【摘要】 <正> §1.根据序数正常表示的唯一性,M.Sierpiński证明了方程ξ~2=η~3+1没有超限的序数解.本文目的在于拓广这一结果,而从事更广一类序数方程求解问题的研究.更多还原

【Abstract】 In the paper[2]Mr.Sierpiski proved the following equation has no solution of transfinite ordinal numbers ξ,η. The main purpose of the present paper is to prove. Theorem 1.The equation ξⁿ=ηⁿ⁺¹+1 has no solution of transfinite ordinal numbers ξ,η,where n denote an arbitrary natural number,n>1. Furthermore,we can generalize theorem 1 as: Theorem 2.The equation has no solution of transfinite ordinal numbers ξ,η,where n>1,m≠2 are arbitrary natural numbers. Neverthless ξ2=η³+1 has an solution: hence,the remaining open problem is to find all such transfinite ordinal numbers ξ,η which satisfyingξⁿ2 =ηⁿ⁺¹+1. On the other hand,Mr.Sierpiski in another paper[3]proved the following equations αβ=βα α^mβⁿ=βⁿα^m are equivalent for ordinals α and β,by the same methode we can prove the following equations α+β=β+α αm+βn=βn+αm are equivalent for ordinal numbers α and β.更多还原