Title

General Equilibrium Model and Set Theoretic
Finiteness

Author(s)Urai, Ken; Murakami, Hiromi

Citation

大阪大学経済学. 64(2) P.259-P.267

Issue Date 2014-09
Text Version publisher
URL

https://doi.org/10.18910/57059

DOI

10.18910/57059

rights
Note

Osaka University Knowledge Archive : OUKA
https://ir.library.osaka-u.ac.jp/repo/ouka/all/

Osaka University

Vol.64 No.2

OSAKA ECONOMIC PAPERS

September 2014

General Equilibrium Model and Set Theoretic Finiteness
Ken Urai † and Hiromi Murakami ‡Abstract
The axiom of infinity in the ZF set theory is usually considered essential for Brouwer’s fixed
point theorem and the proof of the existence of economic equilibria. In this paper, the meaningsof the infinity axiom are reconsidered from the view point of the minimal requirements forthe general equilibrium theory. We provide several tools and conditions under which a settheoretically finite setting is sufficient, at least for an elementary equilibrium existence argumenton the static general equilibrium model.
In view of the social science, the purpose of general equilibrium theory is to describe the whole
structure of the human society based on the price and the market mechanism. However, there
exists a classical problematic feature relating to the objectivity, i.e., the introspection problem for
us to describe the world including ourselves, which also brings about important questions about
the true sense of our individual rationality.
In order to consider about the introspective feature of the social science and the humanrationality under an economic model, there may exist two approaches. One is to attempt todescribe our true rationality into the model, so that we may treat directly the whole society as thetotality of such rational individuals. The other approach is giving up to define the true sense of
rationality (a general intelligence of human) and, instead, to seek a minimal requirement for our
rationality to construct and recognize the world described by a certain economic model. In thispaper, we base our argument on the second approach, and show that the static general equilibrium
theory such as Debreu (1959), could essentially be treated under a finitistic mathematicalargument.
JEL Classification: C62, D51
Keywords: ZF Set-Theory, Axiom of Infinity, Barycentric Subdivision, Sperner’s Lemma,
General Equilibrium Theory†‡

Graduate School of Economics, Osaka University, E-mail: urai@econ.osaka-u.ac.jp
Graduate School of Economics, Osaka University, E-mail: murakami@math.econ.osaka-u.ac.jp

－ 260 －OSAKA ECONOMIC PAPERS

Vol.64 No.2

1 Introduction
In the view of social science, the purpose of general equilibrium theory is to describe the whole
structure of human society based on price and market mechanism. However, there exists a classicalproblematic feature relating to objectivity (e.g. Weber 1904), i.e. the introspection problem for us todescribe the world including ourselves, which also brings about important questions about the true
sense of our individual rationality.
To consider the introspective feature of social science and human rationality under an economic
model, there exist the following two approaches. One is to attempt to describe our true rationality inthe model, so that we may directly treat the whole society as the totality of such rational individuals. In
such a case, we always have to point out the incompleteness of such a description of our real society
and rationality.1 The other approach is giving up defining the true sense of rationality (a general
intelligence of humans) and, instead, to seek a minimal requirement for our rationality to construct andrecognize the world described by a certain economic model. For the latter attempt, the implication, thesoundness, the consistency, etc. of the model would be evaluated under our rationality, so the totalityof the theory describing a certain model is important. Such a soundness is necessary for the theory to
be plausible for agents described in the theory from an introspective viewpoint.
Under the second approach, we have to admit that an economic theory is always related to certain
kinds of value judgments, especially epistemologic and/or methodological ones, which will construct
the fundamental relation between economics and ethics, and also provide the meanings, roles, andlimits of economics.
In this paper, we base our argument on the second approach, and show that the static generalequilibrium theory as in Debreu (1959) could essentially be treated under a finitistic mathematicalargument. From the ZF set-theoretic viewpoint, the theory without using the infinity axiom iscomplete (e.g. Urai 2010), so our argument gives an affirmative answer to the question of whether themodel of static general equilibrium, even when we allow each agent of it to have introspective viewsof the world to which they belong, is possibly complete as a model of the society, at least from the ZF
meta-theoretic viewpoint.
2 Sperner’s Lemma
Begle (1950a, 1950b) gives an algebraic argument on abstract simplices and barycentric
subdivision, as well as the Vietoris Mapping Theorem and the fixed-point theory. These arguments
could be utilized to give a certain kind of purely algebraic development for the abstract reconstructionof the general equilibrium theory.
In this paper, we do not use such a Čech-homological approach since our main concern in thefollowing is restricted on the finiteness in the sense that the infinity axiom of the ZF theory of sets is
not necessary for the development of a theory. Our argument is an extension of the setting in Nikaido(1968) for simplices of order1

to the cases with a general complex and simplices of order. The

This is a similar result to the Gödel-type incompleteness theorem. We usually have to admit that such a description ofindividual rationality must be introspectively incomplete as long as it is consistent. See Urai (2010, Chapter 9).

September 2014

－ 261 －

General Equilibrium Model and Set Theoretic Finitenesspurpose of this section is as follows.
We show that the concept of barycentric subdivision and Sperner’s lemma is possible to be
proved without using any concepts based on the infinity axiom.
2.1 Abstract Simplex
be

Let
(

points ofpoints. For example, if we regard that),will construct

are the-dimensional simplex. However, we do not takesuch a concrete way. In the abstract sense, we consider that these points are in general position and-simplex, which are named

construct abstract. Since these points are associated with

fixed numbers, they are oriented. When we emphasize the orientation, we express these points as.
2.2 Simplex and Vertex
be the oriented complex that consists of all subsimplices of abstract simplex

Let.

(Definition 1: Increasing Sequence)The-increasing sequence (, ofthe sequence,subset of), which is made of(points,non-empty subsets of, is, such thatis a proper).

(Definition 2: Simplex and Vertices of order 0)
the vertex of order-simplex of order , and we call each ofis theAbstract simplex. In addition, we call an abstract simplex,

vertices of order

(

),

as
, which consists of, an-simplex of order .

(Definition 3: Simplices and Vertices of order )Let

which consists of. We callan)-increasing sequence,, constructing an-simplex of orderas vertices of order , and the abstract simplex

asbe greater than or equal to . We consider the (vertices of order,-simplex of order .

Lower-right shaded portion, -simplex of order ,Lower-left shaded portion, -simplex of order ,Figure 1 : Simplices and Vertices of order－ 262 －OSAKA ECONOMIC PAPERSAn

is an abstract subsimplex of an-simplex of order,

the above definitions, each number of elements ofvertices of orderthe set of

Vol.64 No.2

-simplex of order,,, is not unique. If we fix one
vertices of orderon the sets that are made fromcould induce
kinds of

there exist

kinds of. In the following, we fix

vertices of order

and

and

,

, and the above definitions are made from

to

and

belongs to a different rank of sets (since they can become

are completely different, and we can easily make all of the sets
, the vertices of order

totally-ordered. Especially for) to. Hence, with orderthat appear in the above are only finite, and as long as ordernon-empty sets). Thus,order-simplex of order. Then, the, and the above inductive definitions merely imply that we omit theare different, each ofvertex of orderand focus, . That is, we do not need the Peano axioms.

All vertices of order
order-simplex of order-simplices.

Consider a sufficiently large natural number
descriptions of order , ,,.

, the number of such increasing sequences

is exactly the same with the number of permutations of-simplex of order. When we follow

becomes, we could always define. Then, we could consider the mapping,, from, the(the set of vertices of(the set of vertices of order ).

(1)

(

).

Based on this mapping, we could say thatand the above total ordering can be.

considered as typically given on
2.3 Oriented Complex. For each

Let us define, assume that the above total ordering is defined so

is possible. We definethat the identification

as follows.

(2)
It is easy to verify that
Let us define

is an oriented complex with the set of vertices,. For each, assume that the above total ordering is defined so

is possible. We definethat the identification.2

as follows.

(3)
It is easy to verify thatis an oriented complex with the set of vertices,.3

2.4 Carrier
Complex
order
=
2
3

,

represents orderbarycentric subdivision of) is the set havingas-fold parentheses. We write the set of all subsets ofand the set of all subsets of

See Urai and Murakami (2014).
See Urai and Murakami (2014).

. The -simplex (vertices ofas. Thereafter, we could make

September 2014

－ 263 －

General Equilibrium Model and Set Theoretic Finiteness,,, and we have

. Let us define the symbolas the relation ‘‘is an

as the relation ‘‘is an element of an element of anelement of an element of’’ a set, and the symbol. Then, we haveelement of’’ a set,(4)
We could identify

as

and

. (For any

denoted byasis called the carrier of. The set,

is a subset of-simplex of order , i.e.

and is.) In general, for,(5)
is called the carrier of
constructing.(is the union of carriers of vertices of-simplex of order.) Now, we can prove the next theorem, which is a complete finitistic algebraic version

of Lemma 4.2 in Nikaido (1968, p.57). Note that in this theorem, we do not use any topological notion
like the relative boundary of the original simplex.
Theorem 1: For, ()-simplex of(), we have either of the following

conditions.
,(i) ‌When(ii) Otherwise,

is a unique subsimplex of an-dimensional simplex of order .

is a common-subsimplex of exactly two-dimensional simplices of order .

is a subsimplex of a certain

is, by definition, an

Proof:(I) When,-increasing sequence of vertices of order . Hence,order vertices,
Case (i): Since to construct anof theis an-increasing sequence constructed by.
-increasing sequence,different vertices are necessary, the vertex

is unique. Hence, the method for constructing anthat does not belong tosequence from the, and-increasing-increasing sequence is also unique, i.e. to add-increasing sequence; so

is a subsimplex of the uniqueat the end-simplex represented by the-increasing sequence.

uniqueCase (ii): Since
sequence,

contains all order, representing, is necessarily equal toelements, there exists
number of elements of, where-increasing. Since this set has, is . Then, for such anminus the-increasing sequence, the number-increasing sequence from it is two, i.e. to insert

or

and

is

exactly a common subsimplex of these two(II) Let us consider the case with, and we show for the case with-increasing sequence,, of thesuch that the number of elements ofof methods for constructing anin the place between

vertices, the last entry,, where. Therefore,-simplices of order .
. We assume that the assertion of the theorem holds for. First, note that since every, of ordercan be identified with anvertices constructing an-simplex

－ 264 －OSAKA ECONOMIC PAPERS

of order, the last entry of such a sequence,vertices constructing anorder-simplex of order,-increasing sequence,, of order-simplex of order, the last entry of such a sequence,vertices constructing an-simplex of order

exists at least one abstract,

vertices, must contain-simplex of order. Thus, there, whose carrier is a subset oftimes and obtain a sequence of

can repeat such a process fordifferent. Hence, there exists at least one abstract. Then again, sincedifferent orderat least, must contain at least-simplex of order, whose carrier is a subset ofcan be identified with anconstructing an

Vol.64 No.2

-simplices,

. We

,

,

,

. We call such a sequence a resolution ofsuch thatto simplices (and vertices) of orders. The resolution of

may not be uniquemust bebut always exists. It follows, in particular, that the number of elements of the carrier of,greater than or equal to the number of elements ofCase (i): Consider a resolution of,,is unique. Let, the vertex of orderthat does not belong tobe such a unique vertex of order, so the resolution of-increasing sequence of orderof ordervertices,-increasing sequence by adding,,. Then,isis uniquely determined. Moreover, for, the possibility to extend it to theis unique since in that case,. Thus, the resolution ofmust be equal toof orderis also uniquely. We h a v e s h o w n t h a t f o r a n y r e s o l u t i o n o fdetermined as
,

,

,

,

-simplex of order, is an abstract

, the

,

-subsimplex of a certain, since the carrier of anis, the order-simplex of order. Therefore, if the possibility of such

form,

,

is unique. Note that the resolution of

possibility of the last two entries together withconstructed such that given an,

and the number ofmust be greater than or equal toresolution,,,. Sinceelements ofthe.

-simplex is always equal to

together withhaving theis unique, and if, the possibility of

in a certain resolution ofis also unique. By repeating the same argument, it follows that the resolution ofand the possibility of the vertex,
is an abstract subsimplex of, is unique.

Case (ii): We may represent
is anform a certain-simplex of orderof a certain

as an abstract-increasing sequence of order-simplex of order(ii-1): When the last entry,
the only, under which-simplex of order-subsimplex, wherevertices,, which. Let us consider the next two cases.

, is equal to the set,
under which
since this is the only

. In this case,is obtained as an abstract-simplex of orderis-subsimplex

whose set of vertices

includes all. Hence, the argument for case (ii) in (I) would be sufficient to confirm that

there are exactly two-simplices of ordersatisfying thatis an abstract-subsimplex

September 2014

－ 265 －

General Equilibrium Model and Set Theoretic Finitenessof it.
(ii-2): When the last entry,
is an, is not equal to the set,

. In this case, since-increasing sequence, among the order

vertices,. Without loss of generality, we can rename theexists exactly one vertex that does not belong to. Hence, we haveunique vertex as

is an abstract, thereand-subsimplex of order. Then,,such that. Therefore, from ourwhere the last equation follows from the fact that,assumption that the assertion of the theorem holds foris a common abstract

subsimplex of exactly two-simplices of order. Thus, one is, of course,

and there exists another-simplex of order,vertex,order. Except for these two simplices, there is no-simplex of ordersatisfying thatthat for, wherethat

forms an

is an abstract subsimplex of it. It follows, the possibility of, such-increasing sequence of vertices of order-simplex of orderwhich in turn form a certain. Of course, this means that the

to be an abstract subsimplex of a certain-simplex,orto the two cases where, of orderis restricted.

(III) By (I) and (II), we have shown that the assertion of the theorem holds for all.

By using the above theorem, we formulate and prove Sperner’s Lemma about

for allVertex assignment of order

is a function,, i.e. to assign a vertex

certain,, is restricted to the two cases whereor
possibility for,

together with another.

, which gives for eachof orderabelonging to the carrier for each vertex

oforder .
Then, for each-simplex of order

when

(

),
is equal tois equal to, -simplex,(

for eachwhenotherwise. When) has an odd number of regular-simplices (hence,-simplex).

Proof: Fix the order, , and the complex,
Denote by, as, and asis called regular.

Theorem 2: (Sperner’s Lemma)
at least one, we define as

the number of-simplex,. Let us denote by

the number of regular-simplices whose carrier is equal to, denote by

the number of regular subsimplices of-simplices.
. Moreover,. Then, exactly

the same argument as with Nikaido (1968, p.61, Lemma 4.3) for the same proof is applicable. Thus,
we will omit the proof in this paper.4

4Also for details of this proof, see Urai and Murakami (2014)

－ 266 －OSAKA ECONOMIC PAPERS

Vol.64 No.2

3 Economic Equilibrium
In this section, we provide a certain condition under which the economic equilibrium arguments
need not necessarily be constructed on the commodity spaces of the real field.
Assumption 1: There is a positive number

such that any amount less thandoes not haveessential meaning for our equilibrium concept (as well as all concepts necessary to grasp and/or
measure the amounts in defining the equilibrium).
Such an assumption would enable us to define the economic equilibrium as the status that market
excess demand is not greater than

. Moreover, assume that there is a continuous excess demand

function (as an instrument in the meta-theory), and we can takechange less thanThen, if

uniformly so that any pricewould not cause the change of excess demands greater than .

is not a price that leads to an equilibrium as we explained in the above recognizable

sense, it would be possible that there exists no full-labeled simplex in a certain(-neighborhood) of such a price, under the standard labeling for points in the price domain based on the excess

demand function in the meta-theory.5 Therefore, by considering the above Sperner’s Lemma, Theorem
2, the full-labeled simplex when the diameter of the simplex is sufficiently smaller than, every pointin the neighborhood represents the economic equilibrium price in the above meaning.
Although we have only discussed the existence of equilibria, it would be possible to consider the
first and second fundamental theorems of welfare economics in a similar way. These problems require
further research.
REFERENCES
Begle, E. G. (1950a): “The Vietoris mapping theorem for bicompact spaces,” Annals of Mathematics
51(3), 534-543.
Begle, E. G. (1950b): “A Fixed point theory,” Annals of Mathematics 51(3), 544-550.
Debreu, G. (1959): Theory of Value. Yale University Press, New Haven, CT.
Nikaido, H. (1968): Convex Structures and Economic Theory. Academic Press, New York.
Nikaido, H. (1970): Introduction to Sets and Mappings in Modern Economics. North-Holland
Publishing Company, Amsterdam, London. Transrated by Kazuo Sato.
Urai, K. (2010): Fixed Points and Economic Equilibria vol. 5 of Series of Mathematical Economics
and Game Theory. World Scientific Publishing Company, New Jersey/London/Singapore.
Urai, K. and Murakami, H. (2014): “Keizaigakuteki kinkou riron ga yugensei gainen no motode
tojitamono to naru kanousei ni tsuite,” Doshisha shogaku 66(1), 384-396. (Japanese: The
Possibility of Economic Equilibrium Theory to be Closed under the Concept of Set-theoretical
Finiteness.)
Weber, M. (1904): “Die“Objektivität” Sozialwissenschaftlicher und Sozialpolitischer Erkenntnis,”5See, for example, the standard proof of an equilibrium existence in Nikaido (1970, chapter 10).

September 2014

General Equilibrium Model and Set Theoretic Finiteness－ 267 －

Archivfür Sozialwissenschaft und Sozialpolitik 19, 22-87. English translation: “Objectivity” insocial science, in The Methodology of the Social Sciences (Edward A. Shils and Henry A. Finch,tr. & ed.)