Abstract
In this chapter we introduce a calculus of logical deduction, called first-order logic, that makes it possible to formalize mathematical proofs. The main theorem about this calculus that we shall prove is Gödel’s completeness theorem (1.5.2), which asserts that the unprovability of a sentence must be due to the existence of a counterexample. From the finitary character of a formalized proof we then immediately obtain the Finiteness Theorem (1.5.6), which is fundamental for model theory, and which asserts that an axiom system possesses a model provided that every finite subsystem of it possesses a model.
In (1.6) we shall axiomatize a series of mathematical (in particular, algebraic) theories. In order to show the extent of first-order logic, we shall also give within this framework the Zermelo–Fraenkel axiom system for set theory, a theory that allows us to represent all of ordinary mathematics in it.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag London Limited
About this chapter
Cite this chapter
Prestel, A., Delzell, C.N. (2011). First-Order Logic. In: Mathematical Logic and Model Theory. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-2176-3_2
Download citation
DOI: https://doi.org/10.1007/978-1-4471-2176-3_2
Publisher Name: Springer, London
Print ISBN: 978-1-4471-2175-6
Online ISBN: 978-1-4471-2176-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)