A type of subtlety for
Pκλ called “strongly subtle” is introduced to show almost ineffability is consistencywise stronger than Shelah property. The following are also shown:
is strongly subtle” has rather strong consequences. (ii) The ideal
is not strongly subtle} is not
λ-saturated , and completely ineffable ideal is not precipitous. (iii) In case that
λκ=2
λ, almost
λ-ineffability coincides with
λ-ineffability. (iv) It is not provable that
κ is
λκ-ineffable whenever
κ is
λ-ineffable.
Key words or phrases: Pκλ - Subtlety - Shelah property - (almost) ineffability - Partition property
Mathematics Subject Classification (2000) Primary 03E
Research partially supported by “Grant-in-Aid for Scientific research (C), The Ministry of Education, Science, Sports and Culture of Japan 09640299”, and “Japan Society for the Promotion of Science 14540142”.The author is very grateful to the referee for his correcting many errors and helpful suggestions.