The reason for defining possible worlds in terms of maximality or completeness is that not every possible state of affairs is complete enough to be considered a possible world. Plantinga gives as an example the proposition that "Socrates is snubnosed." A possible state of affairs must include or preclude more than that in order to be considered a possible world.
Similarly, Robert C. Koons and Timothy K. Pickavance (2017) define a possible world as a possibility that's maximal insofar as every proposition is either true or false according to it.2 They also describe concretism and abstractionism as two contrasting views about the nature of possible worlds. While concretism is the view that possible worlds are maximal possible concrete objects, abstractionism is the view that possible worlds are maximal possible abstract objects.3
Dale Jacquette (2006) explains that if a logically possible world is taken to be a maximal consistent set of propositions, then it could (theoretically) be constructed by randomly choosing one logically possible proposition and then considering an exhaustive ordering of all other logically possible propositions, and adding each one to the given set if and only if it is logically consistent with the propositions already collected, until there are none left. The propositions in a maximally consistent set of propositions would therefore collectively represent every state of affairs associated with a corresponding logically possible world.4
Another conclusion, however, might be that the actual world is the only maximally consistent set of propositions, and that all other logically possible but nonactual worlds are submaximally consistent.
But what about propositions whose truth or falsehood is indeterminate or undecidable? In the actual world we live in, there are such undecidable propositions. Is the actual world then not a possible world? How then can maximality or completeness be considered a valid criterion for some possible state of affairs to be considered a possible world? Must a possible world be maximal in the sense that every proposition is decidably true or false according to it, and therefore also in the sense that for every proposition there is some rational procedure that can determine in a finite number of steps the truth or falsehood of that proposition according to it?
These questions are motivated by Gödel's first incompleteness theorem, which says, roughly, that for any consistent system S of formal arithmetic in which (1) the set of axioms and the rules of inference are recursively definable, and (2) every recursive relation is definable, there are undecidable arithmetical propositions of the form xF(x), where F is a recursively defined property of natural numbers.5
Thus, it seems that possible worlds can't be both complete and consistent, because the actual world isn't that way. For every possible proposition expressible within a nontrivial formal system of arithmetic to be provable or disprovable, that system has to be in some way inconsistent. All consistent nontrivial formal systems of arithmetic are deductively incomplete.
Of what use then is the concept of maximality or completeness as a means of better understanding the metaphysics of modality?
Patrick Grim (1991) presents an argument similar to the Liar Paradox as a refutation of the maximality of possible worlds. He explains that if possible worlds are taken to be or to correspond to maximal consistent sets or propositions, and if the actual world, on such an account, is taken to be or to correspond to the maximal set of all truths, then we can examine the proposition A: The proposition A is not a member of the maximal set M of all truths. Is A a member of set M or not? If it's a member, then it must not be, and if it's not a member, then it must be.6
Tony Roy (2012) also presents an argument against the maximality of possible worlds, by employing Cantor's Theorem (that the set of all subsets of a given set has a greater cardinality than the set itself):
"Suppose that for any proposition a, some sentence expresses a and some sentence expresses not-a...Then the supposition that worlds are maximal and so include one of a or not-a for every sentence is incoherent. Consider a world w, and the set P(w) which has as members all the subsets of w. By Cantor's Theorem, there are more sets of sentences in P(w) than sentences in w. Trouble.
...And this generates a problem about the maximality of w. Suppose w is maximal; then given our assumption that there are sentences to express any proposition and its negation, for any A in P(w), w includes one or the other of,
a1 Some member of A is true; and
a2 No member of A is true.
So w includes at least one sentence for each member of P(w); so there are not more members in P(w) than w. This is impossible; reject the assumption; w is not maximal.
So given a language with adequate expressive power, the very attempt to say everything about a world is self-defeating."7
FOOTNOTES
1Alvin Plantinga, The Nature of Necessity (Oxford: Clarendon Press, 1974), pp. 44-45.
2Robert C. Koons and Timothy K. Pickavance, The Atlas of Reality: A Comprehensive Guide to Metaphysics (Chichester: John Wiley & Sons, 2017), p. 318.
3Ibid., p. 321.
4Dale Jacquette, "Propositions, Sets, and Worlds," in Studia Logica, Vol. 82, No. 3, April 2006, pp. 338-340.
5Kurt Gödel, "On formally undecidable propositions of Principia Mathematica and related systems," [Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme," 1931] in Kurt Gödel Collected Works, Volume I, edited by Solomon Feferman, et al. (Oxford: Oxford University Press, 2004) p. 181.
6Patrick Grim, The Incomplete Universe: Totality, Knowledge, and Truth (Cambridge: MIT Press, 1991), pp. 6-8.
7Tony Roy, "Modality," in The Continuum Companion to Metaphysics, edited by Neil A. Manson and Robert W. Barnard (London: Continuum, 2012), pp. 51-52.
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