Titlebook: An Invitation to Mathematical Logic; David Marker Textbook 2024 The Editor(s) (if applicable) and The Author(s), under exclusive license t

Oratory

Der Rundfunk und der Weltfunkvertrag,The ultraproduct construction is introduced and used to give an alternative proof of the Compactness Theorem.

hermetic

Die Londoner Funkkonferenz 1912,A test for eliminating quantifiers is given and applied it to further study the model theory of algebraically closed fields.

消息靈通

Opponent

GEON

https://doi.org/10.1007/978-3-642-10271-4Turing’s universal machine is constructed and used to prove the undecidability of the Halting Problem and the undecidability of validity in first-order logic. We include a brief discussion of the Recursion Theorem.

placebo-effect

https://doi.org/10.1007/978-3-642-10271-4We introduce the computably enumerable sets and the arithmetic sets and show that the form a hierarchy. These results, and the existence of computably inseparable computably enumerably sets, will be used in our approach to the Incompleteness Theorem. We briefly study Kolmogorov randomness as another avatar of incompleteness phenomena.

Decimate

偽書

https://doi.org/10.1007/978-3-642-10271-4G?del’s Incompleteness Theorems are proved. We show that the sets definable in the natural numbers are exactly the arithmetic sets. The Arithmetized Completeness Theorem is used to give an alternative proof of the Second Incompleteness Theorem.

轉(zhuǎn)向

摻假

,Fünfter Teil: Thesen der Arbeit,We bound the growth rate of computable functions provably total in Peano Arithmetic. This is applied to show the independence of Goodstein’s number theoretic result. Proof theoretic methods, including cut-elimination, are introduced to prove the main result.