Home > Mathematical Logic, Set Theory > What the axioms do for us (part II)

What the axioms do for us (part II)

In his excellent entry for Set Theory in Gowers’ The Princeton Companion to Mathematics, Joan Bagaria writes: “Set theory […] plays two very different roles at the same time: on the one hand, it is an area of mathematics devoted to the study of abstract infinite sets and their properties; on the other, it provides mathematics with its foundation”.

It is the foundational aspect of set theory that I wish to emphasize in this post. In the previous one I presented the axioms of set theory as describing a (piece of) set theoretic universe, but that was just one side of the coin.

ZFC is an axiomatic formal system based in first-order logic. Two outstanding features of this formal system are:

1. Any mathematical concept or argument is (or at least it should be) formalizable within the system.
2. The formal theorems of the theory, once interpreted, are all of them true (what they say is “the fact of the matter”).

As in the previous post we could call this one the “orthodox view”. ZFC encapsulates the valid principles of mathematical reasoning, and hence the valid principles allowing mathematicians to find mathematical truths.

To accept a collection of axioms as a foundation of mathematics, then, means to accept those mathematical principles they encapsulate, but also to reject whatever principles their acceptance forbids. There are at least two points that deserve special attention here:

1. It is the collection of axioms as a whole that must find justification through the principles it encapsulates, not just each axiom on its own. A coherent framework arises only when they are put to work altogether.
2. It is the equilibrium between what is gained and what is lost by accepting a collection of axioms what actually matters from a foundational point of view. Few mathematicians would like to see hobbled their efforts by arbitrary restrictions. In Shelah’s words (Logical Dreams) “the best framework, the best foundation, is the one that governs you least”.

Hence accepting some axioms in place of some others is clearly more a matter of convenience than a matter of faith. Particularly, when different frameworks are available which encapsulate the very same principles other criteria such as naturalness or simplicity may intervene.

From this point of view, so called extrinsic justifications for the axioms of set theory happen to make perfect sense: to be able to know which principles are available and which are not it is inescapable to explore what follows and what does not from a chosen collection of candidate axioms.

Let us see all this through some examples that can be found in Maddy, Believing the axioms I:

Fraenkel, Bar-Hillel and Levy justify the use of an extensional notion of set “because it is simpler and clearer”, while at the same time arguing that “it can simulate intensional notions when the need arises”.

Zermelo justified Foundation because it gives a useful understanding of the universe of sets. He could also have argued that it does not prevent one to simulate non-well-founded notions of set when the need arises.

In set theory what is mathematically possible or not is formalized in terms of set existence. Pair, union, replacement, separation or even choice are all very immediate examples of what kind of “manipulations” mathematicians might expect to be able perform over given collections in order to obtain new collections. So one might expect that the more sets one can prove to exist in set theory, the more of these “manipulations” will be available. This could sort of justify a rule of thumb such as maximize that, by the way, has as a consequence, as Koellner and Woodin have proved, the attainment of a “significant reduction in incompletess”, at least at certain levels of the arithmetical hierarchy (just below CH).

Determinacy or V=L are not seen as axioms of set theory because they somehow limit the space of available mathematical possibilities (Determinacy is incompatible with choice and V=L with the existence of certain large cardinals, for example measurable cardinals).

Some mathematicians do not accept set theory as a foundation of all mathematics. For example, Andrej Bauer has said “set theory is not the foundation of the mathematics I do, and neither are the mathematical objects I study sets in the sense of classical set theory”. It would be interesting to know which parts of his mathematical activity he thinks are not formalizable within ZFC and why. But this is only one way to question the “orthodox view”.

As I noted above there are two outstanding features of ZFC. Even if we accept that any mathematical concept or argument is (or at least it should be) formalizable within the system, after what we saw in the previous post, namely, that the theorems of ZFC admit different interpretations, it is possible to argue that they cannot just be regarded as true simpliciter. In other words, that it is not clear which is exactly the link between what we think is mathematically possible and set existence theorems in ZFC. The appeal to an uncompromised platonism made by Drake and many set theorists could be then regarded as a means to solve this annoying situation; begging the question by taking what the theorems say at face value.

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