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For the proof of Gödel's Incompleteness Theorem, most versions of proof use basically self-referential statements.

My question is, what if one argues that Gödel's Incompleteness Theorem only matters when a formula makes self-reference possible?

Is there any proof of Incompleteness Theorem that does not rely on self-referential statements?

Sarah Brightman
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6 Answers6

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Roughly speaking, the real theorem is that the ability to express the theory of integer arithmetic implies the ability to express formal logic.

Gödel's incompleteness theorem is really just a corollary of this: once you've proven the technical result, it's a simple matter to use it to construct variations of the Liar's paradox and see what they imply.

Strictly speaking, you still cannot create self-referential statements: the (internal) self-referential statement can only be interpreted as such by invoking the correspondence between external logic (the language in which you are expressing your theory) and internal logic (the language which your theory expresses).

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    The last paragraph here is quite important: The Gödel sentence doesn't know it is speaking about itself; we have to conclude that looking at the formal system from the outside. – hmakholm left over Monica Jul 17 '12 at 10:56
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There is a rather pretty proof of a standard version of Gödel's First Theorem by Kleene, that extracts it as a corollary of his (Kleene's) Normal Form Theorem. The proof involves diagonalization, but not self-reference. There's a two-page exposition here: http://www.logicmatters.net/resources/pdfs/KleeneProof.pdf

There are two further elementary proofs which don't involve self-reference, whose conclusions are something-a-bit-less-than the full Gödelian result, in Chs 6 and 7 of the second edition of my Gödel book.

Again, both those arguments involve diagonalization. It is diagonalization rather than self-reference which might reasonably be said to be characteristic of typical (though not all) incompleteness proofs.

Peter Smith
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There is a weaker version of the first incompleteness theorem that is an almost trivial consequence of an insight from computability theory, namely of the result that 

  there exists a computably enumerable set that is not computable   (*).

Consequence: the set of true first-order sentences (i.e. true about the 'real' natural number sequence) is not axiomatizable by a c.e. axiom set.

Unfortunately, most proofs of ($*$) have themselves a scent of self-referentiality hanging around them. However, you may want to check out 'simple sets'. Simple sets are c.e. and not recursive, and the standard textbook-proof of their existence is, to the best of my knowledge, the argument that comes closest to a non-selfreferential argument for ($*$).

Zev Chonoles
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Anonymous
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There is this one, that I have heard of but not perused myself:

Link

Glorfindel
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argentage
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A low-level answer.
Gödel's own undecidable statement (it is now known from Julia Robinson etc.) can be of the form: "Here is a polynomial equation in many variables with integer coefficients. Does it have a solution in positive integers?" There is nothing self-referential about that. But we GOT that equation by coding something that Sarah considers to be self-referential into the polynomial.

GEdgar
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Here's a direct computability-theoretic argument. Suppose $T$ is an "appropriate" theory. Let $A,B$ be two disjoint c.e. sets, and let $A_T=\{x: T\vdash x\in A\}$ and $B_T=\{x:T\vdash x\not\in A\}$. Each $A_T$ and $B_T$ is c.e. by definition; since $T$ is complete we have $A_T\sqcup B_T=\mathbb{N}$, and hence they're each computable; and since $T$ is "appropriate" we have $A\subseteq A_T$ and $B\subseteq B_T$. But this gives us a contradiction: just take $A,B$ to be a pair of computably inseparable c.e. sets.

Of course, how do we know that computably inseperable c.e. sets exist? Well, this in turn is a straightforward trick with a universal Turing machine: we set $$A=\{e:\varphi_e(e)\downarrow =0\}\quad\mbox{and}\quad B=\{e:\varphi_e(e)\downarrow=1\}.$$ By definition, $A$ and $B$ are disjoint c.e. sets (and note that if $\varphi_e(e)\uparrow$ then $e$ is not in $A$ or $B$).

Now suppose $C$ were a computable separator for $A$ and $B$; that is, $C$ is computable, $A\subseteq C$ and $B\cap C=\emptyset$. Let $\varphi_c$ be the total computable characteristic function of $C$ - that is, $$C=\{x:\varphi_c(x)\downarrow=1\}.$$ We now have two cases:

  • If $c\in C$, then $\varphi_c(c)=1$. But then we have $c\in B$, contradicting the assumption that $C\cap B=\emptyset$.

  • If $c\not\in C$, then $\varphi_c(c)=0$. But then we have $c\in A$, contradicting the assumption that $A\subseteq C$.

Note that crucially "$\varphi_c(c)\uparrow$" is not an option - since $\varphi_c$ is total.

There's no self-reference at all there: the closest we come is in the verification step of our construction of computably inseparable c.e. sets, but that's not actually self-reference (feeding an existing object its own "name" is different from building an object which somehow "knows its own name ahead of time"). This is exactly the diagonalization versus self-reference point.

Noah Schweber
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  • what do you mean by c.e and r.e – David Okogbenin Apr 13 '20 at 17:36
  • @DavidOkogbenin "C.e." stands for "computably enumerable" (and "r.e." for "recursively enumerable" - they're synonymous, but I've edited so I only use one). There are many characterizations of c.e. sets; the one that's most useful here is probably "$X$ is c.e. iff for some $e$ we have $$X={n: \varphi_e(n)\downarrow=1},$$ where $\varphi_e$ is the $e$th partial computable function according to some standard enumeration of such. – Noah Schweber Apr 13 '20 at 18:01
  • Incidentally, you may also find this more broadly useful. – Noah Schweber Apr 13 '20 at 18:04
  • Thank you. I am new to this so I’m not entirely clear. What is the difference between diagonalization and self-reference. When I read about it( and i did not understand all that I read though), it looked like the diagonalization(or the diagonal Lemma- i don’t think they’re the same thing) was a more subtle form of self reference. – David Okogbenin Apr 13 '20 at 22:55
  • @DavidOkogbenin Think about infinite binary sequences for concreteness. Diagonalization can be applied to any array of infinite binary sequences: given such an array $D=(d_i)_{i\in\mathbb{N}}$ (so each $d_i$ is an infinite binary sequence), we get the "diagonal" sequence $n\mapsto d_n(n)$ (or if we prefer, we can get the "antidiagonal" sequence $n\mapsto 1-d_n(n)$). Note that a priori we don't care whether the sequence $e$ this produces is or is not one of the $d_i$s; for example, in Cantor's argument the conclusion is exactly that the antidiagonal sequence is not one of the "rows" of $D$. – Noah Schweber Apr 13 '20 at 23:25
  • Here, we're considering the sequence (of $0$s, $1$s, and "NaN"s - the latter since the functions involved are partial) whose $n$th term is $\varphi_n(n)$. This is a perfectly-defined object, and no self-reference is involved; its construction is just a diagonalization. We now consider what happens with a specific term of this sequence, namely the $c$th term $\varphi_c(c)$, under certain hypotheses on $c$. But at no point do we build an object which actually refers to itself. – Noah Schweber Apr 13 '20 at 23:29
  • Basically, $A$ and $B$ were created "diagonal-wise" - with the status of $n$ depending on the $n$th object in some list (namely the standard enumeration of the partial computable functions). There's no self-reference here, but we are "going down the diagonal" of the appropriate array. We then do a proof by contradiction: supposing we have a $c$ with certain properties (namely: $c$ is an index for a c.e. set such that [stuff]), we then deduce a contradiction. Now that contradiction does involve considering how $c$ relates to the set coded by $c$, but that's not self-referential: (continued) – Noah Schweber Apr 13 '20 at 23:35
  • it's the difference between "This statement is false" and "All statements are false." The referent of the latter is the set of all statements, of which it itself is a member, but we wouldn't really call that genuine self-reference. Similarly, $c$ is the index for some set of natural numbers; we can look at whether any given number is in that set, and there's nothing a priori special about $c$ itself in doing so. Of course this is all fairly subjective, but I think it's an important distinction to understand (even if you ultimately decide that it's bogus). – Noah Schweber Apr 13 '20 at 23:38