Abstract
In previous chapters we introduced mathematical structures, and we followed with a detailed description of basic number structures. Now it is time to look at structures in general. The classical number structures fit very well the definition: a set with a set of relations on it. But what about other structures? Are they all sets? Can a set of relations always be associated with them? Clearly not. Not everything in this world is a set. I am a structured living organism, but I am definitely not a set. Nevertheless, once a serious investigation of set theory got underway, it revealed a fantastically rich universe of sets, and it showed that, in a certain sense, every structure can be thought of a set with a set of relations on it. To explain how it is possible, we need to get a closer look at sets. As we saw in the previous chapter, the deceptively simple intuitive concept of set (collection) leads to unexpected consequences when we apply the well understood properties of finite sets to infinite collections. The role of axiomatic set theory is to provide basic and commonly accepted principles from which all other knowledge about infinity should follow in a formal fashion. There are many choices for such theories. In this chapter we will discuss the commonly used axioms of Zermelo and Fraenkel.
The ontological decision concerning infinity can then simply be phrased as: an infinite natural multiplicity exists.
Alain Badiou Being and Event [ 1 ]
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Notes
- 1.
A free variable in a formula, is a variable that is not within a scope of a quantifier. For example, in ∀x∃y[x ∈ y ∧¬(z ∈ z)] only z is free.
- 2.
Here, and in other axioms, we assume that formulas, such as φ(z), can have other free variables.
- 3.
Cantor defined ω to be the order type of \(({\mathbb {N}},<)\).
- 4.
To see this, you need to look at the set theoretic representation of natural numbers. For numbers so constructed, it makes perfect sense to write expression like 3 ∈ 5.
- 5.
A word of caution: What we described here is the process of generating transfinite counting numbers. They are known in set theory as ordinal numbers or just ordinals. Ordinal numbers are used to count steps in infinite processes, but they are not used to measure sizes of infinite sets. Cantor’s cardinal numbers serve that second purpose.
- 6.
Another word of caution: to say that we would not be able to reconstruct the real number line and other similar objects without the Power Set Axiom is not quite precise. We need that axiom to do it more or less naturally within the Zermelo-Fraenkel set theory. There are other axiomatic systems that do not have such an axiom, in which one can formalize much of modern mathematics, one of the more prominent ones being the second-order arithmetic. Those other systems are interesting, and they are studied for many reasons, but none of them has the status of Z F and its extensions that became a lingua franca of mathematics.
References
Badiou, A. (2010). Being and event. London/New York: Continuum.
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Kossak, R. (2018). Set Theory. In: Mathematical Logic. Springer Graduate Texts in Philosophy, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-97298-5_6
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