Abstract
In the previous chapter we saw how a large portion of mathematics can be formalized in first-order logic. The very fact that the construction of the classical number structures can be formalized this way makes first-order logic relevant, but is it necessary? For centuries mathematics has been developing successfully without much attention paid to formal rigor, and it is still practiced this way. When intuitions don’t fail us, there is no need for excessive formalism, but what happens when they do? In modern mathematics intuition can be misleading, especially when actual infinity is involved. In this chapter, we will see how seemingly innocuous assumptions about actually infinite sets lead to consequences that are not easy to accept. Then, we will go back to our discussion of a formal approach that will help to make some sense out of it.
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Notes
- 1.
This geometric fact has an interesting number theoretic variant: the sum of two equal square numbers is never a square number. 12 + 12 = 1 + 1 = 2 (not a square), 22 + 22 = 4 + 4 = 8 (not a square), 32 + 32 = 9 + 9 = 18 (not a square), and so on. An elegant proof of this is in Appendix A.2.
- 2.
An interesting technical aspect is that we might as well allow a step-by-step construction in which each step is itself a step-by-step construction. Cantor’s theorem applies in this case as well.
- 3.
Real numbers understood as Dedekind cuts, or, in the usual representation, as sequences of digits, are infinite objects. Here we mean that real numbers are finite in the sense that they measure finite quantities. For example, the area of a circle with radius 1 is π. It is definitely finite (less than 4), but π is a real number with an infinite decimal representation.
- 4.
For full details see [37].
References
Smoryński, C. (2012) Adventures in formalism. London: College Publications.
Wagon, S. (1985). The Banach-Tarski paradox. Cambridge: Cambridge University Press.
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Kossak, R. (2018). Points, Lines, and the Structure of \({\mathbb {R}}\). In: Mathematical Logic. Springer Graduate Texts in Philosophy, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-97298-5_5
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