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Maths › 03. Logic and Foundations ›

15. Ordinal Analysis

Ordinal analysis studies the strength of formal theories by measuring how far their methods of proof can justify transfinite induction along well ordered structures. The chapter begins with proof theoretic ordinals, where an ordinal is assigned to a formal theory as a measure of the induction principles and consistency arguments that the theory can support. Transfinite induction is then introduced as a generalization of ordinary induction, allowing properties to be...
logic proof theory ordinal analysis transfinite induction formal systems May 2, 2026
Maths › 03. Logic and Foundations ›

14.5 Extensions

14.5 Extensions and Refinements Gödel’s original theorems admit stronger forms and alternative formulations. These refinements clarify exactly which assumptions are needed and how incompleteness behaves across different systems. Rosser’s Improvement Gödel’s first theorem used a strengthened consistency assumption. Rosser showed that this can be weakened. Instead of the original Gödel sentence, Rosser constructs a sentence $R$ that balances proofs of a statement and its negation. Informally, $R$ says: “if there...
logic godel proof theory computability foundations May 2, 2026
Maths › 03. Logic and Foundations ›

14. Godel Theorems

Godel's incompleteness theorems show that sufficiently expressive formal systems have inherent limits, especially when they are able to represent basic arithmetic. The chapter begins with arithmetization of syntax, where formulas, proofs, and derivations are encoded as natural numbers, allowing a formal system to speak about its own syntactic objects. The first incompleteness theorem is then introduced, showing that any consistent and sufficiently strong formal system contains true arithmetic statements that...
logic proof theory godel theorems incompleteness formal systems May 2, 2026
Maths › 03. Logic and Foundations ›

14.2 First Incompleteness

14.2 First Incompleteness Theorem The goal is to construct a sentence that talks about its own provability. This requires combining arithmetization with a method for controlled self-reference. Representing Provability From the previous section, we have an arithmetic relation: $$ \mathrm{Proof}(x,y) $$ which means “$x$ codes a valid proof of the formula with code $y$”. Using this, define the provability predicate: $$ \mathrm{Prov}(y) ;\equiv; \exists x, \mathrm{Proof}(x,y) $$ So: $$ \mathrm{Prov}(\ulcorner...
logic godel incompleteness diagonalization proof theory May 2, 2026
Maths › 03. Logic and Foundations ›

13.1 Syntax of Proofs

13.1 Syntax of Proofs A proof system needs a precise notion of proof. Informally, a proof is an argument that shows why a conclusion follows. Formally, a proof is a finite object built according to fixed rules. The syntax of proofs describes the shape of these objects. Formulas and Judgments A proof does not manipulate meanings directly. It manipulates formulas. A typical formula is written: $$ A $$ A derivability...
logic proof theory syntax derivations May 2, 2026
Maths › 03. Logic and Foundations ›

13.4 Consistency Proofs

13.4 Consistency Proofs A formal system should avoid proving contradictions. Consistency is the property that a system does not collapse into proving everything. At the simplest level, a system is consistent if there is some formula $A$ such that the system does not prove both $A$ and $¬A$. More commonly, we say that a system is consistent if there is no formula $A$ such that: $$ \vdash A $$ and...
logic proof theory consistency foundations May 2, 2026
Maths › 03. Logic and Foundations ›

13.5 Proof Length and Complexity

13.5 Proof Length and Complexity Proof theory can ask not only whether a formula is provable, but also how hard it is to prove. This leads to the study of proof length and proof complexity. A formal proof is a finite object, so it has a size. We may count the number of lines, the number of inference steps, the total number of symbols, or the size of the proof...
logic proof theory complexity computation May 2, 2026
Maths › 03. Logic and Foundations ›

13.2 Derivability

13.2 Derivability Derivability is the formal relation between assumptions and conclusions. It tells us when a formula can be obtained by applying the rules of a proof system. We write: $$ \Gamma \vdash A $$ to mean that $A$ is derivable from the assumptions in $\Gamma$. Here $\Gamma$ is a collection of formulas. It may contain one formula, many formulas, or no formulas at all. For example: $$ p,\ p...
logic proof theory derivability inference May 2, 2026
Maths › 03. Logic and Foundations ›

13. Formal Proof Systems

Formal proof systems study proofs as mathematical objects, with explicit syntax, fixed rules, and precisely defined notions of derivation. The chapter begins with the syntax of proofs, where a proof is treated as a finite formal object built from formulas, assumptions, axioms, and inference rules. Derivability is then introduced to express when a formula can be obtained from a given collection of assumptions inside a chosen proof system. Normal forms...
logic proof theory formal proofs derivability proof complexity May 2, 2026
Maths › 03. Logic and Foundations ›

13.3 Normal Forms

13.3 Normal Forms Normal forms for proofs are simplified shapes of derivations. The aim is to remove unnecessary detours and expose the essential structure of a proof. In propositional logic, formulas have normal forms such as CNF and DNF. In proof theory, proofs can also have normal forms. A proof may be correct but still contain redundant steps. Normalization rewrites the proof into a cleaner form. Detours in Proofs A...
logic proof theory normalization structure May 2, 2026
Maths › 03. Logic and Foundations ›

3.4 Proof Transformations

Proofs are not only objects that certify the truth of formulas, but also mathematical structures that can be analyzed and transformed, and studying these transformations reveals important properties about the nature of reasoning and the efficiency of different proof systems. A proof transformation takes a derivation and produces another derivation of the same conclusion, often with improved structure, reduced complexity, or clearer logical form, and such transformations are central in...
logic proof systems proof theory normalization May 2, 2026
Maths › 03. Logic and Foundations ›

3. Proof Systems

Proof systems provide a formal method for deriving conclusions from assumptions using explicit rules, and they complement the semantic viewpoint by giving a syntactic notion of reasoning that can be carried out step by step without referring to truth values in a structure. In propositional and first order logic, a proof system specifies a collection of inference rules together with a notion of derivation, and these rules describe how new...
logic proof theory formal systems May 2, 2026
Maths › 03. Logic and Foundations ›

1.5 Soundness and Completeness

Propositional logic can be studied in two complementary ways. The semantic approach studies truth under valuations, while the proof theoretic approach studies formal derivations from axioms and inference rules. Soundness and completeness explain how these two approaches are related. Semantic Consequence Let $\Gamma$ be a set of formulas, and let $A$ be a formula. We say that $A$ is a semantic consequence of $\Gamma$ if every valuation that makes all...
logic propositional logic proof theory soundness completeness May 2, 2026