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Mathematics LibreTexts

5.1: Introduction

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    There is a fair bit of groundwork to cover before we get to the Incompleteness Theorem, and much of that groundwork is rather technical. Here is a thumbnail sketch of our plan to reach the theorem: The proof of the First Incompleteness Theorem essentially consists of constructing a certain sentence \(\theta\) and noticing that \(\theta\) is, by its very nature, a true statement in \(\mathfrak{N}\) and a statement that is unprovable from our axioms. So the groundwork consists of making sure that this yet-to-be-constructed \(\theta\) exists and does what it is supposed to do. In this chapter we will specify our language and reintroduce \(N\), a set of nonlogical axioms. The axioms of \(N\) will be true sentences in \(\mathfrak{N}\). We will show that \(N\), although very weak, is strong enough to prove some crucial results. We will then show that our language is rich enough to express several ideas that will be crucial in the construction of \(\theta\).

    In Chapter 6 we will prove Gödel's Self-Reference Lemma and use that lemma to construct the sentence \(\theta\). We shall then state and prove the First Incompleteness Theorem, that there can be no decidable, consistent, complete set of axioms for \(\mathfrak{N}\). We will finish the chapter with a discussion of Gödel's Second Incompleteness Theorem, which shows that no reasonably strong set of axioms can ever hope to prove its own consistency.