Elementary Functional Arithmetic

Elementary Functional Arithmetic, or EFA, is a weak theory of arithmetic created by removing induction from Peano Arithmetic. Because it lacks induction, axioms defining exponentiation must be added.

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$\\forall x(x^{\\prime}\eq 0)$ ( $0$ is the first number)
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$\\forall x,y(x^{\\prime}=y^{\\prime}\\rightarrow x=y)$ (the successor function is one-to-one)
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$\\forall x(x+0=x)$ ( $0$ is the additive identity)
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$\\forall x,y(x+y^{\\prime}=(x+y)^{\\prime})$ (addition is the repeated application of the successor function)
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$\\forall x(x\\cdot 0=0)$
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$\\forall x,y(x\\cdot(y^{\\prime})=x\\cdot y+x$ (multiplication is repeated addition)
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$\\forall x(\eg(x<0))$ ( $0$ is the smallest number)
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$\\forall x,y(x<y^{\\prime}\\leftrightarrow x<y\\vee x=y)$
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$\\forall x(x^{0}=1)$
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$\\forall x(x^{y^{\\prime}}=x^{y}\\cdot x)$

单词	ElementaryFunctionalArithmetic
释义	Elementary Functional Arithmetic Elementary Functional Arithmetic, or EFA, is a weak theory of arithmetic created by removing induction from Peano Arithmetic. Because it lacks induction, axioms defining exponentiation must be added. • $\\forall x(x^{\\prime}\eq 0)$ ( $0$ is the first number) • $\\forall x,y(x^{\\prime}=y^{\\prime}\\rightarrow x=y)$ (the successor function is one-to-one) • $\\forall x(x+0=x)$ ( $0$ is the additive identity) • $\\forall x,y(x+y^{\\prime}=(x+y)^{\\prime})$ (addition is the repeated application of the successor function) • $\\forall x(x\\cdot 0=0)$ • $\\forall x,y(x\\cdot(y^{\\prime})=x\\cdot y+x$ (multiplication is repeated addition) • $\\forall x(\eg(x<0))$ ( $0$ is the smallest number) • $\\forall x,y(x<y^{\\prime}\\leftrightarrow x<y\\vee x=y)$ • $\\forall x(x^{0}=1)$ • $\\forall x(x^{y^{\\prime}}=x^{y}\\cdot x)$
随便看	ProofOfCaratheodorysLemma ProofOfCaratheodorysTheorem proof of Carathéodory’s extension theorem proof of Carathéodory’s lemma proof of Carathéodory’s theorem ProofOfCasoratiWeierstrassTheorem proof of Casorati-Weierstrass theorem ProofOfCassinisIdentity proof of Cassini’s identity ProofOfCasusIrreducibilisForRealFields proof of casus irreducibilis for real fields ProofOfCatalansIdentity proof of Catalan’s Identity ProofOfCauchyConditionForLimitOfFunction proof of Cauchy condition for limit of function ProofOfCauchyDavenportTheorem proof of Cauchy-Davenport theorem ProofOfCauchyIntegralFormula proof of Cauchy integral formula ProofOfCauchyResidueTheorem proof of Cauchy residue theorem ProofOfCauchySchwarzInequality proof of Cauchy-Schwarz inequality ProofOfCauchySchwarzInequalityForRealNumbers proof of Cauchy-Schwarz inequality for real numbers