The real core model and its scales Original Research Article

This paper introduces the real core model K(View the MathML source) and determines the extent of scales in this inner model. K(View the MathML source) is an analog of Dodd-Jensenʹs core model K and contains L(View the MathML source), the smallest inner model of ZF containing the reals R. We define iterable real premice View the MathML source and show that Σ1(scM)∩View the MathML source(View the MathML source) has the scale property when View the MathML sourcevRAD. We then prove the following Main Theorem: ZF + AD + V = K(View the MathML source) ⇒ DC. Thus, we obtain the Corollary: If ZF + AD +View the MathML source(View the MathML source)⊈L(View the MathML source) is consistent, then ZF + AD + DC + ∀α < ω2 (α-Π11)-ADView the MathML source is also consistent.