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The following theorems are proved about the Frattini-free componentG
Φ of a soluble stable ℜ-group: a) If it has a normal subgroupN with nilpotent quotientG
Φ/N, then there is a nilpotent subgroupH ofG
Φ=NH. b) It has Carter subgroups; if the group is small, they are all conjugate. c)...
It is shown how the strong ordinal notation systems that figure in proof theory and have been previously defined by employing large cardinals, can be developed directly on the basis of their recursively large counterparts. Thereby we provide a completely new approach to well-ordering proofs as...
The notion of a “ptyx” is formalised in second-order arithmetic, and, using proof-theoretic techniques based on sequent-calculus, bounds are obtained for the ptykes of type 1 and 2 which can be proved to be ptykes using arithmetic comprehension.
Assuming the consistency ofZF + “There is an inaccessible number of inaccessibles”, we prove that Kelley Morse theory plus types is not a conservative extension of Kelley-Morse theory.
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