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Mathematical logic and computability
Study formal reasoning and its limits through effective computation, arithmetic representation, completeness, and incompleteness.
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Module 01 research
Mathematical logic and computability
Study formal reasoning and its limits through effective computation, arithmetic representation, completeness, and incompleteness.
- 01
Formal languages, structures, and satisfaction
Separate expressions from their interpretations and calculate quantified truth in a finite structure.
36 min - 02
Soundness and Gödel completeness
Connect finite proofs to all-model consequence without turning completeness into a decision algorithm.
36 min - 03
Compactness and Löwenheim–Skolem
Construct a nonstandard element and state model-size theorems with the language cardinal included.
36 min - 04
Turing machines and computable functions
Trace an explicit unary machine and distinguish total output procedures from partial searches.
36 min - 05
Recursive enumerability and the halting problem
Build fair searches, prove halting undecidable, and distinguish missing evidence from a negative answer.
36 min - 06
Reductions, degrees, and undecidability
Compile one decision problem into another and keep many-one reductions separate from oracle computation.
36 min - 07
Primitive recursion and arithmetic encoding
Build total finite computations and encode sequences with a fully declared prime-exponent convention.
36 min - 08
Representability and first incompleteness
Construct the diagonal sentence and separate Gödel’s original hypothesis from Rosser’s improvement.
36 min - 09
Second incompleteness and consistency strength
Track the derivability conditions and the particular internal sentence expressing consistency.
36 min - 10
Arithmetical hierarchy and Tarski undefinability
Classify quantifier alternations and prove why arithmetic cannot define all of its own standard truth.
36 min