Gödel Incompleteness and the Empirical Sciences

Abstract

We show how widespread are metamathematical phenomena in mathematics and in the sciences which rely on mathematics. We will consider specific examples of undecidable sentences in mathematics, physics and economics. Our presentation is informal; rigorous developments can be found in the references.

Partially supported by CNPq, Philosophy Section; the authors are members of the Brazilian Academy of Philosophy.

Appendices

Appendix

We have added at this point a rather technical appendix where we give rigorous statements of the theorems that either bear on, or which imply, the results described above. It is to be noted that here we even go beyond the arithmetic hierarchy [20] and show how nasty (or how interesting, it depends on one’s viewpoint...) these mathematical phenomena can be.

A First General Undecidability and Incompleteness Theorem

Let S be our formal theory and let $L_S$ be its underlying formal language. We state here (without proof) our main undecidability and incompleteness theorems.

Definition A.1

A predicate P in $L_{S}$ is nontrivial if there are term expressions $\xi , \zeta$ in S such that $S\vdash P(\xi )$ and $S\vdash \lnot P(\zeta )$. $^{\fbox {}}$

If $\xi \in L_{S}$ is any expression in that language (adequately extended), we write $\Vert \xi \Vert$ for its complexity, as measured by the number of letters from S’s alphabet in $\xi$. Also we define (according to [11]) the complexity of a proof $C_{S}(\xi )$ of $\xi$ in $L_{S}$ to be the minimum length that a deduction of $\xi$ from the axioms of S can have, subject to the conditions in the Ehrenfeucht–Mycielski paper.

Let P be any nontrivial predicate, and let $\mathcal{B}$ be an algebra of functions used in our theory; $\mathcal F$ is a subalgebra.

Then:

Proposition A.2

Within S:

• 1. There is an expression $\xi \in \lceil \mathcal{B}\rceil$ so that $S\not \vdash \lnot P(\xi )$ and $S\not \vdash P(\xi )$, while there is a model for our theory with standard arithmetic such that M $\models P(\xi )$.

• 2. There is a denumerable set of expressions for functions $\xi _{m}(x)\in \lceil \mathcal{B}\rceil$, $m\in \omega$, such that there is no general decision procedure to ascertain, for an arbitrary m, whether $P (\xi _{m})$ or $\lnot P(\xi _{m})$ is provable in S.

• 3. Given an arbitrary total recursive function $g:\omega \rightarrow \omega$, there is an infinite number of values for m so that $C_{S}(P(\xi _{m}))> g(\Vert P (\xi _ {m})\Vert )$. $^{\fbox {}}$

That result was our first general incompleteness theorem [8]; it can be derived from Rice’s theorem in computer science [20], which is an equally general result, but the proof we first gave for Proposition A.2 is weaker than Rice’s theorem, since it only leads to unsolvable problems of Turing–degree not higher than $\mathbf{0}'$. However:

Proposition A.3

We can explicitly and algorithmically construct in $L_{S}$ an expression for the characteristic function of a subset of $\omega$ of degree $\mathbf{0}''$. $^{\fbox {}}$

That expression depends on recursive functions defined on $\omega$ and on elementary real–defined and real–valued functions plus the absolute value function and an integration, as in the case of the $\theta$ function (the Halting Function).

We simply use Theorem 9-II in [20] (p. 132). Actually the degree of the set described by the characteristic function whose expression we are going to obtain will depend on the fixed oracle set A; so, our construction is a general one.

Let $A\subset \omega$ be a fixed infinite subset of the integers. An oracle Turing machine $\phi _{x}^{A}$ with oracle A can be visualized as a two–tape machine where tape 1 is the usual computation tape, while tape 2 contains a listing of A. When the machine enters the oracle state $s_{0}$, it searches tape 2 for an answer to a question of the form “does $w\in A$?” Only finitely many such questions are asked during a converging computation; we can separate the positive and negative answers into two disjoint finite sets $D_{u}(A)$ and $D^{*}_{v}(A)$ with (respectively) the positive and negative answers for those questions; notice that $D_{u}\subset A$, while $D^{*}_{v}\subset \omega - A$. We can view those sets as ordered k– and $k^{*}$–ples; u and v are recursive codings for them [20]. The $D_{u}(A)$ and $D^{*}_{v}(A)$ sets can be coded as follows: only finitely many elements of A are queried during an actual converging computation with input y; if $k'$ is the highest integer queried during one such computation, and if $d_{A}\subset c_{A}$ is an initial segment of the characteristic function $c_{A}$, we take as a standby for D and $D^{*}$ the initial segment $d_{A}$ where the length $l(d_{A})=k'+1$.

We can effectively list all oracle machines with respect to a fixed A so that given a particular machine, we can compute its index (or Gödel number) x, and given x we can recover the corresponding machine.

Now let us write $p(n,q,x_{1},\ldots ,x_{n})$ for a 2–parameter universal Diophantine polynomial. We can define the jump of A as follows:

$$A'=\{\rho (z):\exists x_{1},\ldots ,x_{n}\in \omega \, p(\rho (z),\langle z, d_{z,A}\rangle , x_{1},\ldots ,x_{n})=0\}.$$

($\rho$ is an adequate recursive 1–1 function.) With the help of the $\lambda$ map [3], we can now form a function modelled after the $\theta$ function; it is the desired characteristic function:

$$c_{\emptyset ''}(x)=\theta (\rho (x),\langle x, d_{x,\emptyset '}\rangle ).$$

(Actually we have proved more; we have obtained

$$c_{A'}(x)=\theta (\rho (x),\langle x, d_{x,A}\rangle ),$$

with reference to an arbitrary $A\subset \omega$.)

We write $\theta ^{(2)}(x)=c_{\emptyset ''}(x)$.

Let $\mathbf{0}^{(n)}$ be the n–th complete Turing degree in the arithmetical hierarchy.

Corollary A.4

(Complete Degrees.) For all $p\in \omega$, expressions $\theta ^{(p)}(m)$ can be explicitly constructed for characteristic functions in the complete degrees $\mathbf{0}^{(p)}$. $^{\fbox {}}$

1.1 General Incompleteness Theorems

Therefore,

Proposition A.5

For every $n\in \omega$ there is a sentence $\xi$ in S such that a model with standard arithmetic $\mathbf{M}\models \xi$ while for no $k\le n$ there is a $\Sigma _{k}$ sentence in N demonstrably equivalent to $\xi$. $^{\fbox {}}$

Let $m_{0}(\emptyset ^{(m)})=\langle \rho (z),\langle z, d_{y,\emptyset ^{(m)}}\rangle \rangle$ (use of the pairing function $\tau$ is supposed) such that

$$\mathbf{M}\models \forall x_{1},\ldots , x_{n}[p(m_{0},x_{1},\ldots , x_{n})]^{2}>0,$$

for an universal polynomial p.

Let $q(m_{0}(\emptyset ^{(m)}),x_{1},\ldots )=p(m_{0}(\emptyset ^{(m)}),x_{1},\ldots ))^{2}$ be as after Proposition A.5. Then:

Corollary A.6

Within T, for:

$$\beta ^{(m+1)}=\sigma (G (m_{0}(\emptyset ^{(n)})),$$

$$G(m_{0}(\emptyset ^{(n)}))=\int _{-\infty }^{+\infty } C(m_{0}(\emptyset ^{(n)}),x)e^{-x^{2}}dx,$$

$$C(m_{0}(\emptyset ^{(n)}),x)=\lambda q(m_{0}(\emptyset ^{(n)}),x_{1},\ldots , x_{r}),$$

$\mathbf{M}\models \beta ^{(m+1)}=0$ but for all $n\le m+1$, $S^{(n)}\not \vdash \beta ^{(m+1)}=0$ and $S^{(n)}\not \vdash \lnot (\beta ^{(m+1)}=0)$. $^{\fbox {}}$

Then,

Corollary A.7

If $L_{S}$ contains expressions for the $\theta ^{(m)}$ functions as given in Corollary A.4, then for any nontrivial predicate P in N there is a $\zeta \in L_{S}$ such that the assertion $P(\zeta )$ is S–demonstrably equivalent to and S–arithmetically expressible as a $\Pi _{m+1}$ assertion, but not as any assertion with a lower rank in the arithmetic hierarchy. $^{\fbox {}}$

An extension of the preceding result is:

Corollary A.8

For any nontrivial property P there is a $\zeta \in L_{S}$ such that the assertion $P(\zeta )$ is arithmetically expressible and for a model with standard arithmetic $\mathbf{M}\models P(\zeta )$, but it is only demonstrably equivalent to a $\Pi _{n+1}$ assertion and not to a lower one in the hierarchy. $^{\fbox {}}$

Definition A.9

$\emptyset ^{(\omega )}=\{\langle x,y\rangle :x\in \emptyset ^{(y)}\}$, for $x,y\in \omega$. $^{\fbox {}}$

Then:

Definition A.10

$\theta ^{(\omega )}(m)=c_{\emptyset ^{(\omega )}}(m)$, where $c_{\emptyset ^{(\omega )}}(m)$ is obtained as in Proposition A.3. $^{\fbox {}}$

We can assuredly obtain arithmetic expressions for the characteristic functions in those higher degrees. However we are here especially interested in the analytic expressions for those functions.

Still,

Definition A.11

$\emptyset ^{(\omega + 1)}=(\emptyset ^{(\omega )})'$. $^{\fbox {}}$

Corollary A.12

$\mathbf{0}^{(\omega + 1)}$ is the degree of $\emptyset ^{(\omega + 1)}$. $^{\fbox {}}$

Corollary A.13

$\theta ^{(\omega + 1)}(m)$ is the characteristic function of a nonarithmetic subset of $\omega$ of degree $\mathbf{0}^{(\omega + 1)}$. $^{\fbox {}}$

In the next results, M is again a model with standard arithmetic for S:

Corollary A.14

Within S:

$$\beta ^{(\omega +1)}=\sigma (G (m_{0}(\emptyset ^{(\omega )})),$$

$$G(m_{0}(\emptyset ^{(\omega )}))=\int _{-\infty }^{+\infty } C(m_{0}(\emptyset ^{(\omega )}),x)e^{-x^{2}}dx,$$

$$C(m_{0}(\emptyset ^{(\omega )}),x)=\lambda q(m_{0}(\emptyset ^{(\omega )}),x_{1},\ldots , x_{r}),$$

$\mathbf{M}\models \beta ^{(\omega +1)}=0$ but $S\not \vdash \beta ^{(\omega +1)}=0$ and $S\not \vdash \lnot (\beta ^{(\omega +1)}=0)^{(\omega +1)}=0)$. $^{\fbox {}}$

Details are found in [3]. Let’s go beyond it:

Definition A.15

$\emptyset ^{(\omega + 1)}=(\emptyset ^{(\omega )})'$. $^{\fbox {}}$

Corollary A.16

$\mathbf{0}^{(\omega + 1)}$ is the degree of $\emptyset ^{(\omega + 1)}$. $^{\fbox {}}$

Corollary A.17

$\theta ^{(\omega + 1)}(m)$ is the characteristic function of a nonarithmetic subset of $\omega$ of degree $\mathbf{0}^{(\omega + 1)}$. $^{\fbox {}}$

In the next results, M is a model (with standard arithmetic) for S:

Corollary A.18

Within S:

$$\beta ^{(\omega +1)}=\sigma (G (m_{0}(\emptyset ^{(\omega )})),$$

$$G(m_{0}(\emptyset ^{(\omega )}))=\int _{-\infty }^{+\infty } C(m_{0}(\emptyset ^{(\omega )}),x)e^{-x^{2}}dx,$$

$$C(m_{0}(\emptyset ^{(\omega )}),x)=\lambda q(m_{0}(\emptyset ^{(\omega )}),x_{1},\ldots , x_{r}),$$

$\mathbf{M}\models \beta ^{(\omega +1)}=0$ but $S\not \vdash \beta ^{(\omega +1)}=0$ and $S\not \vdash \lnot (\beta ^{(\omega +1)}=0)$. $^{\fbox {}}$

Proposition A.19

(Nonarithmetic intractability.) Given any nontrivial P such that for different $\xi$, $\chi$, $S\vdash P(\xi )$ and $S\vdash \lnot P(\chi )$:

• 1. There is a family of expressions $\zeta _{m}\in L_{S}$ such that there is no general algorithm to check, for every $m\in \omega$, whether or not $P(\zeta _{m})$.

• 2. There is an expression $\zeta \in L_{S}$ such that $\mathbf{M}\models P(\zeta )$ while $S\not \vdash P(\zeta )$ and $S\not \vdash \lnot P(\zeta )$.

• 3. Neither $\zeta _{m}$ nor $\zeta$ are arithmetically expressible. $^{\fbox {}}$