Abstract
In mathematics, or at least in the mathematics inspired by logical methods, to know a structure means to know all sets that are definable in it. In this chapter we will take a look at the smallest nonempty sets—those that have only one element. This a specialized topic, and it is technical, but it will give us an opportunity to see in detail what domains of mathematical structures are made of and in what sense they are “given to us.”
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This is where the envelope metaphor brakes down. Clearly, there are many empty envelopes, but it follows from the axiom of extensionality that there is only one empty set.
- 2.
This is not quite accurate. The axiom of infinity also declares the existence of a set, but, as we have seen, the axiom postulates the existence of a set that contains infinitely many elements that are constructed from the empty set in a particular manner (the set-theoretic natural numbers).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kossak, R. (2024). Definable Elements and Constants. In: Mathematical Logic. Springer Graduate Texts in Philosophy, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-031-56215-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-031-56215-0_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-56214-3
Online ISBN: 978-3-031-56215-0
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)