Given a system of inequalities with ordinal numbers: <mtable columnalign="right left right left

Leland Morrow

Leland Morrow

Answered question

2022-06-21

Given a system of inequalities with ordinal numbers:
a 1 > b 1 a n > b n
Let ϕ be a permutation on { 1 , , n }. Is it true that a ϕ ( 1 ) + + a ϕ ( n ) > b 1 + + b n ?

Answer & Explanation

Paxton James

Paxton James

Beginner2022-06-22Added 25 answers

No. There are already counterexamples with n = 2. Here is one: ω + 1 > ω and 3 > 2, but 3 + ( ω + 1 ) = ( 3 + ω ) + 1 = ω + 1 < ω + 2.
Similar counterexamples can be obtained for any n > 1: just consider ω + 1 > ω and a bunch of inequalities of the form k i > m i for k i , m i finite, with the only requirement that m 2 + + m n > 1. Again, k 2 + + k n + ( ω + 1 ) = ω + 1 < ω + m 2 + + m n

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