Die Widerspruchsfreiheit der reinen Zahlentheorie.¶
Gentzen, G. (1936). Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, 112(1), 493-565.
Cited by¶
2 citations across 2 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Mathematical Induction
- preservation-and-progress proof of type soundness for ML-like languages, and is the workhorse of programming-language metatheory. Well-founded induction on program executions proves termination — pick a well-founded measure (a function from program states to a well-founded set), show every execution step strictly decreases the measure, and conclude that no infinite execution is possible. Transfinite induction in set theory extends induction beyond \(\omega\) into the ordinals; Gentzen 1936
This sourceConsistency proof of Peano Arithmetic via transfinite induction up to ε₀.
- preservation-and-progress proof of type soundness for ML-like languages, and is the workhorse of programming-language metatheory. Well-founded induction on program executions proves termination — pick a well-founded measure (a function from program states to a well-founded set), show every execution step strictly decreases the measure, and conclude that no infinite execution is possible. Transfinite induction in set theory extends induction beyond \(\omega\) into the ordinals; Gentzen 1936
- Well-Foundedness (Well-Ordering)
- Descending-chain-free semantics in domain theory (Scott's pointed CPOs and Plotkin's powerdomains) provide fixed-point theorems for denotational semantics — the recursive equation f(x) = g(f, x) has a unique least fixed point in a CPO, and the construction iterates from ⊥ up the well-founded approximation order.
This sourceConsistency proof of Peano Arithmetic via transfinite induction up to ε₀.
- Descending-chain-free semantics in domain theory (Scott's pointed CPOs and Plotkin's powerdomains) provide fixed-point theorems for denotational semantics — the recursive equation f(x) = g(f, x) has a unique least fixed point in a CPO, and the construction iterates from ⊥ up the well-founded approximation order.
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