On n-Place Strictly Monotonic Functions

✍ Scribed by John Hickman

• Publisher: John Wiley and Sons
• Year: 1985
• Tongue: English
• Weight: 164 KB
• Volume: 31
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19850310904

✦ Synopsis

ON %-PLACE STRIClTLY MONOTONIC FUNCTIONS by JOHN HICXMAN in Canberra (Australia)')

Let O X be the class of ordinals, and let f : ON" -+ ON be a n n-place ordinal funcl tion for some number 91 > 0. We define a partial order <* on ON" by set'ting (a., . . . . j .rn) < * (y, , . . ., y,,) if .ri 5 yi for each i 5 TL and x i < yi for some i 5 n.

Throughout this note we shall he working with finite sequences (.rl , . . . x n ) of ordinals. If i is some index with 1 5 i 5 n., and if we stipulate in a given context that x i is replaced by a n ordinal z, then by (xl, . . .. z , . . ., x,) we shall mean the sequence (.rl . . . . . .ri-, z , x i + . , x,,). Also t,hroughout this not'e n-e shall denote hy (11 the least transfinite ordinal.

We define f to be .rz.or~?iaZ if f is coiitjiiiuous in each variable and strictly monotonic (i.e. f(.rl ~ . . . . x,,) < f ( y , , . . . ~ y,,) whenevever ( 2 , , . . . . x,,) < * (y, , . . . , y,,)). We wish to show that for 71 > 1 there are no normal functions.

Assume that f is a normal function. We show first of all t.hat max(z, ~ .