THE KANSAS UNIVERSITY 

 SCIENCE BULLETIN. 



Vol. V, No. 21] MARCH, 1911. [voTxv'™i 



THE FOUNDATIONS OF ARITHMETIC. 



DISSERTATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY, 



SUBMITTED TO THE FACULTY OF THE GRADUATE 



SCHOOL OF THE UNIVERSITY OF KANSAS. 



By Arthur Bowes Frizell, of Boston. 



This thesis seeks a foundation for arithmetic in the 

 ideas underlying Cantor's formulation of his system of 

 ordinal types. 



It proceeds by postulating, but follows D. Hilbert 

 and G. Peano rather than E. V. Huntington. The 

 search is for postulates possessing heuristic and didac- 

 tic, not merely subsumptive value. 



A motif is found in the notion of an abstract group, 

 which secures the development step by step of all num- 

 ber systems so far studied without further postulates 

 than those needed for the transfinite ordinals. 



As axioms are to be avoided, it is necessary to state 

 carefully the definitions and theorems used even when 

 they are well known to the mathematicians, but in this 

 case the proofs are omitted. 



1. Definition. A set of symbols a, b, . . . will be 

 said to form a K-class if we possess a test which enables 

 us to assert in every case either that a = 6 or that a is 

 not = b subject only to the restrictions that 



(383) 



