A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions

In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions.  If this is right, then the actual world, which is of course one of the possible worlds,  is the maximally consistent set of true propositions.  But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore, Cantor's theorem implies that possible worlds cannot be maximally consistent sets of propositions.

1. Cantor's Theorem states that for any set S, the cardinality of the power set P(S) of S > the cardinality of S. The power set of a set S is the set whose elements (members) are all of S's subsets. Recall the difference between a member and a subset. The set {Socrates, Plato} has exactly two elements, neither of which is a set. Since neither is a set, neither is a subset of this or any set. {Socrates, Plato} has four subsets: the set itself, the null set, {Socrates}, {Plato}. Note that none of the four sets just listed are elements of {Socrates, Plato}. The power set of {Socrates, Plato}, then, is {{Socrates, Plato}, { }, {Socrates}, {Plato}}.

In general, if a set S has n members, then P(S) has 2ⁿ members. Hence the name power set. Cantor's Theorem that the power set of a set S is always strictly larger that S is easily proven. But the proof needn't concern us now.

2. Suppose there is a set T of all truths, {t₁, . . . , t_i, t_{i + 1}, . . .}. Consider the power set P(T) of T. The truth t₁ in T will be a member of some of T's subsets but not of others. Thus, t₁ is an element of {t₁, t₂}, but is not an element of { }. In general, for each subset s in the power set P(T) there will be a truth of the form t₁ belongs to s or t₁ does not belong to s. But according to Cantor's Theorem, the power set of T is strictly larger than T. So there will be more of those truths than there are truths in T. It follows that T cannot be the set of all truths.

3. Given that there cannot be a set of all truths, the actual world cannot be the set of all truths.  This implies that possible worlds cannot be maximally consistent sets of propositions.   I learned the Cantorian argument that there  is no set of all truths from Patrick Grim. I don't know whether he applies it to the question whether worlds are sets.

4. As far as I can see, the fact that possible worlds cannot be maximally consistent sets does not prevent them from from being maximally consistent conjunctive propositions.

Comments

[ocham]

Since there are no sets, there cannot of course be a set of all possible propositions.

Note, even in Cantor's world, we can still speak of *every* truth.

[Account Deleted]

Bill,

An interesting argument. I must reflect upon it a bit and see how to deflect it.

I have a collection of papers I downloaded on the internet about this topic. I think Grim was one of the disputants. I must find it and review it again.

peter

[Mike]

I've thought a little here http://prosblogion.ektopos.com/archives/2008/12/grimful-omnisci.html about Grim's result that there is no set of all truths (it is basically Russell's proof that there is no set of all sets) and so no maximally consistent set (and, so again) no possible worlds so defined. I wonder how large the worry is. Take any set S, we know that P(S) will contain a subset of S that is not in S, but does it contain any more propositions? For instance, does the set S= {p, q} contain fewer propositions than the set S'= {p, q, {p, q}}? I'm not sure it does, if propostions are not themselves sets. But apart from subsets of the set of propositions in S, what other propositions might S' contain? It might contain true, self-referential propositions. S' might contain the proposition q = the propositions p an q are in set S. But these propositions are necessarily true. I'm inclined to believe that all necessarily true claims express the same proposition, since they are all true in the same set of worlds. I know this is controversial and I know the standard counterexamples. But if that is true, then adding self-referential propositions will not add more propositions to S. All this by way of saying that Grim might have shown nothing more than that we can always add sets of propositions to a consistent set S, but the additional set of propositions might be logically equivalent to propositions already in S. I think that would be no worry at all.

[Bill Vallicella]

Peter,

Given your purposes, nothing hinges on whether worlds are sets. Think of them as maximally consistent propositions or as maximally consistent states of affairs. Indeed, I don't think you need to commit yourself at all on the question what exactly possible worlds are. You can talk the talk without metaphysical commitment.

[Bill Vallicella]

Mike,

Maybe I'm dense, but I don't recognize Grim's argument in your account. Peter, do you see what Mike is driving at?

[Account Deleted]

Bill,

I know that my argument does not hinge on defining possible worlds as maximally consistent sets; still Grim's argument is interesting to explore. I found an interesting debate between Plantinga and Grim on this proof regarding omniscience. I also found one other paper that might be of interest by Gary Mar, but I have not read it yet.

As for Mike's idea. I think (but I am not sure) he wants to say this. Suppose all necessary propositions are taken to express the same truth; also suppose a proposition that expresses whether a given proposition belongs to one of the subsets of the power set of P or not is necessarily true; then all the propositions you generate by asking whether this truth belongs to this subset or not and so forth are going to express just one truth. Therefore, the set of all true propositions is not going to be bloated to be larger than its power set because you do not add any new propositions; you only consider propositions that express the same truths over and over again.

At least this is what I get from his post. I may be wrong.

peter

[Josh]

Hmmm. So does this mean that God, even if he knew all truths, could not form a large conjunctive proposition that contained all those truths? Or am I off the reservation here?

[Dave Gudeman]

Bill, it seems you could you get around this problem by defining a possible world as a maximally consistent set of first-order propositions. Or do you know of some reason that this move doesn't work?

Also, since I can't think of any relevant difference between a maximally consistent set of propositions and a maximally consistent conjunctive proposition, I wonder if with some effort this argument could not be converted to a sort of diagonalization argument that would work against a maximally consistent conjunctive proposition as well.

[Vlastimil Vohánka]

Bill,

Q. Smith, in his Ethical and Religious Thought in Analytic Philosophy of Language (1997, pp. 100 and 121), in order to obviate set-theoretical paradoxes, suggests to define possible worlds neither as maximally consistent SETS of propositions, nor as maximally consistent conjunctive propositions, but as maximally consistent PROPER CLASSES of (ATOMIC) sentences.

But, first, what is "proper class"? (In fact, what the heck is "set"?)

Secondly, "atomic" is there, presumably, to avoid that some proper class is its own member. But what is "atomic sentence"?

[Vlastimil Vohánka]

Thirdly, is not "consitent" ultimately modal, and thus resting on "possible"? Then all the above definitions of possible worlds are circular.

[ocham]

My remark above was tongue in cheek, but there was a serious point. The notion of a set is a clearly defined term in mathematics, and has a precise meaning. Whether it corresponds slightly or even at all to what in ordinary English we call a ‘set’ is a different matter.

Cantor said, in a letter to Hilbert on 2 October 1897 that the ordinals are an 'absolutely infinite' or 'inconsistent' multiplicity (Vielheit): a multiplicity whose elements cannot be assumed to 'be together' without contradiction, or such that it is impossible to think of the multiplicity as a unity (Einheit).

“I say of a set that it can be thought of as finished (and call such a set, if it contains infinitely many elements, 'transfinite' or 'suprafinite') if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and to think of the set itself as a compounded thing for itself; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements”.

“… In contrast, infinite sets such that the totality of their elements cannot be thought of as "existing together" or as a "thing for itself" … and that therefore also in this totality are absolutely not an object of further mathematical contemplation, I call 'absolutely infinite sets'”.

This suggests that what in ordinary English we call ‘the set of all truths’ certainly does exist (it is what Cantor would call an ‘absolutely infinite set’). But it is not a ‘set’ in the way that term is used in mathematics.

[Mike]

Maybe I'm dense, but I don't recognize Grim's argument in your account.

What account are you referring to? Grim gives two arguments, or two versions of the same argument. One of them is against there being no set of all truths, the other is against worlds as maximally consistent sets of propositions. I don't fuss much the details of his presentation, since we know how the argument goes. What does concern me is whether the larger sets include propositions that are not logically equivalent to those in the smaller sets. If they do, there's not much to worry about.

[Bill Vallicella]

Vlastimil writes, "Thirdly, is not 'consistent' ultimately modal, and thus resting on 'possible'? Then all the above definitions of possible worlds are circular."

Indeed, I have recently argued that BL-consistency is a modal concept. And yes, 'consistent' cannot be defined without invoking 'possible.' Circularity is an objection only if one's aim is to reduce the modal to the nonmodal. But it may be that this reduction cannot be achieved. Consider:

1. p is possible =df there is a possible world in which p is true.

2. p is necessary =df there is no possible world in which p is false.

(Assume Bivalence) I find these definitions illuminating and helpful despite the fact that they fail to reduce the modal to the nonmodal.

If I say that a possible world is a maximally consistent proposition, i.e., one that entails every proposition with which it is consistent, then again 'possible' or an equivalent occurs on both sides of the definition. But this is a problem only if one is attempting a reduction of the modal to the nonmodal.

David Lewis attempts a reduction, but his system is too crazy to be believed. He is (or was) living proof that being a genius does not guarantee having sound ideas! And I am living proof that having sound ideas does not require genius. [Grin]

[Bill Vallicella]

Josh writes, "Hmmm. So does this mean that God, even if he knew all truths, could not form a large conjunctive proposition that contained all those truths? Or am I off the reservation here?"

As I recall, Patrick Grim argues first that there is no set of all truths, and then tries to show that, if so, there cannot be an omniscient being. Whether the second argument works depends on what exactly omniscience is. If there were a Cantorian argument against the existence of a true maximally consistent proposition -- and I don't know whether there is or not -- then this argument could be used as the first stage in an argument against the possibility of an omniscient being. But again, it depends on the definition of 'omniscience.'

So you're on the reservation.

[Vlastimil Vohánka]

Bill,

Thanks.

Where have you argued that BL-consistency is a modal concept? "BL" means Bivalent Logic?

[Vlastimil Vohánka]

A propos,

A. Pruss has his dissertation on possible worlds, written under N. Rescher, online (except the part on the cardinality objection against D. Lewis which was published as a paper in a fine journal). Interesting, I haven't read all of it though.

bearspace.baylor.edu/Alexander_Pruss/www/papers/PhilThesis.html

[Vlastimil Vohánka]

I hate to to be such a chouse and nag,

still, you say that a maximally consistent proposition is one that ENTAILS every proposition with which it is consistent.

Is the entailment here strict implication or some intuitive, but stronger than material, conditional connection?

Anyway and again, the notion of entailment appears as modal.

Yes, I concede that some non-reductive definitions are illuminating, in a sense. But is the illumination worth striving for? How much light can something in need of an explication shed on something with the same property?

[ocham]

>>Whether the second argument works depends on what exactly omniscience is.

If omniscience means 'knows every truth', then the fact, if true, that there is no set of all truths, does not prevent one knowing 'every' truth.

For we can speak of 'every truth' just as we can speak of 'every ordinal'.

[ocham]

>>Hmmm. So does this mean that God, even if he knew all truths, could not form a large conjunctive proposition that contained all those truths? Or am I off the reservation here?

Cantor's proof forbids an ordering of the reals, so to the extent a long conjunction is precisely such an ordering, even God could not form such a proposition. On the other hand, as I have argued, this does not rule out God's knowing every truth.

You know there is a story that Cantor approached the Catholic church with his ideas about the Absolute Infinite. I don't know much more than that.

[Vlastimil Vohánka]

Bill,

I've found out you mean by "BL" broadly logical, and that you've treated the notion of entailment here:
maverickphilosopher.typepad.com/maverick_philosopher/2009/02/further-modal-concepts-consistency-inconsistency-contradictoriness-and-entailment.html

[Vlastimil Vohánka]

Ocham,

Why, in sum, there are no "sets"?

[ocham]

>>Why, in sum, there are no "sets"?

Well, that is probably a bold claim. There is no proof they exist, no proof that they do not exist. Given I can safely say there is no flying spaghetti monster, I can safely say there are no such things as sets. Or let's say: there is no reason to believe in their existence.

Note: As I have said before on this forum, there are such things as pluralities, by which I mean things that are logically many, but which we denote by collective nouns which a grammatically singular. Thus, there are pairs, centuries, dozens, 'a number of' and so on.

These differ logically from sets. Whereas the existence of exactly a dozen things implies no more than the existence of a dozen things, the existence of a set of 12 things implies the existence of at least 13 things, namely the 12 things plus the set.

[ocham]

Here is a link to a very good essay by Dauben explaining the theological and philosophical underpinnings of Cantor's set theory. Re-reading, one is struck by how much this owes to outdated Teutonic metaphysical speculations.

http://www.math.wisc.edu/~gomez/m473/f06/Handouts/Dauben1977pope.pdf

[ocham]

I strongly recommend the article linked to above. Interesting that the first people to become interested in and indeed support Cantor's work were neo-scholastic theologians and philosophers. "In the absolute mind [of God] the entire sequence [of real numbers] is always in actual consciousness, without any possibility of increase in the knowledge or contemplation of a new member of the sequence" says Gutberlet. Gutberlet also argued that since God's thought is unchanging, the collection of God's thoughts must comprise an absolute, infinite, complete, closed set.

There was also a worry that Cantor's view amounted to a heresy on the lines of Spinoza's view of nature. This led Cantor to distinguish between an 'Infinitum aeternum increatum sive Absolutum', reserved for God and his attributes, and an 'Infinitum aeternum creatum sive Transfinitum' evidenced throughout created nature and exemplified in the actual infinite number of objects in the universe. Cardinal Franzelin agreed with this distinction, saying that only the Absolute Infinite is properly infinite.

Note, again, that the Absolute Infinite is not and cannot be a set, according to Cantor's theory. It cannot be thought of as a unity without contradiction.

[Vlastimil Vohánka]

Thank you, Ocham.

I read the paper you link to a couple of years ago.

Here's WL Craig's paper on infinity in theology and math, I haven't read it yet, though: http://www.reasonablefaith.org/site/News2?page=NewsArticle&id=5531

One more question, if I may:
Why, in sum, the Absolute Infinite, in contrast to the Transfinite Infinite, cannot be thought of as a unity without contradiction?

[ocham]

>>Why, in sum, the Absolute Infinite, in contrast to the Transfinite Infinite, cannot be thought of as a unity without contradiction?

See my quotes above from Cantor's letter to Hilbert:

"I say of a set that it can be thought of as finished (and call such a set, if it contains infinitely many elements, 'transfinite' or 'suprafinite') if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and to think of the set itself as a compounded thing for itself; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements.
[...]

"In contrast, infinite sets such that the totality of their elements cannot be thought of as "existing together" or as a "thing for itself" … and that therefore also in this totality are absolutely not an object of further mathematical contemplation, I call 'absolutely infinite sets'".

[ocham]

I realise that I have just quoted Cantor, without really answering your question. See the Burali-Forti paradox.

http://en.wikipedia.org/wiki/Burali-Forti_paradox

Although this does not altogether answer the question either. The problem is that while mathematics itself is very precisely defined and well-formed, once mathematicians start talking philosophy they get very vague and almost incoherent.

Indeed, it is well known scientists who have an excellent track record in their own discipline tend to be very embarrassing when they start on philosophy.