Acta Scientiarum Universitatis Pekiniensis (Naturalum)

a1 Ai — MX KE RH AH 88

ES 6 xX HK

Cl] PABA abe KS (ARS), 2, 429142, 1956.

ON A SYSTEM OF AXIOMS FOR THE POSITIVE NUMBERS Leng Sen-ming (Department of Mathematics and Mechanics) Axpsrract

We have proposed a system of axioms (I—XII) for the positive numbers. [This journal, 2, 441—442 (1956). Axiom V there should bs supplemented by the: additional condition impcsed on the positive number 1: such that +=1.] Here we establish the Icgical equivalence among this system and two others. The first is

obtained from I—XII by modifying V and XI as follows.

Y". 1 is a pesitive number.

XI’. If p is a positive number, then z ‘i <i and 1<p+1.

P The second system is obtained from J—XII by writing p* and p7 for p+1 and

= and by modifyingV and VIII as follows (so that the individual positive number 1 is dispensed with in the statements of the axioms).

V". There exists at least one pcsitive number.

VII’. If p and-q are pesitive numbers such that p<q, and if a set of positive numbers contains at least one positive number and contains r* and r~ for each of its elemenis r, then the set contains a pcsitive number r such that p<r and

r<q. '