ZF + DC + AX4

We consider mainly the following version of set theory: “ZF + DC and for every λ, λℵ0 is well ordered”, our thesis is that this is a reasonable set theory, e.g. on the one hand it is much weaker than full choice, and on the other hand much can be said or at least this is what the present work tries to indicate. In particular, we prove that for a sequence ¯δ = δs : s ∈ Y , cf(δs) large enough compared to Y , we can prove the pcf theorem with minor changes (in particular, using true cofinalities not the pseudo ones). We then deduce the existence of covering numbers and define and prove existence of true successor cardinal. Using this we give some diagonalization arguments (more specifically some black boxes and consequences) on Abelian groups, chosen as a characteristic case. We end by showing that some such consequences hold even in ZF above. Date: November 1, 2014. 2010 Mathematics Subject Classification. Primary 03E04, 03E25; Secondary: Key words and phrases. set the...

• Type: Journal
• Year: 2016
• Where: AML
• Authors: Saharon Shelah
• Keywords: Algorithms, AML 2016