Semi-formal finitist proof of the transfinite induction in an initial segment of Cantor's second number class

1981 ◽  
pp. 67-79 ◽  
Author(s):  
Shôji Maehara
Keyword(s):  
Initial Segment ◽  
Transfinite Induction ◽  
Number Class

Systems of notations and the ramified analytical hierarchy

1974 ◽  
Vol 39 (2) ◽  
pp. 243-253 ◽  
Author(s):  
Joan D. Lukas ◽  
Hilary Putnam
Keyword(s):  
Initial Segment ◽  
Turing Degrees ◽  
Power Set ◽  
The Church ◽  
Insight Into ◽  
Degrees Of Unsolvability ◽  
Number Class

The purpose of this paper is to show that arithmetically minimal systems of notations can be constructed which provide notations for all ramified analytical ordinals (all the ordinals in the minimum β-model for analysis). This is a much larger section of the second number class than the Church-Kleene constructive ordinals (although still only an initial segment of the ordinals). Arithmetic minimality means that if H is an “H-set” associated with an ordinal α in our system and H′ is an H-set associated with the same ordinal α in an arbitrary system of notations S, then H is arithmetical in H′. Thus the arithmetical degrees associated with ordinals in our system are as low as possible.In order to clarify the structure of degrees of unsolvability and, more generally, to gain a deeper insight into the power set of the integers, coarser but neater classifications than the structure of Turing degrees have been sought. Several hierarchies of sets of integers have been studied, each of which organizes a certain class of sets (or their degrees of unsolvability) into a well-ordering of levels with increasing complexity of nonrecursiveness appearing at each new level. The best known of these hierarchies is the Kleene hierarchy of arithmetical sets.

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