Abstract
We consider the implications for general equilibrium theory of the problems of consistency and completeness as shown in the Gödel-Rosser theorems of the 1930s. That a rigorous consistent formal system is incomplete poses serious problems for dealing with unresolved problems in a fully formal system such as general equilibrium theory. We review the underlying mathematical issues and apply them to this problem for general equilibrium theory. We also consider alternative approaches such as agent-based modeling founded on empirically estimated behavioral assumptions for agents that may allow for a better way to model non-equilibrium evolutionary economic dynamics.
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Notes
The “as if” argument of Friedman has been criticized as abused and ‘instrumentally’ simplified or misunderstood as the logic of ‘any hypothesis is fine if it helps to explain data’. Friedman was an eminent proponent of Instrumentalism. The previous statement is too impressionistic to represent correctly the logic of Friedman, however this is how the “as if” argument is often over-simplified and misunderstood. Boland gives a sound description of Friedmans’ Instrumentalism; here, we summarize it with two quotes from Boland (1979): (i) “Theories do not have to be considered true statements about the nature of the world, but only convenient ways of systematically generating the already known ‘true’ conclusions”; (ii) “He [Friedman] says that as long as the observed phenomenon can be considered to be a logical conclusion from the argument containing the false assumption in question, the use of that assumption should be acceptable. In particular, if we are trying to explain the effect of the assumed behavior of some individuals (e.g. the demand curve derived with the assumption of maximizing behavior), so long as the effect is in fact observed and it would be the effect if they were in fact to behave as we assume, we can use our behavioral assumption even when the assumption is false. That is, we can continue to claim the observed effect of the individual’s (unknown but assumed) behavior is as if they behaved as we assume”. For a critique of Friedman’s methodology, see Caldwell (1980).
Antonio Palestrini nicely summarizes the problem with the expression: sometimes in economics one says “let’s assume”, meaning actually “let’s axiom”. Economic models often involve “as if” claims to “specify what kind of evidence is relevant” to the truthfulness or falsehood of assumptions or “to specify which claims ought or ought not to be evaluated for truth” (Lehtinen 2013).
As an example, think of the representative agent paradigm. There is no consensus on what the representative agent is or should represent (Jerison 1984, 2006; Kirman 1992; Hartley 1997), nor have there been attempts to prove or disprove its real existence. A statements such as “the representative household maximizes utility...” is not a hypothesis nor an assumption, but an axiom because it undoubtedly states how a household behaves.
This is not a subtlety for a discipline that is grounded on mathematical economics as a formal system. Formally, hypotheses and assumptions are but temporary truths that allow one to proceed with the modeling, in advance of testing for them. Axioms are statements that do not follow from other statements as their premises and do not need to be proved.
By following the formalistic approach of Hilbert, economic terms become pure syntactic symbols to be managed with the principles of mathematics. In all this process, the core notion is that of formal system and economics becomes a mathematized science. A formal system is a conceptual framework to formalize knowledge upon a decidable set of axioms as principles of deduction and inferential rules for deducing theorems of a formal axiomatic theory. Every statement in the formal system is developed according to a language that owns its alphabet of typographic symbols to write words according to grammatical rules which, according to syntactic rules, generate strings to be combined into expressions called well formed formulæ (wff) or statements. Therefore, the grammar generates words from an alphabet, syntax generates statements, and semantics gives meaning to expressions. Every formal system owns a set of axioms, i.e. a structure of wff equipped with an effective method that is an algorithm which, in a finite number of steps, returns an outcome or nothing. If the algorithm may return a yes-no answer about a question, then the question is said to be decidable; if the answer may be only yes or nothing then the question is said to be semi-decidable: notice that the question itself can be a statement in the formal system. If the question is about a formula or a statement, for it to be an axiom, then the effective method must prove that it does not depend on other axioms (or any other expression) as its premises. Notice that deciding whether a formula is an axiom or not does not require truthfulness or falsehood of its content, because an axiom is a statement accepted as an evident or revealed truth. From here onward only decidable sets of axioms are considered. A theory, as a set of theorems deduced from a (decidable) set of axioms, is recursively axiomatizable because, while developing a theory (a) the effective method operates on any wff stated upon the set of axioms, (b) which retrieves itself by means of the effective method to add or not a new formula in the theory and (c) the cognitive contents of a theory are enclosed in theorems deduced from axioms. Finally, inferential rules are instructions operating on wff in the formal system to develop demonstrations ending with a theorem.
Let us disambiguate that we consider a theory (economic or else) is scientific not only if it is consistent, indeed a theory might tell something false without contradiction. A theory is scientific if predictions follow without contradiction from its premises and the consequences of its predictions are in agreement with facts, at least up to an acceptable degree of approximation. This means that a theory is scientific if it is logically sound and somehow empirically falsifiable according to the scientific method. Moreover, beyond falsification, there is validation. An ‘applied’ theory should be validated in terms of effectiveness in explaining facts, without certainty of having found a law that is confirmed by experience. A ‘pure’ theory should be validated in terms of consistency of its reasoning and inferential or deductive method in solving (mathematical) problems without contradiction. The optimal situation for a theory that has both pure and applied approaches would be that the two planes of discourse intersect while reinforcing each other, as happens in physics contrary to economics. Sornette et al. (2017) highlight the strict relationship among validation, hypothesis testing and statistical significance testing of mathematical statistics while proposing (algorithmic) validation as a constructive iterative process.
This worrying trend comes not without implications. No plausible critique is possible from the outside, say from the empirical evidence: the methodology strictly follows the late procedure of adding new epiciycles to fit the data. As it became clearer and clearer during the late Middle Ages that the Ptolemaic theory of the Sun going around the Earth was not predicting celestial mechanical dynamics well, its advocates kept adding small “epicycles” to the predicted main cycles to get it to explain the data. In analogy with this, we use the epicycle argument to mean that if a consistent theory is not powerful enough to prove a statement (incompleteness) that has been formulated within the theory, or to match with facts (incorrectness) one might be tempted assuming additional hypotheses to reduce the level of incompleteness/incorrectness: if such hypotheses are assumed without proof they play as axioms, hence one is adding epicycles. As will be discussed in the following, augmenting a consistent set of axioms by means of new axioms has two drawbacks: (i) it restricts the scope of the theory and, to prove the augmented set axioms is still consistent, (ii) it takes a more powerful theory based on a different set of axioms. On the contrary, tested hypotheses may be assumed in the theory, which becomes falsifiable, without modification of the axiomatic set; in this case the drawback is that the theory is less general but less incorrect.
For a clear and rigorous historical description of the Hilbert program see the Stanford Encyclopedia of Philosophy at https://plato.stanford.edu/entries/hilbert-program (retrieved in February 2017).
In the same year Walras published the Elements of Pure Economics, 1874, Cantor proved that some not finite sets can have different cardinalities, for example the cardinality of real numbers is not the same as that of natural numbers. In 1897 Cantor completed his theory of sets. The big impact was that he brought the actual infinity to the attention of mathematicians who, on the basis of Aristotelian logic, had excluded it as unreasonable, as absurd (Galileo) or a “way of saying” (Gauss) with respect to the potential infinity. Not without controversies the Cantor theory of sets was finally accepted: such theory proves that the power set P(N) of the natural numbers N has a cardinality that is greater than the cardinality of N. Cantor proved many things but one is of great interest: there does not exist a cardinal number that is greater than every other cardinal number and that the set of not finite sets is itself not finite but it is the greatest of all the not finite sets it enumerates. The actual infinity was presented in the temple of mathematics.
The strict avoidance of infinity of any sorts is advocated by the constructivist school of mathematics that began with Kronecker in the late 1800s, who specifically criticized Cantor sharply. Moreover, the notion of limit was introduced by Weierstrass to overcome the ambiguity about the infinitesimal left by Leibniz and Newton: curiously enough, Weierstrass himself believed in potential infinity.
Founded by Frege, logicism tried to reduce mathematics to being a subset of logic, with the term initially coined by Bertrand Russell but then spread in the late 1920s by Carnap.
Translation by the authors from Piñero (2014).
Founded by Heyting’s mentor, L.E.J. Brower, famous for his fixed point theorem, intiutionism can be seen as a part of constructivism, which came from Kronecker’s work. Fundamentally intuitionism rejects the law of the excluded middle. While it was viewed at this conference that it was defeated by formalism, many have since further developed and advocated the intuitionist approach (Kleene and Vesley 1965).
Einstein (1921) reinforces this aspect by saying “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality”, to mean that mathematics is useful but it hides no truth of the real world.
In modern formalism, everything resolves around the notion of demonstration, as well as the need to empty statements of all semantic content to develop theories at the syntactic level. Axiomatics and finitism are still present but some distinctive traits have been relaxed (Lolli 2002): some post-Hilbert streams of formalism do not consider the Principle of Non-Contradiction as indispensable because Gödel proved that coherence cannot be proved from the inside of the formal system. What matters is that formal theories are decidable, without possibility of syntactic errors but, at the same time, without the ambition of providing any absolute certainty about results.
Debreu’s Bourbakist mentor Cartan (1943) wrote: “The miracle of science is that we can build an abstract mathematics that can be lately effectively applied to the laws of nature”. (Translation by the authors of the quotation from Cartan (1943) reported in the Italian edition of Ingrao and Israel (1991) on page 269.) Furthermore, Bourbaki (1949) clarifies the position: “Why do such applications [of mathematics] ever succeed? Why is a certain amount of logical reasoning occasionally helpful in practical life? Why have some of the most intricate theories in mathematics become an indispensable tool to the modem physicist, to the engineer, and to the manufacturer of atom-bombs? Fortunately for us, the mathematician does not feel called upon to answer such questions, nor should he be held responsible for such use or misuse of his work”.
While Debreu allowed the use of Cantorian infinity in proofs, his (and Arrow’s) version of GET assumed a finite space of agents and commodities. These limits would later be expanded. Aumann (1964) would allow for a continuum of agents to prove directly equivalence of game theoretic core and competitive equilibria for many agents. Bewley (1972) assumed a continuity-imposing Mackey topology to allow for considering an infinite set of commodities. Fishburn (1970) and Kirman and Sondermann (1972) considered Arrow’s impossibility theorem with infinite agents, finding that results at the finite level did not carry over straightforwardly to the infinite level. Sometimes assuming actual infinity strengthens conventional results; sometimes it does not.
Probably the greatest difference between formalism and intuititionism is that the latter allows for excluded middle, i.e., statements to be both true and false simultaneously. Thus in modern nonstandard analysis (Robinson 1966) that allows for infinite real numbers and their infinitesimal reciprocals, those infinitesimals are both equal to zero and not equal to zero (Hellman 2008; Rosser Jr 2012).
While logicism accepts the law of the excluded middle, it does not demand bivalence, allowing for systems of many valued logics such as those based on Boolean algebra (Rosser and Turquette 1952).
The notions of syntactic consistency, syntactic completeness, semantic consistency and semantic completeness, we refer to are those reported in Berto (2008).
We recognize that, increasingly, parts of physics have come to resemble economics in not being easily tested empirically; this is most seriously the case with string theory, which at the current time remains an untested purely theoretical model competing with various alternatives, with some physicists noting the similarity to the situation in theoretical mathematical economics (Smolin 2006).
Both economics and physics can be pure (formal or theoretical) or applied (or empirical). Theoretical and applied physics consider phenomena as “matters of fact” and share the same questions to understand “how the world works”. Mathematical models in physics have a lot of physical content because the theoretical and applied plane of the discourse intersect. Mathematical models of applied or empirical economics do not have axioms (apart from instrumental ones implied by the mathematical technique involved) and have a lot of economic content. Mathematical models of formalistic theoretical economics have many axioms that formalize economic notions into mathematical notions, so transforming the economic problem into a mathematical problem. Formal and applied economics do not share the same questions: the latter is mainly concerned with describing or explaining “how the world works”, while the former is mainly interested in explaining “why the world should work as we want it to be”. Therefore, mathematical models of formalistic theoretical economics have a lot of mathematical content, while the economic content is less or not relevant.
The standard set is usually called ZFC, for “Zermelo-Frankel-Choice” axioms (Kleene 1967). Competing forms of mathematics may not assume all those axioms, as is the case with intuitionistic mathematics already mentioned.
Of course, not all economists follow this principle. Nevertheless, many agree that the economic discourse is more scientific if formalistically spoken with the language of mathematics, seemingly unaware of the consequences in terms of consistency and incompleteness.
Consider having isolated all and only the primitive entities in which we are interested, for instance prices and action plans in any given place and time, all the functions appropriated to represent exchange preferences and, moreover, being able effectively to compute the values of all the functions at any value of their arguments. Then consider the representations of the general economic equilibrium according to the specification of Debreu: a finite number of goods and a finite number of individuals, each choosing the best action plan in terms of production, if a firm, or consumption, if a household, all defined in a convex and closed space-time on the multi-dimensional field of real numbers. The representations may be exhaustive.
The proposed statements consider “consistent” as synonymous with “coherent” and they are retrieved from the Stanford Encyclopedia of Philosophy at https://stanford.library.sydney.edu.au/archives/spr2014/entries/goedel-incompleteness (retrieved in February 2017). For a rigorous logic-formal explanation of such theorems, the interested reader is addressed to Smorynski (1977) and Beklemishev (2010).
In Gödel’s original version, he was unable to show that simple consistency implied incompleteness within a sufficiently effective system but had to assume the specific ω-consistency. Rosser’s “trick” was to use the “Rosser sentence” that “If this sentence is provable, there is a shorter proof of its negation.” This allowed for showing the neat connection between simple consistency and incompleteness, despite the apparently more complicated form of the sentence. The combined theorem is more widely known in Europe as the “Gödel-Rosser Theorem” than in the United States, even though discussions there are based on its results, which also more readily link to the discussions regarding computability raised by Turing in connection with the incompleteness theorems.
A classic example of this is the Cretan Liars Paradox: “All Cretans are liars, and I am a Cretan,” with Gödel’s own version of this informally being “This sentence is not provable.” Such neverending self-referencing loops are tied to the halting problem of computer science that is at the heart of the problem of non-computability as posed by Turing (1936). This is the deep link between computational complexity and the incompleteness theorems, and Kleene (1967) provided a Turing machine-based proof of Rosser’s version of the incompleteness theorem.
The combined reading of results due to Sonnenschein (1972, 1973), Mantel (1974) and Debreu (1974) is known as the SDM Theorem; see also Arrow and Intrilligator (1982), Kirman (1989) and Rizvi (2006). As formulated by Debreu, the SMD Theorem states that, for a continuous, homogeneous of degree zero function, with Walras’ Law holding, there exists an economy with at least as many agents as goods for which the function is an aggregate demand function with prices bounded away from zero. In short, aggregate demand based on microfoundations is highly arbitrary and variable, only loosely bounded in form. That is to say, “there are no restrictions on market demand functions if no restrictions are placed on agents’ preferences” (Shafer and Sonnenschein 1989) but, as Kirman (2004) writes, “there is no hope of a general result on stability since the only conditions on the aggregate demand excess demand function that can be derived from strengthening the assumptions on individual preferences are the four that we gave above”. Therefore, the SMD Theorem reveals a limitation to the general equilibrium analysis, that is, standard axiomatic restrictions on individual behavior and preferences do not guarantee the stability of equilibrium. Blume and Durlauf (2001) put emphasis on the semantic point of view, concluding that, due to the SMD theorem, “the theory, at least as classically conceived, is incomplete as a way of understanding economic phenomena”. Finally, a further limitation this theorem puts forth is that aggregate data are useless to test microfounded theories, as Kirman and Koch (1986) proved the SMD Theorem still holds even with identical individual preferences and the same share of total income.
For a comprehensive exposition of all such topics, the reader is addressed to Arrow and Intrilligator (1982).
Thus despite its repeated empirical failures (Phuong et al. 2016), more axioms are added to the flawed DSGE theory, much as epicycles were relentlessly added to the failed Ptolemaic theory.
Winrich (1984) argues that self-reference is the cause of incompleteness of neoclassical economics. Smith et al. (1999) extend the problem to ordinal utility theory, Bayesianism and decision theory. See also Bookstaber (2017) for a less formalized discussion about the implications of self-reference in economics.
A broader effort to establish a school of thought on “computable economics” that studies the limits of economic theory based on computability issues arising from the Incompleteness Theorem and issues associated with Turing machines has been due to Velupillai especially, who provides excellent broad overviews (Velupillai 2000, 2009). However, it should be noted that a part of his program is to try to reconstruct economic theory based on non-classical mathematical foundations using constructivist approaches such as intuitionism that assume fewer of the classical axioms than do the works cited here that also find non-computability limits profoundly associated with various models of economic equilibria. While we view this research program with interest (Rosser Jr 2012), we do not pursue its implications for this discussion further here.
Over-simplification means that the general axiomatic conditions, useful to prove a given statement or to solve a certain problem (e.g. the existence of equilibrium), may need to be “restricted” to simplify further the world in order to prove a new statement or to solve a new problem (e.g. concerning the structure of the equilibrium set) that cannot be proved or solved within the original axiomatic set. In a sense, over-simplification may lead to the formal construction of an artificial world where new properties hold while being too far from being realistic or applicable.
Quotation of Debreu reported in Bryant (2010).
See chapters XI and XII of Ingrao and Israel (1991) for a narrative and historical review.
As Ingrao and Israel (1991) report, Hildebrand said that Debreu considered that the axiomatic development of the existence theory was not sufficient for a theory of uniqueness and stability. Debreu was obviously concerned with these issues but not from the global point of view; rather he understood that uniqueness and stability could be entangled as local properties of an equilibrium set.
Dierker (1974) showed that, if they are transversal the number of equilibria is odd.
In game theoretic models, the conditions can be far more stringent, with Maskin and Riley (2003) spending over two decades to work these out for sealed-bid auctions. Of course, in the simple two good Walrasian model, the gross substitutability condition also provides stability. Since the famous three good example of Scarf (1960), it has been known that uniqueness does not guarantee stability for more than two goods for such models.
It is a “meta-example,” which represents the evolution of the axiomatics of the GET: a historical narrative about how the axiomatic set of the GET evolved -ideally, say, from AX for existence, to AX1 for uniqueness and AX2 for stability- can be found in Ingrao and Israel (1991), mainly chapters X (existence), XI (uniqueness) and XII (stability).
Notice that this implies passing from the actual infinity of the GET to the potential infinity of the DGSE. In the GET, the equilibrium is a sequence in the topological space-time; it realizes all at once, forever and everywhere instantaneously, i.e. not progressively. On the other hand, in the DGSE approach, the equilibrium dynamically changes, progressively evolving along some optimal path, identified by means of recursive methods such as dynamic programming with infinite time horizon. Several distinctions can be found between GET and DSGE, among others: (i) the GET is microeconomics while DSGE is macroeconomics, (ii) GET is deterministic while DSGE is stochastic, (iii) GET admits actual infinity while DSGE admits potential infinity, (iv) in GET rationality of agents is solely related to the maximization principle while in DSGE rational expectations are introduced, (v) GET is uniquely concerned with the existence of equilibrium while DSGE also deals with uniqueness and stability, (vi) in GET the price vector clearing the market (i.e. the equilibrium) comes as a theorem while in DSGE it plays as an hypothesis which, if assumed (e.g. as an axiom), improves the capability of the models to match with data.
See the “Kornai’s vs. Hahn’s Critiques” in Móczár (2017).
This set of categories has been discussed with Federico Giri: the scheme GET-DGET/CGE-DSGE is of the authors.
We note that there have been efforts to introduce heterogeneous agents into DSGE modeling. Curiously this often takes the form of assuming a continuum of agents a la Aumann in an interval that varies on some characteristic (Krusell and Smith 1998). This leads to similar dynamics of a representative agent model, with the interval effectively acting as a single agent without any actual interactions between the agents.
Let us now put some caveat on this aspect quoting from Friedman (1953): “The hypothesis is rejected if its predictions are contradicted; it is accepted if its predictions are not contradicted […] Factual evidence can never ‘prove’ a hypothesis; it can only fail to disprove it, which is what we generally mean when we say, somewhat inexactly, that the hypothesis has been ‘confirmed’ by experience”.
In his detailed review and systematization of the reasons that made general equilibrium (still) dead, Ackerman (2002) explores some ‘Alternatives for the Future’ and finds the ABM as a valuable route to simulate worlds dominated by chaos and complexity. Bookstaber (2017) supports the idea of ABM as a new paradigm: “It [ABM] operates without the fiction of a grand leader, or a representative consumer or investor who is unerringly right as a mathematical model can dream. […] agent-based economics arrives ready to face the real world, the world that is amplified and distorted during times of crisis. This is an new paradigm rooted in pragmatism and in the complexities of being human”.
The contrast between evolutionary and equilibrium approaches was at the foundation of Veblen’s (1898) critique of neoclassical equilibrium theory and his proposal to replace it with the study of the evolution of economic institutions.
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Acknowledgements
The authors are grateful to the reviewers for detailed reports that improved the quality of the paper. The authors thank the participants to the CRISIS conference (Ancona, September 2016) for comments, and special thanks go to Alan Kirman, Duncan Foley, David Colander, Pedro Garcia Duarte, Gian Italo Bischi, Antonio Palestrini and Federico Giri for fruitful discussions. Mauro Gallegati (Polytechnic University of Marche, Ancona, Italy) gratefully acknowledges the support from the European Union, Seventh Framework Programme FP7, under grant agreement FinMaP n0: 612955. Simone Landini (IRES Piemonte, Turin, Italy) and J. Barkley Rosser Jr. (James Madison University, Harrisonburg, Virginia, USA) gratefully acknowledge the support from their institutions for the participation to the conference. The opinions of the authors are their own and do not involve the responsibility of their institutions.
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Landini, S., Gallegati, M. & Barkley Rosser, J. Consistency and incompleteness in general equilibrium theory. J Evol Econ 30, 205–230 (2020). https://doi.org/10.1007/s00191-018-0580-6
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DOI: https://doi.org/10.1007/s00191-018-0580-6
Keywords
- Axiomatic formal system
- Consistency
- Incompleteness
- Gödel-Rosser’s theorems
- General equilibrium theory
- Agent-based modeling