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Proof that Wittgenstein is correct about Gödel

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Abstract

The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic property of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide valid the deductive inference. Conclusions of sound arguments are derived from truth preserving finite string transformations applied to true premises.
Proof that Wittgenstein is correct about Gödel
The conventional notion of a formal system is adapted to conform to the sound deductive
inference model operating on finite strings. Finite strings stipulated to have the semantic
property of Boolean true provide the sound deductive premises. Truth preserving finite string
transformation rules provide valid the deductive inference. Conclusions of sound arguments
are derived from truth preserving finite string transformations applied to true premises.
An “analytic” sentence, ... has historically been characterized as one whose truth depends
upon the meanings of its constituent terms (and how they’re combined) alone...
https://plato.stanford.edu/entries/analytic-synthetic/
Analytical_Knowledge
The set of expressions of language verified as true entirely on the basis of their semantic
meaning specified as stipulated relations between expressions of this language.
When these stipulated relations between expressions language are encoded as stipulated
relations between finite strings a Turing Machine would decide membership in this recursive
language, thus the Truth of every finite string.
Analytical_Knowledge
Expressions of language verified as true entirely on the basis of their semantic meaning
encoded as stipulated relations between the finite strings of this language.
Axioms, rules-of-inference, syntax, and truth conditional semantics are all fully integrated
together into the single formalism of finite string transformation rules.
Validity and Soundness https://www.iep.utm.edu/val-snd/
A deductive argument is said to be valid if and only if it takes a form that makes it impossible
for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive
argument is said to be invalid.
A deductive argument is sound if and only if it is both valid, and all of its premises are actually
true. Otherwise, a deductive argument is unsound.
Sound_Deductive_Formalism conforms to the sound deductive inference model:
(a) True Premises are finite strings stipulated to have the semantic property of Boolean true
or are derived from truth preserving operations on such strings.
(b) Valid Deduction is the application of truth preserving finite string transformations to True
Premises or finite strings derived from truth preserving operations on such strings.
(c) Conclusions of a sound argument are any final or intermediate finite strings derived from
truth preserving finite string transformations applied to True Premises or finite strings derived
from truth preserving operations on such strings.
---1---
To provide a simple intuitive grasp of the Sound Deductive Formalism (SDF) we define a
very simple formal system named Simple_Arithmetic.
Simple_Arithmetic evaluates the infinite set of finite strings representing this relationship:
Natural_Number “+” Natural_Number “=” Natural_Number
defined by this AWK regular expression: /[0-9]+[\+][0-9]+[=][0-9]+/
to determine whether or not a formal proof exists that derives the semantic propery of
Boolean true for the finite string. (see appendix).
∀F ∈ Sound_Deductive_Formalism ∀X ∈ WFF(F) (True(F, X)) ↔ Theorem(F, X))
When it is understood that every element of the set of analytical knowledge is either a
semantic tautology (defined to be true) or deduced from semantic tautologies then we see
that these semantic tautologies and deductive rules-of-inference can be expressed as
relations between finite strings.
This unifies sound deduction with formal proofs to theorem consequenes, thus making every
element of the set of analytical knowledge provable.
AK = Analytical_Knowledge(as defined above)
∀x AK (( AK ⊢ x) ↔ True(AK, x))
No analytical expression of language is ever actually true unless there are a connected set of
ideas that make it true. What-so-ever connected set of ideas that make an expression of
language true can always be expressed as a connected set of relations betweeen finite
strings. This connected set of relations between finite strings is the formal proof of the orginal
expression of language.
Wittgenstein definitions of True() and False()
‘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. (Wittgenstein
1983,118-119) Formalized by Olcott as:
LHS := RHS means LHS is defined as the RHS
∀x (True(RS, x) := (RS ⊢ x)) // x is a theorem of RS
∀x (False(RS, x) := (RS ⊢ ¬x)) // ¬x is a theorem of RS
Wittgenstein’s minimal essence of the 1931 Incompleteness Theorem sentence
“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’. (Wittgenstein 1983,118-119)
Formalized by Olcott: P ↔ (RS ⊬ P)
When we sum up the results of Gödel's 1931 Incompleteness Theorem by formalizing
Wittgenstein’s verbal specification such that this formalization meets Gödel's own sufficiency
requirement: “Every epistemological antinomy can likewise be used for a similar
undecidability proof.” then we can see that Gödel's famous logic sentence is only unprovable
---2---
in PA because it is untrue in PA because it specifies the logical equivalence to self
contradiction in PA.
Since the Wittgenstein-Olcott axiom schema define True(RS, x) as Provable(RS, x) then
¬Provable(RS, x) would be defined as ¬True(RS, x). This means that the Wittgenstein-Olcott
minimal essence of the 1931 Incompleteness Theorem <IS> The Liar Paradox.
The Formalized Liar Paradox says that P is materially equivalent to Not True.
The truth table shows that this is self-contradictory.
P ↔ ¬True(P) P ↔ RS⊬P
T F F T F F
F F T F F T
The truth table of minimal essence of the 1931 Incompleteness theorem is identical to the
truth table of the Liar Paradox because the third columns of these truth tables are stipulated
by the Wittgenstein-Olcott axiom schema to mean exactly the same thing.
The failure of logical equivalence shows that both P and ¬P are contradicted (false)
(in the above formula) thus meeting the [epistemological antinomy] sufficiency condition that
Gödel stipulated for proof equivalence: “14 Every epistemological antinomy can likewise be
used for a similar undecidability proof.” (Gödel 1931:40)
The fact that self-contradictory sentences specified in the language of a formal system cannot
be proven in that formal system does not make the formal system itself incomplete or
inconsistent as long as unprovable (from axioms) is construed as untrue.
Formalizing the liar paradox in the “C” programming language:
void main()
{
bool LP = (LP != true);
}
Even the “C” compiler recognizes the value is tested before it has been intialized.
liarparadox.cpp(3) : warning C4700: uninitialized local variable 'LP' used
Microsoft (R) Incremental Linker Version 9.00.30729.01
Copyright (C) Microsoft Corporation. All rights reserved.
At the most abstract level of analysis:
Conceptual Truth is ONLY semantic relations between concepts that can always be
expressed as[1] syntactic relations between finite strings[2] thereby logically entailing that
truth cannot possibly ever diverge from provability.
[1] Forming an isomorphism between semantic and syntactic relations:
∀x (True(x) ≅ Provable(x))
[2] Such as words, word phrases or predicate logic expressions.
---3---
Examples:
"one" [is a] "Integer"
"cats" [are] "Animals"
"cats" [have] "legs"
"2 + 3" [equals] "5"
"A B" "↔" "B A"∧ ∧
To make the above abstraction more concrete we focus on the single relation between
concepts of [sound deduction] from the sound deductive inference model. Sound deduction
begins with stipulated truth, applies a sequence of truth preserving operations, thus
necessarily ends up with truth.
Truth ONLY comes from:
(1) Stipulated truth (the definitions of the meaning of words)
(2) Applying a sequence of truth preserving operations to stipulated truth.
Truth ALWAYS comes from:
(1) Stipulated truth (the definitions of the meaning of words)
(2) Applying a sequence of truth preserving operations to stipulated truth.
When we construe a formal systems axioms to essentially be stipulated truth then this same
formal systems theorems would also be true because they were derived by applying truth
preserving operations to its axioms. Since this is the way that Truth really works we have
proven that true can never diverge from provability.
The bottom line of all this is that the only reason that G is not provable in PA is that G is not
true in PA, because as Wittgenstein states true requires provable.
'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. (Wittgenstein
1983:118)
Furthermore as Curry states True in Tarski's metatheory does not carry over to his theory as
Tarski claims.
The terminology which has just been used implies that the elementary statements are not
such that their truth and falsity are known to us without reference to {T}. (Curry 1977:45)
Gödel indicates the exact same inescapable contradiction that has been elaborated above.
The difference is that he concludes that some truths are unprovable rather than concluding
that unprovable entails untrue. (Gödel1931:39-41).
Godel, Kurt 1931. On Formally Undecidable Propositions of Principia Mathematica And
Related Systems I, page 39-41. Footnote 14.
---4---
Wittenstein, Ludwig 1983. Remarks on the Foundations of Mathematics (Appendix III), 118-
119. Cambridge, Massachusetts and London, England: The MIT Press (quoted in full
below).
Tarski, Alfred 1983. “The concept of truth in formalized languages” in Logic Semantics,
Metamathematics. Indianapolis: Hacket Publishing Company, 275-276.
Curry, Haskell 1977. Foundations of Mathematical Logic. New York: Dover Publications, 45
Montague, Richard, 1970, “Universal grammar”, Theoria, 36: 373–398.
Reprinted in Thomason (ed.) 1974, pp. 7–27.
Copyright 2018, 2019, 2020 PL Olcott All rights reserved
---5---
Appendix
Curry, Haskell 1977. Foundations of Mathematical Logic. New York: Dover Publications, 45
We begin by postulating a certain non void, definite class 𝓕
of statements, which we call elementary statements...
The statements of are called elementary statements to 𝓕
distinguish them from other statements which we may form
from them...
Let be such a theory. Then the elementary statements 𝓣
which belong to we shall call the elementary theorems 𝓣
of ; we also say that these elementary statements are 𝓣
true for . Thus, 𝓣given , an elementary theorem is an 𝓣
elementary statement which is true...
A theory is thus a way of picking out from the statements
of a certain subclass of true statements...𝓕
the elementary statements are not such that their truth and
falsity are known to us without reference to . 𝓣
Wittgenstein, Ludwig 1983. Remarks on the Foundations of Mathematics (Appendix III), 118-119.
Cambridge, Massachusetts and London, England: The MIT Press
8. 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...
---6---
Montague, Richard, 1970, “Universal grammar”, Theoria, 36: 373–398.
Reprinted in Thomason (ed.) 1974, pp. 7–27.
There is in my opinion no important theoretical difference between natural
languages and the artificial languages of logicians; indeed I consider it possible
to comprehend the syntax and semantics of both kinds of languages with a
single natural and mathematically precise theory. (Montague 1970, 373)
Tarski, Alfred 1983. “The concept of truth in formalized languages” in Logic Semantics,
Metamathematics. Indianapolis: Hacket Publishing Company, 275-276.
According to Thesis A we can construct, on the basis of the
enriched metatheory, a correct definition of truth concerning
all the sentences of the theory studied.
The formulas (8) and (9) together express the fact that x is an
undecidable sentence; moreover from (7) it follows that x is a
true sentence.
By establishing the truth of the sentence x we have eo ipso
-by reason of (2)-also proved x itself in the metatheory.
Since, moreover, the metatheory can be interpreted in the
theory enriched by variables of higher order (cf. p. 184) and
since in this interpretation the sentence x, which contains no
specific term of the metatheory, is its own correlate, the proof of
the sentence x given in the metatheory can automatically be
carried over into the theory itself: the sentence x which is
undecidable in the original theory becomes a decidable sentence
in the enriched theory.
Godel, Kurt 1931. On Formally Undecidable Propositions of Principia Mathematica And Related
Systems I, page 39-41.
We now obtain an undecidable proposition of the system PM, i.e. a proposition A, for which
neither A nor not-A are provable, in the following manner:
...
S = R(q)
holds for some determinate natural number q. We now show that the proposition [R(q); q]13 is
undecidable in PM. For: supposing the proposition [R(q); q] were provable, it would also be
correct; but that means, as has been said, that q would belong to K, i.e. according to (1), Bew
[R(q); q] would hold good, in contradiction to our initial assumption. If, on the contrary, the
negation of [R(q); q] were provable, then n ε K , i.e. Bew [R(q); q] would hold good. [R(q);
q] would thus be provable at the same time as its negation, which again is impossible.
The analogy between this result and Richard’s antinomy leaps to the eye; there is also
a close relationship with the “liar” antinomy,14 since the undecidable proposition [R(q); q]
states precisely that q belongs to K, i.e. according to (1), that [R(q); q] is not provable. We
are therefore confronted with a proposition which asserts its own unprovability.15
14 Every epistemological antinomy can likewise be used for a similar undecidability proof.
---7---
15 In spite of appearances, there is nothing circular about such a proposition, since it begins by
asserting the unprovability of a wholly determinate formula (namely the q-th in the alphabetical
arrangement with a definite substitution), and only subsequently (and in some way by accident) does
it emerge that this formula is precisely that by which the proposition was itself expressed.
---8---
/****************************************************************************
This code snippet demonstrates [truth conditional semantics] for the subset of
analytic knowledge involving “=” relational expressions of the arithmetic
operation of “+” applied to finite strings OF ASCII digits of arbitray length
representing natural numbers.
Truth-conditional semantics is an approach to semantics of natural
language that sees meaning (or at least the meaning of assertions)
as being the same as, or reducible to, their truth conditions.
https://en.wikipedia.org/wiki/Truth-conditional_semantics
The finite string transformation rules specified by this source-code provide the
means to formally prove whether a finite string of the language of the
Simple_Arithmetic formal system has the semantic property of Boolean true.
This AWK regular expression: specifies the entire language of the
Simple_Arithmetic Sound Deductive Formalist formal system:
/[0-9]+[\+][0-9]+[=<>][0-9]+/
****************************************************************************/
char AddWithCarry(char D1, char D2, char& Carry)
{
char SUM = ADD_Digit[D1][D2];
if (Carry == '1' && SUM == '9')
{
SUM = '0';
Carry = '1';
}
else if (Carry == '1' && SUM < '9')
{
SUM = ADD_Digit[SUM][Carry];
Carry = ADD_Carry[D1][D2];
}
else // Carry == '0'
Carry = ADD_Carry[D1][D2];
return SUM;
}
std::string Add(std::string& OP1, std::string& OP2)
{
std::string SUM;
char Carry = '0';
for (int N = OP1.length() - 1; N >= 0; N--)
SUM += AddWithCarry(OP1[N], OP2[N], Carry);
if (Carry == '1')
SUM += '1';
std::reverse(SUM.begin(), SUM.end());
return SUM;
}
//
// (Proven && True) || (Unproven && Untrue)
//
bool ProveInput(std::string& OP1,std::string& OP2,
std::string& SUM, char Relational_OP)
{
std::string RESULT;
RESULT = Add(OP1, OP2);
return (RESULT == SUM);
}
---9---
If we adapt Prolog to be our representational system then we test the capabilities on its
expressive power to answer queries based on natural language knowledge. Empirical
knowlege can be encoded as axioms.
https://en.wikipedia.org/wiki/Two_Dogmas_of_Empiricism#Analyticity_and_circularity
marital_status(bill, married).
marital_status(sam, single).
bachelor(X) :- \+ marital_status(X, married).
?- bachelor(bill).
false
?- bachelor(sam).
true
The above simple Prolog shows how to define bachelor(X) as synonymous with not married(X)
without any cycles that the Wikipedia article about Quine's objection indicated would be required.
---10---
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