Incompleteness Theorems: Logical Necessity of Inconsistency

WittgensteinvsGödelonFoundations of Logic
Carl Hewitt

Abstract
This talk (in four parts) explains how some of the writings of Ludwig Wittgenstein can be interpreted as precursors of important developments in the foundations of mathematical logic for information systems applications. These Wittgenstein writings stand in almost exact opposition to the views of Kurt Gödel.
First part: the current state of foundations of mathematical logic for information systems applications is overviewed with regard to issues of expressiblity, incompleteness, and inconsistency tolerance.
Second part: The above developments have precursors in the following writings of Wittgenstein: *There can’t in any fundamental sense be such a thing as meta-mathematics. . . . Thus, it isn’t enough to say that p is provable, what we must say is: provable according to a particular system.True in Russell’s system” means, as we have said, proved in Russell's system; and “false in Russell's system” means that the opposite has been proved in Russell's system.Have said-with pride in a mathematical discovery [e.g., inconsistency of Russell’s system because incompleteness is self-proved]: “Look, this is how we produce a contradiction.”Indeed, even at this stage, I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from consistency.
Third part:the above views are contrasted with the almost opposite ones of Gödel.The situation between Gödel and Wittgenstein can be summarized as follows:
Gödel
Mathematics is based on objective truth.
Roundtripping proves incompleteness but (hopefully) not inconsistency.
Theories should be proved consistent.
Wittgenstein
Mathematics is based on communities of practice.
Self-proof of incompleteness leads to inconsistency.
Theories should use inconsistency tolerant reasoning.
Fourth part:How do the above provide framework and guidance for the further development of logic for information systems applications?
*The (posthumously edited) writings of Wittgenstein are idiosyncratic and fragmentary. Interpretations in this talk are directed to putting them in their best possible light as precursors.

Further reading
Common sense for concurrency and inconsistency tolerance using Direct Logic™ ArXiv:0812.4852

Overview
State of the art
Cherry-picking Wittgenstein
Gödel on Wittgenstein
Inventing the future

Pervasive Inconsistency
“find bugs faster than developers can fix them and each fix leads to another bug”--Cusumano & Selby 1995

Wittgenstein 1930
Indeed, even at this stage, I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from consistency.

Predicalc
Inconsistency Tolerant Spreadsheets
Michael Kassoff, Lee-Ming Zen, Ankit Garg, and Michael Genesereth. PrediCalc: A Logical Spreadsheet Management System 31st International Conference on Very Large Databases (VLDB). 2005.

Direct Logic
A minimal fix
to
Classical Logic

Logical Invariance

DirectArgumentationin Direct Logic
Sense Making
,(  ├T) ├T
“├:The Ultimate”

Transitivity in Argumentation
Scientist and engineers speak in the name of new allies that they have shaped and enrolled; representatives among other representatives, they add these unexpected resources to tip the balance of force in their favor.Bruno Latour
(├T) , (├T)├S(├T)

 if an argument holds and furthermore if the antecedent of the argument holds, then the consequence of the argument holds.
Soundness in Argumentation
(├T)├S((├T)├T (├T))
whenan argument holdsand
furthermore the antecedent of theargument holds,infer thatthe consequence of the argument holds

Adequacyin Argumentation
(├T)├S(├T(├T))
when an inference holds,
infer it holds that the inference holds

Faithfulness in Argumentation
(├T(├T))├S(├T)
when it holds that an argument holds,
infer that the argument holds.

Roundtripping
The execution of code can be dynamically checked against its documentation. Also Web Services can be dynamically searched for and invoked on the basis of their documentation.
Use cases can be inferred by specialization of documentation and from code by automatic test generators and by model checking.
Code can be generated by inference from documentation and by generalization from use cases.

Propositionsversus Sentences
Sentence
“Gallia estomnisdivisa in partestres.”
starts with “Gallia”
Proposition
All of Gaul is divided into three parts.
believed by Caesar

Reification and Abstraction
Reification
Gallia estomnisdivisa in partestres. 
“All of Gaul is divided into three parts.”
Gallia estomnisdivisa in partestres. 
“Todos Galia está dividida en tres partes.”
Abstraction
“Gallia estomnisdivisa in partestres.”⇔All of Gaul is divided into three parts.

Logical Roundtripping
├Algebra (y = x+x⇔y = 2*x )
because
├Algebra(y = x+x⇔y = 2*x)

RoundtrippingLogical Connectives
⇔
  ⇔()
├T⇔  ├T
∀ ⇔∀

Hilbert on Incompleteness
that within us we always hear the call: here is the problem, search for the solution: you can find it by pure thought, for in mathematics there is no ignorabimus.

The Gödelian Proposition
UninferableT≡ Fix(Diagonalize)
where Diagonalize ≡λ(s) ⊬T s

What is UninferableT?
Lemma:UninferableT⇔⊬TUninferableT
Proof:
UninferableT⇔Fix(Diagonalize)
⇔Diagonalize(Fix(Diagonalize))
⇔λ(s)⊬Ts (Fix(Diagonalize))
⇔⊬TFix(Diagonalize)
⇔ ⊬TUninferableT 
⇔⊬TUninferableT

Absolute Incompleteness
⊬TUninferableT
⊬TUninferableT

Self-Refutation
Self Infers Opposite:
(├T)├S(├T)
Self Infers Argument for Opposite:
(├T(├T)) ├S(├T)
Argument for Self Infers Opposite:
((├T)├T) ├S(├T)
Argument for Self Infers Argument for Opposite:
((├T)├T(├T)) ├S(├T)

Proof of Incompleteness
1) To prove: ⊬TUninferableTUninferableT⊢T (⊬TUninferableT) lemma (⊢TUninferableT) ⊢T (⊢T⊬TUninferableT))  Soundness⊢T⊬TUninferableT Self-refutation
2) To prove: ⊬TUninferableT(UninferableT⊢T ( ⊬TUninferableT) ) lemma (UninferableT) ⊢T(⊢TUninferableT)) Double Negation (⊢TUninferableT) ⊢T (⊢T⊢TUninferableT))Soundness (⊢TUninferableT) ⊢T(⊢TUninferableT)) Faithfulness⊢T⊬TUninferableT Self-refutation

Incompleteness Redux
Inconsistency tolerant proof:
├T(UninferableT⇔⊬T UninferableT )
├T(⊬TUninferableT)
├T(⊬TUninferableT)

Absolute Inconsistency
├TUninferableT
├TUninferableT

Proof of Inconsistency
⊢TUninferableTis immediate from:
the incompleteness theorem ⊢T ⊬TUninferableT
the lemma ⊢T(UninferableT⇔⊬TUninferableT)
⊢TUninferableTis immediate from:
immediate above ⊢TUninferableT
contrapositive of lemma
⊢T(UninferableT⇔⊢T UninferableT)

Overview
State of the art

Ludwig Wittgenstein

Wittgenstein on “meta-theory”
There can’t in any fundamental sense be such a thing as meta-mathematics. . . . Thus, it isn’t enough to say that p is provable, what we must say is: provable according to a particular system.

Wittgenstein onTruth in PM
“True in Russell’s system” [Principia Mathematica (PM)] means, as we have said, proved in Russell's system; and “false in Russell's system” means that the opposite has been proved in Russell's system.

Wittgenstein onIncompleteness and Inconsistency
Let us suppose I prove the unprovability (in Russell’s system[Principia Mathematica (PM)] ) of P[⊢PM⊬PMPwhereP⇔⊬PMP,i.e. P⇔⊢PMP];then by this proof I have proved P[⊢PMP].
Now if this proof were one in Russell’s system [⊢PM⊢PMP]—I should in this case have proved at once that it belonged [⊢PMP] and did not belong [⊢PMP]to Russell’s system.—That is what comes of making up such sentences.
But there is a contradiction here!—Well, then there is a contradiction here[in PM]. Does it do any harm here?

Wittgenstein onInconsistency Tolerance
Can we say: ‘Contradiction is harmless if it can be sealed off’? But what prevents us from sealing it off?
Let us imagine having been taught Frege’s calculus, contradiction and all. But the contradiction is not presented as a disease. It is, rather, an accepted part of the calculus, and we calculate with it.
Have said-with pride in a mathematical discovery: “Look, this is how we produce a contradiction.”

Overview
State of the art

Kurt Gödel

Gödel onWittgenstein
It is clear from the passages you cite that Wittgenstein did ''not'' understand it [1st incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox*, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics).
*in Principia Mathematica

More Gödel onWittgenstein
He [Wittgenstein] has to take a position when he has no business to do so. For example, “you can’t derive everything from a contradiction.” He should try to develop a system of logic in which that is true.It’s amazing that Turing could get anything out of discussions with somebody like Wittgenstein.

Gödel versusWittgenstein
Gödel
Mathematics has objective truth
Continuum hypothesis?
Roundtripping proves incompleteness but (hopefully) not inconsistency
Theories should be proved consistent
Wittgenstein
Mathematics is a community of practice
Proof of incompleteness leads to inconsistency
Theories should use inconsistency tolerant reasoning

Tarski onInconsistency
I believe everybody agrees that one of the reasons which may compel us to reject an empirical theory is the proof of its inconsistency. . . . It seems to me that the real reason of our attitude is...: We know (if only intuitively) that an inconsistent theory must contain false sentences.

Frege onTruth
when we say that it is true that seawater is salty, we don’t add anything to what we say when we say simply that seawater is salty, so the notion of truth, in spite of being the central notion of [classical]logic, is a singularly ineffectual notion. It is surprising that we would have occasion to use such an impotent notion, nevermind that we would regard it as valuable and important.

Overview
State of the art

Building “a new box”
Direct Logic
DirectArgumentation (argumentation directly expressed)
Direct Inference (no contrapositive bug for inference)
Self-refutation
Incompleteness self-provable
Inconsistency Tolerance
Two-way Deduction Theorem for natural deduction
Boolean Equivalences hold
Concurrency
Actor Model
iScript TM
scriptJ TM

Boltzman onBloodandTreasure
What the poet laments holds for the mathematician. That he writes his works with the blood of his heart.

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Incompleteness Theorems: Logical Necessity of Inconsistency

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