by Wes Hansen

 

“Deconstruction was a basic device of Derrida’s post-structuralism. It asserts undecidability, the endless deferral of meaning. [T]his debilitating word-play is an attempt to defeat any notion of foundational truth or meaning; it trivializes any such pursuit as baseless.” 

—John Zerzan, Lévi-Strauss Revisited, Graffiti #8 [1] 

Technically speaking, undecidability relates to syntax – Proof Theory, not semantics – Model Theory. But if we accept the rather straight-forward first-order proofs of Soundness – every formal sentence true in some model has a proof, and Completeness – every formal sentence which has a proof is true in some model, then syntax and semantics are, at the very least, complementary; there is a logical equivalence between proof and truth, at least on the first-order level. So, what, then, is the foundation of undecidability? Historically speaking, this would be Kurt Goedel’s infamous 1931 paper, On Formally Undecidable Propositions of Principia Mathematica and Related Systems [2], and it is easily shown that this paper, which is also the foundation for a 90 year-old Proof Theory, suffers from numerous logical gaps and outright omissions. In essence, it is a poorly argued, though creative, Platonist polemic directed at Ludwig Wittgenstein and the Logical Positivists. The irony here, is that this Platonist polemic led rather directly to post-structuralism, and even informs, to a large degree, the debilitating secular humanism of post-modernism! This, the intellectual dishonesty surrounding Goedel’s infamous work, is the subject of my rant.  

 In Section 1 of his infamous paper [2], Goedel states (page 38): 

For metamathematical purposes it is naturally immaterial what objects are taken as basic signs, and we propose to use natural numbers for them.” 

The key idea expressed here is, the metamathematical analysis, at least as it is constrained to Proof Theory, is wholly and entirely syntactical, hence, the basic signs need be nothing more nor less than distinguishable placeholders. Semantics – meaning, plays no role. Is this premise, taken for granted by many, true? Goedel presents no support whatsoever in his paper, and I can quite easily show that his paper, and the history since, resoundingly refute the premise.  

Goedel’s “Proof” 

Goedel begins his formal argument by specifying his system P, which “is essentially the system obtained by superimposing on the Peano Axioms the logic of PM” (Principia Mathematica). He then, beginning on page 46 of [2], introduces “a parenthetic consideration having no immediate connection with the formal system P,” this being his definition of recursively defined number-theoretic functions (relations) and recursive number-theoretic functions (relations). On pages 47 and 48 he provides and discusses five Propositions related to recursive functions (relations) and uses these to confirm that each of the 1–45 functions (relations) listed on pages 49–55 are recursive. Relevant to the present discussion, item #31 on page 53, Sub(x (v, y)), which simply says to replace the free variable v in the formula x by the entity y, is recursive. But nowhere in his paper does he discuss the reflexivity of this function, i. e. Sub(x (v,┌x┐)), where ┌x┐ is the Goedel number encoding the formula x itself. Given the central role this function, and its reflexivity, play, not only in his “proof” of Proposition VI but in his entire project (it IS his undecidable Proposition), this is a very curious omission. 

In his “proof” of Proposition VI, the function Sub(y (19,┌y┐)) enters in relation 8.1, page 58 

Q(x, y) ≡ ¬(x B_c(Sub(y (19,┌y┐)))) 

and with very little discussion he claims that ¬ (x B_c(Sub(y (19,┌y┐)))) is recursive, hence, per his discussion on page 56, there is a recursive 2-place relation sign q such that (his formula 9 and 10, page 58) 

¬ (x B_c(Sub(y (19,┌y┐)))) Bew_c(Sub(q (17,┌x┐)(19,┌y┐))); and, 

x B_c(Sub(y (19,┌y┐)))) Bew_c(¬ Sub(q (17,┌x┐)(19,┌y┐))). 

He then defines the recursive (yes, it is recursive provided Sub(y (19,┌y┐)) is recursive, see page 56, but that IS the question) 1-place class-sign (his formula 11, page 58) 

p = 17 Gen q. 

This is recursive provided his Sub(y (19,┌y┐)) is recursive per his item #15 on page 51. He next reduces Sub(y (19,┌y┐)) by p defining the recursive 1-place class-sign (his formula 12, page 58) 

r = Sub(q (19,┌p┐)). 

He then derives the all-important syntactical identity (his derivation 13, page 58) 

Sub(p (19,┌p┐)) = Sub((17 Gen q)(19,┌p┐)); 

= 17 Gen Sub(q (19,┌p┐)); 

= 17 Gen r. 

And this, then, is the formal sentence that he proves is undecidable from his limited class c using the system P.  

Okay, the problem really enters with his relation 8.1, because it includes Sub(y (19,┌y┐)), but we’ll look at Sub(p (19,┌p┐)), since p is specifically defined as 17 Gen q and, per Goedel’s own derivation, Sub(p (19,┌p┐)) is syntactically identical to 17 Gen r, the undecidable sentence. Goedel’s claim, quoted above, is that “for metamathematical purposes it is naturally immaterial what objects are taken as basic signs,” so, to test this, we’ll use p and 17 Gen q showing 

Sub(p (19,┌p┐)) =Sub((17 Gen q(17,19))(19,┌p┐)); 

= 17 Gen q(17, 17 Gen q(17, 17 Gen q(17,…))). 

And we see here that we INEVITABLY end up with two possible situations, if we give a Universal Turing machine Sub(p (19,┌p┐)) (equivalently, 17 Gen r) as input: 

Case 1: The Universal Turing Machine does not halt when given Sub(p (19,┌p┐)) as input, because it continuously finds free variables 19 that it replaces by a proposition including free variable 19. Then it leads immediately to a nested regress and its very existence in system P is impossible without supplementing system P with, say, Peter Aczel’s Anti-Foundation Axiom, i. e. it is not recursive, hence, does not exist in Goedel’s system P, as defined. 

Case 2: The Universal Turing Machine does halt when given Sub(p (19,┌p┐)) as input, because it doesn’t acknowledge the free variable 19 in the proposition being substituted. Then r = Sub(q (19,┌p┐)) is NOT a class-sign because it still contains two free variables, 17 and 19, which means 17 Gen r is not a sentence, in that IT contains the variable 19 free, and it makes no sense to discuss decidability without additional information [3].   

In both cases, Goedel’s “proof” is rendered, well, meaningless, and the FACT that this logical error has (cough, hack, spit) remained “undiscovered” for 90+ years calls into question the validity of Goedels methods in general, certainly his arithmetization of first-order logic, the very foundation of Proof Theory. Okay, let’s be honest, this is intellectual dishonesty on a global scale; certainly Derrida was guilty of it. I mean, it’s not like I just happen to be smarter than everyone else, Goedel included. And if you find this episode of intellectual dishonesty – Academy wide, profoundly disturbing, then you should read my rant about so-called “quantum computation.” I mean, what a scam! Everything funded by SPACs and taking place on “the cloud.” 

  1. https://graffiti-magazine.com/ 
  1. https://monoskop.org/images/9/93/Kurt_G%C3%B6del_On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems_1992.pdf 
  1. http://euclid.trentu.ca/math/sb/pcml/pcml-16.pdf