Consistency and Existence
by Andrew Boucher
v1.00 Last updated: 1 Oct 2000
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On the one hand, first-order theories are able to assert the existence of objects. For instance, ZF set theory asserts the existence of objects called the power set, while Peano Arithmetic asserts the existence of zero. On the other hand, a first-order theory may or not be consistent: it is if and only if no contradiction is a theorem. Let us ask, What is the connection between consistency and existence?
In one direction, the answer is obvious: the existence of a model for a theory demonstrates that it is consistent. The world, so to speak, is coherent; it does not allow for something both to be and not be. So if there exists a model for a theory, then it is sure that the theory is consistent, because otherwise the world would harbor a contradiction. In this way Riemann showed that non-Euclidean geometry was consistent, by modelling it in terms of a Euclidean sphere. (Granted, this was a relative consistency proof, that is showing one theory consistency if another was. But the point remains, since the assertion of existence was also relative, namely of objects existing provided that a Euclidean sphere exists.)
In short, existence implies consistency.
Since at least Hilbert, there are those who make the opposite claim, that one can deduce existence from consistency. In a limited sense, this is again trivial. After all, Godel's Completeness Theorem states that, if a first-order theory is consistent, then it has a model.
But one has to be careful about what actually exists. In the proof of the Theorem, one constructs an artificial and contrived model. The objects which the theory assert to exist, are actually an arbitrary infinite sequence, which (to simplify matters) may be taken to be the Natural Numbers. The answer to the question, "Does the power set of the empty set exist?" is therefore only, "Yes, and here's the Natural Number which it is."
Worse, relationship symbols are essentially defined as particular sets of the Natural Numbers, having to do with whether there is a certain proof in a particular extension of the relevant theory. For example, a* has the relationship R* to b* if and only if there is a proof of aRb. So the answer to the question, "What is the membership relationship?" is simply, "Something to do with whether particular proofs exist."
Now to logicians, who have had it drilled into them that they are supposed to be studying "structure", this probably does not seem bizarre. But to the rest of us poor mortals, it is less than satisfactory. Being told there is a model of ZF set theory and then being shown the Natural Numbers, is a bait-and-switch operation of cosmic proportions. And what I mean by that is: the axioms of ZF are asserted because they are supposed to describe abstract objects called sets and a fundamental binary relationship called membership. If the only thing the theory is actually about is the Natural Numbers, considered in some funny way, well then there would seem to be a problem. For one thing, the justification for the axioms--our ability to abstract and so forth--has gone completely out the window.
So the assertion that consistency implies existence is not immediate. Yes, there are always some things which can serve as an interpretation of the theory. But all we can be sure of, is that these things are just the Natural Numbers (or some other already existing infinite set X) and sets of Natural Numbers. Nothing new has been shown to exist. So no, a consistent theory does not show that there must be things as the axioms are intended to assert.
And I stress what I just said: what the axioms are intended to assert. Because, despite all the formalist blah-blah, when mathematicians (as opposed to logicians) actually sit down and state axioms, they intend them to mean something. The axioms are trying to describe reality in some way. Whether and how it connects to the world determines, to a large extent, whether and how much a particular branch of mathematics is important.
Natural numbers are used everywhere, real numbers are used in science. If the scientist is not talking gibberish, then his terms must have some prior meaning. The mathematician stating axioms for real analysis must use the same meaning, if he wants to connect his theorems to what the scientist is talking about and so render his theory important.
Real analysis can also, again by the Completeness Theorem, be considered to be about the Natural Numbers and about proofs in the theory of real analysis. But if this were all it was about, it would be extremely uninteresting, and it is clear that the value of real analysis does not derive from this interpretation. The world and science do not care about what proofs hold in a certain mathematical theory.
Hilbert once made the remark that it doesn't matter what a theory models: if it's the Natural Numbers or the things in my room, the theory is indifferent. And again, he is right, but in only one direction. Mathematicians study a theory because they think it is about something. If it can also be about something else, then of course this is a plus; no one throws away a free lunch. But the theory has to be about something important in the first place, or else it has no interest. Having a model and having a useful or important model, are two quite very different things.
So, consistency does not imply existence, in any reasonable way. The belief in the existence of objects must come from somewhere else. It is therefore appropriate, when an axiom system is presented, to ask what the theory is intended to model. And an appropriate follow-up is why we should believe there are things like that. Whether ZF (and, even more, ZFC) can withstand this kind of questioning and scrutiny, I leave the reader to his own unbiased judgment.