An axiomatic approach to first law thermodynamics

MICHAEL A. DAY

AN AXIOMATIC APPROACH TO

FIRST LAW THERMODYNAMICS*

I. INTRODCCTION

Being a macroscopic theory of physics, thermodynamics makes no hypo- theses about the structure of matter, and deals with physical systems that are large compared to atomic dimensions. Thermodynamics seeks to estab- lish relationships among macroscopic parameters which represent such observable properties as pressure, volume, temperature, mole concentrations, heat capacities, etc., when the system under consideration is in one of its equilibrium states or is undergoing a process which involves equilibrium states. Also, thermodynamics is very general in scope; however, it has defi- nite limitations to its predictive and explanatory power. For example, thermodynamics can predict many relationships between the properties of a physical system, but not the actual magnitudes of these properties.

Today thermodynamics is usually set forth as being based upon four general laws of nature; and traditionally it has been presented as being based upon two of these laws - the first and second laws of thermodynamics. In this paper, I will examine the first law of thermodynamics using the methods developed by such philosophers of science as Patrick Suppes (1957, 1967), Ernest Adams (1955,1959), Joseph Sneed (1971) and Werner Leinfellner (1965). ‘The kernel of this methodology consists in the following: (a) a definition of a basic set-theoretical predicate which, in some sense, characterizes the formal structure of the law or theory under consideration; (b) a specification of the intended interpretation of the set- theoretical structure defined by the basic set-theoretical predicate; and (c) derivations of theorems and specifications of further definitions which involve the basic set-theoretical predicate. Further, this paper will consider the first law of thermodynamics from a phenomenological point of view, in the sense that the ‘empirical primitives’ of this formulation will only require an understanding of the following notions: thermodynamic system, equilibrium state, process, adiabatic process (i.e., a process in which no heat is transferred between the system and its surroundings), and work.

Journal of Philosophical Logic 6 (1977) 119-134. All Rights Reserved. Copyright 0 1971 by D. Reidel Publishing Company, Dordrecht-Holland.

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II. INFORMAL PRESENTATION OF THE FIRST LAW OF THERMODYNAMICS

To say what the first law is and what is its content in a formal way, presupposes at least an informal conception of what the first law is and its content. For the sake of this paper, let us consider the following quotations from three physicists as putting forth the essential content of the first law, and then attempt to formulate this content in a more precise manner.

Energy and First Law of Thermodynamics. - There exists a function of the state of any thermodynamic system called its ‘energy’ (or ‘total energy’) with the following property: In any process the sum of the heat 4 absorbed by the system from its surroundings and the work w done on the system by its surroundings is equal to the increase of the energy E of the system. For any infinitesimal process we write this

(1) q+w=dE,

and for any finite process

(2) q+w = AE,

where the operator A denotes the excess of final over initial value. . . . The law con- tained in (1) and (2) is known as the ‘First Law of Thermodynamics’. (Guggenheim, 1933, pp. 3-4.)

The internal energy of a body (or of a system of bodies) in a given state is measured by the mechanical work required to bring the body into the given state by means of an adiabatic process. The work required is independent of the intermediate states through which the body may pass and depends only upon the initial and final states provided that the processes considered are adiabatic. The internal energy is, therefore, a function of the state and is uniquely defined except for an additive constant which is determined by the internal energy ascribed to the initial state. (Wilson, 1966, p. 72)

One more aspect of the first law should be mentioned at this point. When a system is carried through a cyclic process (not necessarily reversible) r/, = U, and Q = W. That is, the net heat flowing into the system equals the net work done by the system. This means that it is impossible to construct a machine operating in cycles which, in any number of complete cycles, will put out more energy in the form of work than is absorbed in the form of heat. A machine which would do this is called a perpetual motion machine of the fvst kind . _ . . The first law is sometimes stated, ‘A perpetual motion machine of the first kind is imposstble’. (Sears, 1953, p. 45)

For other sources concerning the first law of thermodynamics and thermodynamics in general see Callen (I 960) Giles (1964), Landsberg (1961), and Zemansky (1968).

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III. FORMALIZATION OF FIRST LAW THERMODYNAMICS

This formalization is based upon six primitive notions which are denoted by ‘S’, ‘C’, ‘P, ‘A’, ‘D’, and ‘IV’, and their intended interpretations are as follows.

(1)

(2)

(3)

(4)

(5)

(6)

S is a set whose only member is a thermodynamic system.

C is a function from S taking on the value of an open region of Re” (where Re” = Rer x Rez x . . . x Re,). C(s) is to be interpreted as representing the possible equilibrium states of the thermodynamic system, or more informally, as representing the constraints on the thermodynamic system. Also, the integer n is such that the equilibrium states of the system can uniquely specified in n parameters.

P is a set of possible processes (not necessarily reversible) which the thermodynamic system can undergo. Furthermore, each member of P is a process which begins with an equilibrium state and ends with an equilibrium state.

A is a subset of P such that each member of A is an adiabatic process.

D is a function from P taking on as values ordered pairs from the set C(s) x C(s). Now if D(p) = (c, c’), then c and c’ are ordered n-tuples of real numbers which are to be interpreted, respectively, as representing the initial and final equilibrium states of the process p.

W is a function from P taking on as values real numbers such that W(p) is the real number which represents the net work done by the thermodynamic system in the process p with respect to some futed units of measurement.

Let us now set forth the basic set-theoretical predicate of this formalization.

DEFINITION 1. Let r‘ = (S, C, P, D, W, A 1. Then r is ajh-st law thermodynamic structure (henceforth abbreviated as ‘FLTS’) if and only if L’ satisfies conditions Dl-D6.

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Di.

D2.

S is a unit set.

C is a function whose domain is S and C(s) is a subset of Re” (where )I is a positive integer) such that C(s) is an open region.’

1>3.

D4.

P is a nonempty set.

D is a function whose domain is P and whose range is C(s) x C(s).

DS. W is a function whose domain is P and whose range is the set of real numbers.

D6. A is a subset of P such that (a) for any c’, C” E C(s), there exists an Q’ E A such that

either II = cc’, c”) or D(a’) = Cc”, c’); (b) for any a’, a” E A, if D(d) = D(a”) then W(a’) = W(a”);

and (c) for any c’, c”, c”’ E C(s) and for any a’, a” E A, if

D(Q’) = (c’, c”) and D(a”) = cc”, c”‘) then there exists an a”’ E A such that D(a”‘) = (c’, c”‘) and !#(a”‘) = W(d) + W(a”).

Now under the intended interpretation, we notice that D6 contains what might be called the ‘empirical content’ of the first law of thermodynamics. For under the interpretation, D6 requires the following conditions to be satisfied: (a) for any two equilibrium states of the system, there must exist an adiabatic process which connects these two states of the system, but the direction of the process (i.e. which state is the initial state and which state is the final state) is unspecified; (b) for any two adiabatic processes which have the same initial states and the same final states, the net work done by the system in one of these processes is equal to the net work done by the system in the other process; and (c) if there exist adiabatic processes such that the final state of one process is the initial state of another process, then there exists an adiabatic process which connects the initial state of the former process with the final state of the latter process, and the net work done by the system in this process is equal to the sum of the net works done by the system in the first two processes.

AXIOMATIC APPROACff TO THERttfODYNAMICS 123

Let us now consider certain theorems and definitions which will be based around Definition 1. The most important aspect of this discussion will be the theorem which asserts that for any r which is a first law thermodynamic structure, there exists what we will call an internal energy structure with respect to r. First of all, we prove the following theorem which will be use- ful in proving other theorems.

THEOREM 1. Let r = (S, C, P, II, IU, A) be a FLTS. Then for any c’, c“, c”’ E C(s) and for any a’, a”, a”’ EA if @a’) = (c’, c”), D(a”) = (c”, c”‘) and II@“‘) = (c’, c”‘), then W(a”‘) = W(a’) + Ma”).

Proof: Assume D(a’) = (c’, c”), f&z”) = cc”, c”‘) and l)(a”‘) = (c’, c”‘). Therefore (by D6(c)), there exists an a”” EA such that O(a”“) = Xc’, c”‘) and W(U”“) = W(u’) + IV@“). Now since D(a”‘) = D(u”“), W(u”‘) = W(u”“) (by D6(b)). Therefore, W(u”‘) = W(u’) + W(u”).

Now we give a definition of ‘cyclic with respect to r’ which according to the intended interpretation is a definition of a cyclic process. Furthermore, we prove a theorem which upon interpretation says that the net work done in any cyclic, adiabatic process is zero. (For notational convenience, we will use the expression ‘D(p)’ to denote the ordered pair which is obtained by reversing the members of the ordered pair D(p).)

DEFINITION 2. Let I‘ = (S, C. P, D: WI A ) be a FLTS. Then p is cyclic with respect to I’ if and only if p E P and D(p) = D(p).

THEOREM 2. Let r = (S, C, P, D, W. A > be a FLTS. Then if p is cyclic with respect to I‘ and p EA, then IV@) = 0.

Pr~f: Assume p is cyclic with respect to 1’ and p EA. Let D(p) = (cl, cl) where c1 and c2 are the appropriate members of C(s). Since p is -- cyclic with respect to I’, D(p) = D(p) = (c, ? cl ). Now let c’ = c” = c”’ = c1 ad 0’ zx a“ I a”’ = p. Therefore (by Theorem I), W(p) = W(p) + W(JJ). For this to be the case, it must be that IV@) = 0.

Next, we prove a theorem which upon interpretation says that if two adiabatic processes are such that one is the reverse of the other with respect to initial and final states, then the net work done by the system in the one

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process is the negative of the net work done by the system in the other process.

THEOREM 3. Let I’ = (S, C, P,D, W,A) be a FLTS. Then ifp,p’ EA and D(p) = D(p’), then IV(p) = - IV(p).

Proof: Assume p, p’ E A and D(p) = D(p’).Let D(p) = (cr , cz) where cl and cz are the appropriate members of C(s). Now D(p’) = (c2, cl >, therefore (by Do(c)), there exists an a’ E A such that I@‘) = IV@) + IV(p’) where D(Q’) = (cr , cl ). But a’ is cyclic with respect to F, so IV@‘) = 0 (by Theorem 2). Therefore, IV@) + IV@‘) = 0, or equivalently IV@) = - IV@‘).

We will now give a definition of ‘an internal energy structure with respect to I”, and then prove that for any I’ which is a first law thermodynamic structure, there exists an internal energy structure with respect to F.

DEFINITION 3. Let F = (S, C, P, D, IV, A > be a FLTS. Then A is un internal energy structure with respect to r if and only if A = 6, C, U) where U is a function whose domain is C(s) taking on real numbers as values such that for any c, c’ E C(s) if p E A and D(p) = (c, c’), then U(c) - U(c’) = W(p).

THEOREM 4. Let I’be a FLTS. Then there exists an internal energy structure with respect to F.

Proof: The complete proof of this theorem is given in the appendix. Below is an outline of that proof.

Step 1. Let A = (S, C, U) where U is the following relation: (1) take any cl E C(s) and let (cl, 0) E U, (2) take each c’ E C(s) such that there exists an a’ EA such that either D(a’) = (cl, c’) or D(a’) = tc’, cl>, and let (c’,- W(~‘))EUifD(a’)=(c~,c’) and let (c’, W(a’))EUifD(u’) = (c’, cl), and (3) allow U to have no other members.

Step 2. Prove that U is a function (a well-defined relation). Step 3. Prove that the domain of U is C(s). Step 4. Prove that for any c, c’ E C(s), if p E A and D(p) = Cc’, c’) then

U(c) - U(c’) = W(p).

Let us now prove a uniqueness theorem concerning the internal energy structures with respect to F. If we refer to the third member of an internal


energy structure with respect to r as an internal energy function with respect to I’, then what the following theorem asserts is that an internal energy function with respect to I’ is unique up to an additive constant.

THEOREMS. Letr=(S,C,P,D, W,A)beaFLTSandA=G,C,U>be an internal energy structure with respect to r. Then 7 = 6, C, U’) is an internal energy structure with respect to r if and only if the domain of U’ is C(s) and there exists a real number a such that for all c EC(s),

U(c) = U’(c) +a.

Proof: Assume y = (S, C’, U’) is an internal energy structure with respect to r. Now, it immediately follows from Def. 3 that the domain of II’ is C(s) Now let c’ be an element of C(s) and cx be the real number such that a = U(c’) - U’(c’). Consider an arbitrary c E C(s). Now there exists a p E A such that D(p) = k, c’) or D(p) = (c’, c) (by w(a)). If D(p) = (c, c’> then

U(c) - Il(c’) = W(p) and U’(c) - U’(c’) = W(j), or

U(c) - U(c’) = U’(c) - U’(c’), or

U(c) = U’(c) + (U(c’) - U’(c’)), or

U(c) = U’(c) + a.

If o@) = cc’, c), then

U(c’) - U(c) = W(p) and U’(c’) - U’(c) = W(p), or

U(c’) - U(c) = U’(c’) - U’(c), or

U(c) = U’(c) + (U(c’) - U’(c’)), or

U(c) = U’(c) + CY.

Therefore, the domain of U’ is C(s) and there exists a real number (Y such that for all c E C(s),

U(c) = U’(c) + CY.

Now to prove the converse. Assume U’ is a function whose domain is C(s) such that for all c E C(s),

U(c) = U’(c) + fY.

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where a is a real number. Now we must show that y = G, C, U’) is an internal structure with respect to r. To show this we must demonstrate that for any c, c’ E C(s), if p E A and D(p) = (c, c’) then U’(c) - U’(c’) = W(p). Now if p EA and D(p) = (c, c’), then

U(c) - U(c’) = iv@), or

(U(c) + a) - (U(c)) + a) = W(p), or

U’(c) - U’(c’) = Iv@).

Therefore, y = tS, C, .!I’) is an internal energy structure with respect to r.

Next, we will give a definition of ‘a heat function with respect to f” which according to the intended interpretation will be a definition of a function which associates with each process the net heat absorbed by the thermodynamic system in that process with respect to the same units of measurement used in determining the work function IV. Also, we will prove a theorem concerning the uniqueness of a heat function with respect to r.

DEFINITION 4. Let T’ = (S, C, P, D, IV, A) be a FLTS and A = (S, C, I/) be an internal energy structure with respect to I’. Then Q is u heat function with respect to r if and only if Q is a function whose domain is P such that for’ all p E P, if D(p) = (c, c’> then Q(p) = CI(c’) - U(c) + We).

THEOREM 6. Let r be a FLTS. Then there exists one and only one heat function with respect to r.

Proof: Now it is obvious that there exists at least one heat function with respect to r since there exists an internal energy structure with respect to r (by Theorem 4).

Now we must prove that there is at most one heat function with respect to r. Assume Q and Q’ are non-identical heat functions. Then there must exist a p E P such that Q@) # Q’@) and

Q@> = WC’) - W> + Wp) Q'(p) = U'(c') - U'(c) + W(p)

where D@) = (c, c’) and U and CJ’ are internal energy functions with respect to r. Therefore,


cqc’) - U(c) # U’(c’) - U’(c).

But (by Theorem S), there exists an 01 such that

U(c’) = U’(c’) + a and U(c) = U’(c) + (Y,

Therefore, it follows that

U(c’) - U(c) # (U(c’) - a) - (U(c) - a), or

U(c’) - U(c) # U(c’) - U(c).

But this is impossible, so Q and Q’ are the same function.

Now we will prove a theorem which upon interpretation says that the net heat absorbed by the thermodynamic system in any adiabatic process is zero.

THEOREM 7. Let r = (S, C, P, D, W, A) be a FLTS and Q be a (the) heat function with respect to r. Then for all p E P, if p E A then Q(p) = 0.

Proof: Assume p EA such that D@) = (c, c’). Then (by Definition 4)

Q(P) = ‘7~‘) - u(c) + f+‘@)

where U is an internal energy function with respect to r. But (by Definition 3), we know

U(c) - U(c’) = W(p), or

U(c’) - U(c) + Wcp) = 0.

Therefore, Q@) = 0.

Finally, we will prove a theorem which upon interpretation says, roughly speaking. that a perpetual motion machine of the first kind is impossible.

THEOREM 8. Let I? = (S, C, P, D, W, A) be a FLTS and Q be a (the) heat function with respect to r. Then for all p E P, if p is cyclic with respect to r, then Q@) = IV@).

fioof: Assume p is cyclic with respect to r. Therefore (by Definition 2), p is such that D(p) = (c, c). Now since Q is a heat function with respect t0 r,

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Q(P) = U(c) - WC) + W-J) where U is an internal energy function with respect to r. Therefore, Qc?J> = w@).”

1V.FORMULATlONOFTHEFlRST LAWOFTHERMODYNAMICS

Let us begin by considering in what sense the previous formalization captures the content of the first law of thermodynamics. One might first notice that certain definitions and theorems of the previous section, upon interpret. ation, in some sense capture the content of the quotations given in the second section. For example, Theorem 4 and Theorem 6, when interpreted, might be construed as containing Guggenheim’s claim that there exists a certain state function U (called the internal energy) with certain properties (for example, AU = Q - IV). Theorem 5, when interpreted, might be construed as Wilson’s claim that the internal energy function is unique up to an additive constant. Theorem 8, when interpreted, might be construed as containing the claim that a perpetual motion machine of the first kind is impossible.

But a fundamental question is still with us: “What is the first law of thermodynamics?” One might be tempted to say that Definition 1 is the first law. But do we want to hold that the first law of thermodynamics is a definition? If so, perhaps all scientific laws are definitions. To avoid this result, one might be tempted to say that one of the previous theorems is the first law of thermodynamics, for example, Theorem 4. But then we are left with the result that not only can the first law of thermodynamics be proved (rigorously at that), but that it can be proved without any reference to the empirical world. One might try to avoid this result by holding that a previous theorem, when interpreted, is the first law of thermodynamics. But this is irrelevant to the objection since the previous theorems can be proved, interpretation or not, without appeal to the empirical world.

At this point, one may begin to wonder in what sense the previous section is a formalization of the first law of thermodynamics. In one sense it is not, since nothing (no theorem or definition) in the previous section can plausibly be identified as the first law of thermodynamics. But in another sense it is, since we can use the results of the previous section to give a

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reasonable, and hopefully enlightening, formulation of the first law of thermodynamics. Consider the following remark by Joseph Sneed.

The essential, distinguishing feature of theories of mathematical physics is that each has associated with it a formal, mathematical structure. This structure forms, so to speak, the COW of the theory, or the mathematical formalism characteristic of the theory. It is this formal, mathematical core which is precisely described by the definition of a set- theoretic predicate which ‘axiomatizes’ the theory. This predicate is then used to make the empirical claims of the theory. (Sneed, 197 1, p. 16)

Now thermodynamics is certainly a theory of mathematical physics, and we would naturally hold that the first law of thermodynamics is one of the empirical claims of thermodynamics. Therefore, following Sneed, we should expect that a formulation of the first law of thermodynamics, based upon the previous section, would use one of the set-theoretical predicates defined in the previous section. And indeed, we can give such a formulation.

Consider the following sentence which uses the set-theoretical predicate defined in Definition 1.

For any thermodynamic system s, there exists a first law thermodynamic structure (S, C, P, D, W. A ) such that S = {s) and C, P, D, W, and A satisfy the intended interpret, ation conditions (2)-(6) (SW page 121) with respect to the thermodynamic system s.~

This sentence certainly satisfies what many philosophers have held to be essential characteristics of scientific laws; for example, this sentence is empirical, universal in form, ‘unrestricted in scope’, supports relevant counterfactuals, etc. Furthermore, this sentence captures, in a reasonable sense, the content of the quotations given in the second section. For example, from this sentence we can not only conclude that for any thermodynamic system s that there exists a first law thermodynamic structure (S, C, P, D, IV, A > where S = {s) and C, P, D, W. and A satisfy the intended interpretation conditions (2)-(6) but we can also conclude that there exist internal enemy structures with respect to this first law thermodynamic structure and that these internal energy structures are unique up to an additive constant.

Now I do not want to concern myself with the question whether a scientific law is a sentence of certain type or what is expressed by a sentence of a certain type (a proposition). But if one is inclined to hold that scientific laws must be sentences then the above sentence is a reasonable candidate

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for the first law of thermodynamics. If one is inclined to hold that scientific laws are not sentences, but what can be expressed by sentences, then what is expressed by the above sentence is a reasonable candidate for the first law of thermodynamics.

V.COMMENTS

Let us now consider some objections which might be raised against the formalization of first law thermodynamics given in the third section of this paper. One might object for the following reason: Physicists hold that not only does an internal energy function exist for each thermodynamic system, but that this internal energy function is, in some sense, continuous and differentiable; and the formalization does not account for this. The objection is vdid, but the formalization can be modified to account for the continuity and differentiability of the internal energy function. Now certain conditions would have to be added to Definition 1; but it is possible to find conditions which involve no new primitive notions with respect to the formalization. However, these conditions require the use of a concept of a mathematical limit which involves a slight variation of the ordinary c-6 limit and the introduction of this mathematical limit would be tedious and not directly related to the purpose of this paper.

One might object to this formalization because it does not account for the fact that the internal energy function is extensive. Again, the objection is valid, and we are faced withproblems that not only would new conditions have to be added to Definition 1, but new primitive notions would have to be added to the formalization. For example, we would have to add a primitive notion which would be interpreted as the operation of combining two thermodynamic systems. Now such additions to the formalization would most likely be complex in nature and therefore, I will leave such problems to the more formalistic reader.

Further, one might feel that the formalization is inadequate since the basic approach of the formalization might not allow for the formalization of both the first and second laws of thermodynamics. I would conjecture, however, that this basic approach could be so extended. Certainly, new conditions would have to be placed on the basic definition and new primitive notions would have to be added. For example, a primitive notion which would be interpreted as a set of reversible processes would have to be added.

AXIOMATIC APPROACH TO TIIERMODYNAMICS 131

Also, a description of these processes would have to be included in the formalization which associated with each reversible process not only its initial and final states, but also a path in state space corresponding to the intermediate states of the process.

Also, I would like to bring to the attention of the reader an article which has recently appeared by C. U. Moulines (1975). The purpose of this article is

to show that it is possible to apply J. D. Sneed’s metatheory of physical theories to the logical reconstruction of a thermodynamic theory. (Moulines, 1975, p. 101)

Moulines restricts his analysis to what he calls “simple equilibrium thermodynamics” which is, roughly speaking, classical equilibrium thermodynamics for systems of a particular type, e.g. systems which are homogeneous, chemically inert, not acted upon by electric fields, etc. At this point, a question could arise as to the relationship between Moulines’ formalization of ‘simple equilibrium thermodynamics’ and the formalization of first law thermodynamics given in this paper. Certainly a detailed answer to such a question is beyond the scope of this paper, but a few remarks are in order. First of all, Moulines’ formalization is concerned with the ‘complete’ thermodynamics of systems of a particular type where the formalization given in this paper is only concerned with systems of this particular type insofar as they are instances falling under the first law of thermodynamics. But it should be realized that Moulines’ formalization is restricted to thermodynamic systems of this particular type where the formalization given in this paper is not. Also, within his formalization, Moulines takes the internal energy function (as well as the entropy function) as a primitive notion where the formalization given in this paper does not. As a matter of fact, the main theorem of this formalizaiton of first law thermodynamics can be interpreted as an existence theorem for the internal energy function.

In conclusion, a remark must be made about using the basic approach behind the formalization, given in this paper, to formalize thermodynamics in the hopes of setting forth the relationships between thermodynamics and statistical mechanics. Using such an approach would not be advisable for at least the following reason: A different formalization, perhaps like Moulines’, which took the internal energy and entropy functions as primitives and placed conditions on these functions (such as differentiability) would

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certainly be less complex than the approach taken in this paper, and such an approach would probably be as fruitful in exploring the possible relationships between thermodynamics and statistical mechanics as any other approach. But in some sense, a formalization (like Moulines’) which takes the internal energy and entropy functions as primitives, presupposes the formalization given in this paper. This is because the formalization given here is, in part, a formalization of what is required for the existence of the internal energy function.

University of Nebraska, Lincoln

APPENDIX:PROOFOFTHEOREM4

THEOREM 4. Let I’ be a FLTS. Then there exists an internal energy structure with respect to r.

Proof: Assume r = (S, C, P, D, FV, A) is a FLTS. Now let A = (S, C, U) where U is the following relation: (1) take any cr E C(s) and let (cl , 0) E U, (2) take each c’ E C(s) such that there exists an a’ E A such that either D(a’) = (cr , c’> or D(a’) = (c’, cl ), and let (c’, - W(a’)> E U if D(a’) = (cr , c’) and let (c’, Ma’)> E (I if D(a’) = (c’, cl ), and (3) allow U to have no other members. To show A is an internal energy structure with respect to r, we must prove that U is a function whose domain is C(s) such that for any c, c’ E C(s) if p E A and D(p) = (c, c’), then U(c) - U(c’) =

w(p)- Step 1. Let us prove that U is a function (a well-defined relation).

Assume (c’, r), (c’, r’) E U such that r # r’. Now by the construction of U, we know that there exist a’, a” EA such that one of the following is the case.

(1) D(a’) = (cl, c’) and D(a”) = (c, , c’>.

(2) D(a’) = (cl , c’) and D(a”) = Cc’, cl ).

(3) D(a’j = (c’, cl) and D(a”) = <cl , c’).

(4) D(a’) = (c’, cl) and D(a”) = (c’, c1 ).

It can be easily shown that each of these four alternatives leads to a contra- diction; therefore, the assumption of this step must be false. Hence, U is a function.


Step 2. Let us prove that the domain of Ii is C(s). Assume c’ E C(s). Now there is an a’ EA such that either D(a’) = cc,. c’) or D(a’) = (c’,c, 1 ,(by D6(a)). Therefore, either cc’, - W(d)> E ZJ or cc’, W(Q) E U; hence c’ E Dom U. Also, by the construction of U all elements of the domain of If are elements of C(s). Hence, the domain of U is C(s).

Step 3. Let us prove that for any c, c’ f C(s) if p E A and D(p) = (c, c’), then U(c) - U(c') = IV@). Assume p f A such that D(y) = (c, c’). Now there exist a’, a” EA such that one of the following is the case.

(1) @a’) = cc,, c’) and D(a”) = (cl 9 c’).

(2) @a’) = (cl, c’) and o(a”) = (c’, cl).

(3) D(a’) = cc’, cl) and @a”) = (cl, c’)

(4) O(u’) = cc’, c1 ) and O(a”) = cc’, cl ).

It can easily be shown that each of these four alternatives leads to the conclusion that U(c) - U(c’) = I#@). Therefore, we may conclude that U(c) - U(c') = W(p) where p E A and D(p) = (c, c’).

NOTES

* The research for this paper was supported in part by a fellowship from the &laude Hammond Fling Foundation (University of Nebraska). Also, for helpful discussions and comments concerning this paper, I wish to thank Robert Hardy, Werner Leinfellner, and David Grow. ’ Actually, requiring C(s) to be an open region could be dropped without affecting any of the results that follow. It is included because if one attempts to strengthen this formalization so as to be able to prove that the internal energy function L’ is continuous and/or differentiable, then one must add certain conditions to this definition which involve mathematical limits. Now requiring C(s) to be an open region simplifies what conditions must be added. 2 One might believe that this condition can be strengthened to the following: For any c, c’ E C(s), there exists an a’ E A such that D(Q’) = (c, c’). But this cannot be done since it would, roughly speaking, require the first law of thermodynamics to contradict the second law of thermodynamics, but we know that these two laws are mutually consistent. 3 I would like to show that there exists at least one first law thermodynamic structure. Now let r = (S, C,P, D, IV, A ) where (1) S = { 3 ); (2) C is the function whose domain isS such that C(3) = (0, 1); (3) A is the set C(3) X C(3); (4) P is identical to A; (5) D is the function whose domain is A such that for all a E A, D(n) = a; and (6) IV is the function whose domain is A such that for aI1 a E A, W(a) = a2 -a, where a = (a,, oz). Now we can prove the following theorem.

134 MICHAEL A. DAY

THEOREM 9. Let p = (S, C, P, D, W, A 1 satisfy conditions (l)-(6) given above. Then r is a FLTS.

Proof: Obvious from Definition 1.

Also, it should be realized that the above T shows that conditions Dl-D6 of Definition 1 are consistent, or speaking more strictly, are consistent if set theory is consistent. Further, it should now be obvious that there can be first law thermodynamic structures which in no way are related to the so-called empirical world. 4 If one holds that scientific laws must be sentences, then I do not wish to imply that this sentence is the only reasonable candidate for the first law of thermodynamics. As a matter of fact, some philosophers defend the view that empirically nonequivalent sentences can be called the same law; hence they would hold that it is possible for two empirically nonequivalent sentences to be both called the first law of thermodynamics (Achinstein, 1971, pp. 17-18)

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Achinstein, Y., Law and Explanation, Oxford University Press, 197 1. Adams, E. W., ‘Axiomatic Foundations of Rigid Body Mechanics’, Unpublished Ph.D.

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(Second Edition), Addison-Wesley, 1953. Sneed, J. D., The Logical Structure of Mathematical Physics, D. Reidel, 1971. Suppes, P., Introduction fo Logic, Van Nostrand, New York, 1957. Suppes, P., ‘What Is a Scientific Theory?‘, in Philosophy of Science Today (ed. by

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An axiomatic approach to first law thermodynamics