A method is proposed for classification to ordinal categories by applying the search partition analysis (SPAN) approach. It is suggested that SPAN be repeatedly applied to binary outcomes formed by collapsing adjacent categories of the ordinal scale. By a simple device, whereby successive binary partitions are constrained to be nested, a partition for classification to the ordinal states is obt...

Let us observe that there exists an ordinal number which is non empty and natural. Let a be a natural ordinal number. Note that succa is natural. The scheme Omega Ind concerns a unary predicate P , and states that: For every natural ordinal number a holds P [a] provided the following conditions are met: • P [ / 0], and • For every natural ordinal number a such that P [a] holds P [succa]. Let a,...

We define a functional interpretation of KPω using Howard’s primitive recursive tree functionals of finite type and associated terms. We prove that the Σ-ordinal of KPω is the least ordinal not given by a closed term of the ground type of the trees (the Bachmann-Howard ordinal). We also extend KPω to a second-order theory with ∆1-comprehension and strict-Π 1 1 reflection and show that the Σ-ord...

We study mechanism which operate on ordinal preference information (i.e., rank ordered lists of alternatives) on the full domain of weak preferences that admits indifferences. We present a novel decomposition of strategyproofness into three axioms: separation monotonic, separation upper invariant, and separation lower invariant. Each axiom is a natural restriction on how mechanisms can react wh...

We show that that if every real has a sharp and there are 1 2-deenable prewellorderings of R of ordinal ranks unbounded in ! 2 , then there is an inner model for a strong cardinal. Similarly, assuming the same sharps, the Core Model K is 1 3-absolute unless there is an inner model for a strong cardinal.