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Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician , as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ε0, which is the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ωω, where ω is the smallest transfinite ordinal. PRA is sometimes called Skolem arithmetic.

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  • 原始帰納的算術 (ja)
  • Primitive recursive arithmetic (en)
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  • 原始帰納的算術（げんしきのうてきさんじゅつ、英: primitive recursive arithmetic）またはPRAは自然数の理論の量化子なしの形式化である。これはトアルフ・スコーレムによって数学基礎論におけるの形式化として提案されたもので、PRAの推論が有限の立場の範疇にあることが広く承認された。また有限の立場がPRAによって捉えきれていると信ぜられているが、有限の立場においても原始再帰よりも強い形の再帰を認めることで（PRAから）拡大することができると信ずる向きもある。それはエプシロン・ノート までの超限再帰であって、これはペアノ算術のに等しい。PRAの証明論的順序数は である。PRAはしばしばスコーレム算術とも呼ばれる。 PRAの言語は自然数と原始帰納的関数からなる算術的命題を表現できる。原始帰納的関数としては例えば加法、乗法、指数関数などが含まれる。PRAは自然数上を走る明示的な量化はできない。PRAはしばしば基本的な証明論（とりわけののような）のための超数学的なとされる。 (ja)
  • Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician , as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ε0, which is the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ωω, where ω is the smallest transfinite ordinal. PRA is sometimes called Skolem arithmetic. (en)
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  • Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician , as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ε0, which is the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ωω, where ω is the smallest transfinite ordinal. PRA is sometimes called Skolem arithmetic. The language of PRA can express arithmetic propositions involving natural numbers and any primitive recursive function, including the operations of addition, multiplication, and exponentiation. PRA cannot explicitly quantify over the domain of natural numbers. PRA is often taken as the basic metamathematical formal system for proof theory, in particular for consistency proofs such as Gentzen's consistency proof of first-order arithmetic. (en)
  • 原始帰納的算術（げんしきのうてきさんじゅつ、英: primitive recursive arithmetic）またはPRAは自然数の理論の量化子なしの形式化である。これはトアルフ・スコーレムによって数学基礎論におけるの形式化として提案されたもので、PRAの推論が有限の立場の範疇にあることが広く承認された。また有限の立場がPRAによって捉えきれていると信ぜられているが、有限の立場においても原始再帰よりも強い形の再帰を認めることで（PRAから）拡大することができると信ずる向きもある。それはエプシロン・ノート までの超限再帰であって、これはペアノ算術のに等しい。PRAの証明論的順序数は である。PRAはしばしばスコーレム算術とも呼ばれる。 PRAの言語は自然数と原始帰納的関数からなる算術的命題を表現できる。原始帰納的関数としては例えば加法、乗法、指数関数などが含まれる。PRAは自然数上を走る明示的な量化はできない。PRAはしばしば基本的な証明論（とりわけののような）のための超数学的なとされる。 (ja)
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