they differ in intended meaning. In the 
pseudo-statement, the statement itself (that a particular number has a 
particular property) is a part of the meaning of the pseudo-statement, 
while in the genuine (but false) statement, it is not.
	An analogous situation obtains in the case of other classes involving 
semantic predicates. If the term 'heterological' that figures in the 
Grelling Paradox were defined as applying to those words of which it is 
false that they are heterological, then the resulting Grelling statement 
(the statement that 'heterological' is heterological) could be plausibly 
interpreted to be self-evaluating. It would be analytically equivalent 
to the statement that it is false that 'heterological' is 
heterological--an equivalence that may be read as saying that the 
Grelling statement says of itself that it is false. This second 
statement would, of course, find itself expressed by a sentence that 
leads a double life.
	Of particular interest for the purpose of understanding the error that 
invalidates Gödel's theorem is the case of the class-number that names 
the class of class-numbers that are not provably members of the class 
they name. Once again, the question as to whether the class-number that 
names this class is a member of the class it names is unanswerable. The 
statement that the class-number possesses the required defining 
characteristic is a criterially deficient predication, and hence a 
pseudo-statement. In addition, the statement is analytically equivalent 
to the statement that the class-number's possession of the defining 
characteristic is not provable, and so may be viewed as saying of itself 
that it is not provable. It is thus self-evaluating, and when stated in 
this form, it is expressed by a sentence that leads a double life. As a 
result, any formal system that admits and purports to accommodate a 
criterially deficient predication of the sort will also require the 
elaborate supplementary machinery found necessary to accommodate 
self-referential statements: a three-valued logic, a procedure for 
determining which instantiations of predicates (or substitutions into 
propositional functions) produce pseudo-statements, and some notational 
device for distinguishing pseudo-statements from the genuine statements 
that are their sentential doubles. As we shall now see, in view of the 
similarity in structure of the above statement to the Gödel sentence, 
analogous remarks apply to the latter.
  	VII. THE GÖDEL SENTENCE
	In his well-known theorem Kurt Gödel purports to show that any formal 
system of classical logic equivalent to that of Principia Mathematica to 
which arithmetic constants and the axioms of arithmetic (Peano's) have 
been added, will contain sentences that are undecidable--that is, 
sentences such that neither they nor their negations are provable within 
the system.  To this end he introduces a provability predicate defined 
syntactically as membership in the set of sentences that are immediate 
consequences of the axiom-sentences. Since the provability predicate 
applies to sentences rather than statements, to avoid confusion it is 
best termed 'a derivability predicate'. As in the arguments of the 
previous section, Gödel has a number assigned as a name to each class of 
numbers according to its rank in an ordering of the various classes of 
numbers. Roughly characterized, the undecidable sentence figuring in the 
theorem (the Gödel sentence) states that a particular class-number 
satisfies a particular one-place propositional function that defines a 
class of numbers. A little more precisely, it states that a particular 
class-number has the defining characteristic of the class it numbers, 
which class is the class of class-numbers such that the sentences 
stating that the class-numbers possess the defining characteristics of 
the classes they name are not derivable.  In his informal introduction 
to his theorem, Gödel points out that the sentence may be read as 
stating via its Gödel number that a particular sentence, itself, is not 
derivable.
	The crucial line of reasoning in the theorem strongly resembles the one 
found in the Liar. It runs roughly as follows: if the sentence were 
derivable, it would have to be true, hence say something true, and 
hence, as it says, not be derivable--which contradicts the assumption of 
its derivability; if the negation of the sentence were derivable, then 
since the sentence states its underivability, it would have to be not 
underivable, hence derivable--with the result that both the sentence and 
its negation would be derivable, a contradiction. As with the Liar, each 
of two possible alternatives generates a contradiction, although in the 
present case the consequence is not paradox but undecidability-- 
undecidability in the form of a sentence of which neither its truth nor 
its falsity is derivable in the system. Gödel reasons that since the 
undecidable sentence apparently states something true, its own 
underivability,  the system contains underivable true sentences, and 
hence is incomplete.
	The Gödel sentence is concerned with derivability rather than 
provability, or sentences rather than statements. As a result one may 
plausibly question whether it is vulnerable to the criticisms directed 
above against criterially circular predications and self-evaluations. 
While the Gödel sentence clearly differs from the latter, it is possible 
nevertheless to raise the question of its legitimacy. Gödel himself 
simply assumes that the sentence is legitimate--which, of course, it is 
in the narrow sense that it conforms to the formation rules of the 
system in which it figures. However, it does not follow that it is 
legitimate in the broader sense that the interpreted sentence makes 
sense. As we saw earlier with self-referential statements and 
criterially circular predications, sentences that are apparently 
well-formed may in fact express nonsense. The Gödel sentence may well 
express just such a pseudo-statement, and have nevertheless been 
admitted into the formal system through an inadequacy of the formation 
rules.  Gödel dismisses the possibility of faulty circularity on the 
grounds that the sentence states only that a certain well-defined 
formula is unprovable, which formula turns out after the fact to be the 
one that expresses the proposition itself.  Yet, an answer of the sort 
will not do. Where circularity results from a substitution, being 
adventitious and well-formed according to the rules do nothing to remove 
the circularity. A statement with a circularly defined predicate is 
semantically vacuous, and hence not a genuine statement. Thus, the 
question of the meaning of the sentence, the statement it expresses, 
calls for serious examination.
	A first rather curious fact that more careful scrutiny brings to light 
is that the most obvious reasons for thinking the sentence meaningful 
are actually inconclusive. For instance, it might be found tempting to 
argue as follows: that any particular string of symbols is either 
derivable from the axiom-strings or not, and hence since the Gödel 
sentence asserts that a particular string is not derivable, whether true 
or not, it must at least be meaningful. However, the reasoning begs the 
point at issue. If the Gödel sentence is not meaningful, then its 
assertion that it is not derivable is not meaningful. It is a 
pseudo-statement that may appear to state something but cannot in fact 
state anything.
	For the same reason, it would be question-begging to reason that since 
the Gödel sentence states something true, its own underivability, it 
must be a genuine statement. If the sentence makes a pseudo-statement, 
it does not state anything, and so cannot state anything true. Reasoning 
of the sort simply assumes (as does Gödel) that the sentence is 
meaningful, and so fails to show that it is.
	In contrast, there are two compelling reasons for deeming the Gödel 
sentence not to be meaningful.	The first of these reasons is that any 
attempt to explicate the meaning of the string of symbols of which the 
Gödel sentence is composed finds that meaning to be a complex whole of 
which the meaning of that same string of symbols is a constituent. Any 
explanation of its meaning turns out to presuppose what it is supposed 
to explain. The situation differs from those discussed earlier in that 
the explanation is given in terms of a string of symbols, a sentence, 
rather than the purported meaning of the symbols. The presence of a 
sentence creates the illusion that there is no vacuity; a statement may 
be vacuous but a sentence is something perceptibly concrete. 
Nevertheless, the situation remains essentially the same as those 
considered earlier. The question being asked is whether the sentence is 
meaningful, and that question cannot be answered by appeal to the 
concreteness of the sentence. Such a line of reasoning would rule any 
string of symbols whatever to be meaningful. Ultimately the situation 
comes down to the following: the Gödel sentence is meaningful if and 
only if the Gödel sentence is meaningful. Despite the shift from 
statement to sentence, the meaning has been given a circular definition, 
which, as we have seen, can only generate semantic vacuity and a 
pseudo-statement.
	The second reason for denying meaningfulness springs from a more 
general consideration. The formalization of arithmetic together with its 
metalanguage is presumably a formalization of the arithmetic and 
metalanguage that occur in natural languages, in particular, in English. 
Its translation back into English must be possible, and make good sense. 
In English, one does not speak of sentences being true or of sentences 
being derivable, but of statements being true, and of statements being 
provable. The only cogent translation of the Gödel sentence back into 
English is a statement that asserts its own unprovability from the 
axioms of arithmetic and the laws of logic. Precisely such a 
self-evaluation of unprovability was examined earlier and found to be a 
criterially deficient predication, a pseudo-statement that is neither 
true nor false. On its intended interpretation, the Gödel sentence does 
not express a meaningful statement.
	The basic point is that for a formal system to qualify as a 
formalization of some discipline, it must admit of translation back into 
the language of the discipline it purports to formalize. The point is 
one that it is easy for logicians to overlook. The logic practiced in 
formal systems is a form of what Husserl terms 'consequence-logic' or 
'logic of non-contradiction', that is, the concern is with what follows 
from certain statements in accordance with given rules, and not with the 
truth of the statements (Hua XVII, pp. 15-6, 58-9).  In addition, as 
Husserl notes with regard to mathematics, it is customary for the formal 
system to be treated somewhat like a game in which strings of symbols, 
depending on their form, are derivable or not derivable from other 
strings according to rules. The signs in the system have, like chess 
pieces, "a games meaning" that replaces the arithmetic or statemental 
meaning for which the signs are actually doing duty (Hua XVII, p. 104).  
Nevertheless, if the game is to allow any conclusions to be drawn about 
the discipline being formalized, its strings of symbols and its rules 
must be interpretable, which means translatable back into the original 
language. In the case of Gödel's formalization of arithmetic, a 
particular sentence, the Gödel sentence, translates into a 
pseudo-statement. Such a sentence can hardly provide a sound basis on 
which to build a persuasive proof of the incompleteness of formalized 
arithmetic.
	Matters are not improved if the Gödel sentence is replaced with a 
simpler one, one of the sort suggested by Kripke that uses a proper name 
to refer to itself and to say that a particular sentence, itself, is not 
derivable.  Any such sentence has nothing to do with either arithmetic 
or the metalanguage of arithmetic, and so its presence in a system of 
formalized arithmetic is quite unwarranted. More importantly, the 
definition of the name it contains is circular. It defines the name in 
terms of a sentence that contains the name, which name is not as yet a 
name since the point of the definition is to make it one. It would be no 
less nonsensical to declare 'Gorg' to be a name for the word 
'Gorg'--although in fact there is no such word since, prior to the 
definition, 'Gorg' is a mere string of letters. Furthermore, the 
sentence in question should in principle be translatable back into 
English if it is to be considered a proper formalization of what it 
purports to formalize. On translation, the sentence becomes a 
nonsensical self-evaluation of unprovability. The Kripke sentence is 
thus no improvement on the Gödel sentence.

EXCERT FROM:
THE LIAR SYNDROME
SATS Nordic Journal of Philosophy, vol. 3, no. 1

VI. GÖDEL AND SENTENTIAL SELF-REFERENCE
	  Kurt Gödel's well-known theorem, widely termed 'Gödel's Theorem', 
demonstrates that any formal system of classical two-valued logic 
augmented with the axioms of arithmetic and a portion of its own 
metalanguage will contain sentences that are undecidable in the 
system--sentences for which neither they nor their negations are 
provable within the system.  The metalinguistic evaluations are made 
possible through a provability predicate defined syntactically as 
membership in the set of sentences that are immediate consequences of 
the axiom-sentences. Since the provability predicate applies to 
sentences rather than statements, to avoid confusion it is better termed 
'a derivability predicate'. The undecidable sentence figuring in the 
theorem, the Gödel sentence, says that a particular sentence, itself, is 
not derivable. Thus, the undecidable sentence responsible for the 
incompleteness apparently states something true, its own underivability.
	The paradoxical line of reasoning central to the theorem also strongly 
resembles the one found in the Liar. It runs roughly as follows: if the 
sentence were derivable, it would have to be true, hence say something 
true, and hence, as it says, not be derivable--which contradicts the 
assumption of its derivability; if the negation of the sentence were 
derivable, since the sentence states its underivability, it would have 
to be not underivable, hence derivable--with the result that both the 
sentence and its negation would be derivable. As with the Liar, each of 
two possible alternatives generates a contradiction, although in the 
present case the consequence is not paradox but incompleteness.
	The Gödel sentence figuring in Gödel's proof  states that a particular 
number satisfies a particular one-place propositional function that 
defines a class of numbers. In Gödel's formal system a number is 
assigned as a name to each class of numbers according to its rank in an 
ordering of the various classes of numbers. Roughly characterized, the 
Gödel sentence states that a particular class-number (the class-number 
of the class of class-numbers for which the sentences stating they 
possess the defining characteristics of the classes they number are not 
derivable) has the defining characteristic of the class it numbers (that 
of the non-derivability of the sentence stating its possession of the 
defining characteristic of the class it numbers).
	Clearly, since the Gödel sentence, on its intended interpretation, 
states the underivability of a certain string of symbols, rather than 
the unprovability of what is said, it is not vulnerable to the reasoning 
presented earlier against statemental self-reference. Sentential 
self-reference is widely and plausibly esteemed to be a harmless 
operation. In this spirit, Saul Kripke has contended that by 
interpreting elementary syntax in number theory, "Gödel put the issue of 
the legitimacy of self-referential sentences beyond doubt; he showed 
that they are as incontestably legitimate as arithmetic itself."  Kripke 
is obviously right when 'a legitimate sentence' is taken to mean a 
formula of the formal system that is a well-formed formula according to 
the formation rules of the system. However, the important issue is 
whether such sentences are legitimate in the sense that they make good 
sense on their intended interpretation, rather than express dubious 
statements that inadequate formation rules have failed to exclude. Gödel 
makes no attempt to show that the interpreted Gödel sentence makes sense 
(nor does Kripke); he seems simply to assume that it makes sense given 
that it is well-formed according to the rules of the system. The 
assumption hardly commands automatic endorsement, since, as we saw 
earlier with statemental self-reference and criterially circular 
predication, sentences considered to be well-formed may in fact express 
nonsense. The issuing of a certificate of legitimacy should be 
contingent upon the results of closer scrutiny of the meaning of the 
Gödel sentence.
	To clarify matters, let 'E' and 'e' represent some normal class of 
numbers (such as the class of even numbers) and its class-number, and 
let 'D' represent an underivability predicate. Let 'N' and 'n' 
represent respectively the class and class-number of all classes such 
that the sentence stating that the number has the defining property for 
membership in the class it numbers, is not derivable. The necessary and 
sufficient conditions for each of the two class-numbers, e and n, to be 
members of the class of class-numbers, N, may then be stated 
respectively as follows:
	(10)	Ne df D('Ee')
	(11)	Nn df D('Nn')
	The statement of membership conditions in (10) is clearly not circular. 
The same is not obviously the case for the statement of membership 
conditions in (11). Indeed, on further inspection, the alleged 
legitimacy of the Gödel sentence, the left-hand side of (11), becomes 
quite suspect.
	For instance, it might be found tempting to argue as follows in favor 
of the claim that Nn, the left-hand side of (11), should make perfectly 
good sense. What it states is equivalent to what is stated by the 
right-hand side, the underivability of a particular string of symbols, 
'Nn'. Since a string of symbols is either derivable from the 
axiom-strings or not, a statement asserting it is not derivable must be 
meaningful, and hence be a genuine statement. Given the equivalence of 
the right-hand and left-hand statements, the Gödel sentence must also 
express a genuine statement. However, such a line of reasoning begs to 
point at issue. The question is whether the sentence 'Nn' makes sense. 
If it does not, then the left-hand statement of (11) does not, and so 
neither does the statement equivalent to it, the right-hand side of 
(11). The latter must then be a pseudo-statement, one that appears to 
assert the underivability of a particular string of symbols, but one 
that in fact cannot assert anything. Thus, in assuming that the 
right-hand side of (11) asserts something, the argument presupposes what 
it purports to establish.
	For the same reason, it would be fallacious to claim (as Gödel does) 
that the Gödel sentence states something true, its own underivability, 
and then to argue that since it states something true, the left-hand 
statement of the equivalence must also be true, and hence a genuine 
statement. If the sentence makes a pseudo-statement, it states nothing, 
and so cannot state anything true. Such an argument simply assumes (as 
Gödel does) that the sentence makes a genuine statement, and so fails to 
show that it does.
	In point of fact, there are two excellent reasons for thinking the 
sentence cannot make a genuine statement. First, the predication on the 
left-hand side of (11) is meaningful only if the statement on the 
right-hand side is meaningful. The latter is meaningful only if the 
string of symbols 'Nn' is a string of symbols that expresses a 
meaningful statement. If the string 'Nn' expressed nonsense, then since 
it is also the Gödel sentence, the latter would not make a meaningful 
statement. Thus, the meaningfulness of the predication, Nn, is 
conditional upon the meaningfulness of the statement expressed by 'Nn', 
which is to say, itself. As a result, the predication is criterially 
circular. The situation echoes that of the Grelling paradox: the 
attribution of a particular predicate to a particular individual fails 
to make sense. In the case of the Gödel sentence the circularity is less 
apparent because the relevant statement is defined in terms of its 
sentence rather than in terms of itself. However, the shift from 
statement to sentence fails to avoid circularity since the question 
still arises as to whether the particular string of symbols is 
legitimate in the sense of expressing a genuine statement.
	The second reason for thinking the sentence illegitimate is no less 
decisive. If the formalization of arithmetic-plus-metalanguage is to be 
considered a faithful rendition of arithmetic-plus-metalanguage in 
English, its translation back into English must make good sense. The 
exception could only be a situation where the formal system employs some 
peculiar idiom in order to correct an incoherent English one. Such 
appears not to be the case. It is true that English speaks of the 
provability of statements rather than of the derivability of sentences, 
but it manages to do so without collapsing into incoherence. Talk of 
sentences being true, or false, or derivable, has its source in what is 
convenient for logicians, and not in the incoherence of some English 
idiom. In these circumstances, the only cogent translation of the Gödel 
sentence back into English is a statement asserting its own 
unprovability, as in (7). Such a statement is a pseudo-statement 
afflicted with the Liar Syndrome, one the negative effects of which are 
neutralizable in English with appropriate precautions.
	Thus, the Gödel sentence is properly judged to be illegitimate. It 
makes a pseudo-statement, and consequently should never have been 
admitted into a formal system that is two-valued, and hence unequipped 
to accommodate such sentences. Moreover, since a pseudo-statement says 
nothing, the argument in Gödel's Incompleteness Theorem fails appealing 
as it does at two crucial points to what the statement says.
	The theorem cannot be rescued by an appeal to the services of the 
simplified version of Gödel sentence suggested by Kripke, a sententially 
self-referential sentence constructed through the use of proper names 
for sentences.  The definition of such a sentence may be represented as 
follows, with 'n' representing a sentence name:
	(12)	n =ds D'n'
	Clearly, the statement expressed by 'n' has nothing to do either with 
arithmetic or with the metalanguage of arithmetic, so its presence in a 
system of formalized arithmetic is quite unwarranted. In addition, a 
definition as in (12) succumbs to charges analogous to those directed 
above against (11). First of all, 'n' is a meaningful name of a sentence 
in a two-valued system only if the right-hand side of (12) is a sentence 
that expresses a meaningful statement, and the latter is the case only 
if the 'n' on the right-hand side is the name of a sentence that 
expresses a meaningful statement. Thus, the meaningfulness of the name 
'n' has been made to depend in circular fashion upon the name 'n' being 
meaningful. The situation is not unlike that of declaring the word 
'Gerg' to be a name for the word 'Gerg', whereas prior to a definition 
it is a mere string of letters, and not a word. Likewise, in (12) 'n' 
may name a name only if 'n' is already a name and hence designates 
something.
	Secondly, a formal system that is a formalization of the arithmetic and 
metalanguage given in a natural language should in principle be 
translatable back into that language if it is to be considered a proper 
formalization of what it purports to formalize. Since the only cogent 
translation back into English of the concept of derivability is that of 
provability, the interpreted Gödel sentence becomes a nonsensical 
self-evaluation of unprovability as in (9) above.
	Thus, the shift from statemental self-reference to sentential 
self-reference is, from the point of view of present concerns, of less 
than dubious utility. Statements that are self-referential and 
predicates that are criterially circular in the sentential mode may be 
represented as follows, where the predicate '' represents any 
sentential semantic predicate:
	(13)	p =ds 'p'
	(14)	Nn df 'Nn'
	(13) is, as it were, the sentential rendition of (2), while (14) is 
that of (8). When transformed into their sentential correlates, the 
pseudo-statements that instantiate (2) and (8) become sentential 
evaluations that instantiate (13) and (14). Certainly, in discussing 
formal systems it may be useful to speak of sentences rather than of the 
statements they make, but otherwise the transformation yields no 
significant gain. If syntax faithfully reflects semantics, as it should, 
the formation rules of the system must screen for definitions and 
instantiations that generate sentences expressing statements afflicted 
with the Liar Syndrome. Contradiction is the price of failure to do so.
	Any system that contains both semantic predicates of some sort (of 
truth, provability, possibility, necessity) and names or designators of 
statements, sentences, or classes, must, if it is to avoid unnecessary 
problems, screen for failures of instantiation and substitution salva 
significatio. It must be suitably equipped either with formation rules 
that eliminate any resulting nonsensical and irrelevant statements, or 
with a notation that prevents confusion of the pseudo-statements with 
the genuine statements that evaluate them. The system that figures in 
Gödel's Theorem fails to do any of this.
	VII. IMPLICATIONS
	The puzzles attendant upon self-reference have over the years generated 
a wide variety of extravagant claims. Although in view of the above 
findings the error of these claims is obvious enough, a brief spelling 
out of the obvious is perhaps not amiss.
	The widespread tenet that a formal language cannot contain its own 
metalanguage without generating paradox is quite overstated. It is true 
only of certain formal languages, those lacking the machinery necessary 
either to eliminate certain pseudo-statements or to accommodate them in 
a three-valued system equipped with disambiguators. The Liar provides no 
grounds to speak, as has Hilary Putnam, of "giving up the idea that we 
have a single unitary notion of truth applicable to any language 
whatsoever ... ,"  and hence of giving up any notion of a God's Eye View 
of the world, and embracing a general Antirealist or non-Objectivist 
account of human knowledge. Indeed, it would be astounding to find such 
claims warranted. English has been serving as its own metalanguage for 
an impressive length of time without requiring the services of hermetic 
levels of truth, and without collapsing into incoherence.
	Gödel's Theorem is often understood to show that any system of 
formalized arithmetic must be incomplete. In addition, it is not 
infrequently touted to have other far-reaching implications. John 
Stewart, for one, has argued that Gödel's Theorem undermines an 
Objectivist epistemology and supports transduction, the view that 
subject and object exist only in their relationship to each other.  
Michael Dummett deems the theorem to show "that no formal system can 
ever succeed in embodying all the principles of proof that we should 
intuitively accept."  Likewise, Roger Penrose takes it to show that in 
mathematical thinking "the role of consciousness is non-algorithmic," 
and that "human understanding and insight cannot be reduced to any set 
of computational rules."
	As concluded above, Gödel's Theorem is made possible by a failure to 
either exclude or accommodate sentences that express pseudo-statements 
on their intended interpretation. Such a situation provides no obvious 
support for the claim that mathematics has no firm foundation, and hence 
none for Antifoundationalism or for Antirealism. Nor does it reveal some 
deep feature of mathematical thinking, a feature that eludes capture in 
a formal system. Such a feature may well exist, but evidence for it must 
be sought elsewhere. Finally. it cannot reasonably be claimed to reveal 
some remarkable capacity of the human mind: self-reference. The latter 
simply generates nonsense. A capacity to lapse into nonsense, however 
proficiently exercised, is hardly a very awe-inspiring human trait.