Journal directory listing - Volume 11-20 (1966-1975) - Volume 11 (1966)

On Buiiding an Arstract mathfmatical System
Author: Hoong-Yuan Kong


We have mentioned that the properties of independence is not necessary for a postulate set. The consistency of a postulate set is the most important property which guarantees that not "both c and~c" can be obtained. The property of completeness is more difficult than the others. It guarantees that either c or~c can be obtained of course, not both. Concerning completeness, we need to introduce the idea of isomorphism, Let us assume that we have a system W1 of objects (such as the points, lines and planes of gepmetry) and certain appertaining basie relations R, R' ... Let Jthere be a second system W2 with corresponding basic relations which (They may have entirely different meanings.) are correlated, say, by the use of the same names, to the relations R, R' --- within the first domain of objects. Then, if it is possible to state a rule by which the elements of the system W1 are paired in a mutually unique manner with the elements of system W2, so that the elements, in W1 between which R (or R'...) holds corresponding to elements in W2 between which the relation with the same name R. (or R' ... respectively) holds; then the systems are said to be isomorphic. The correlation in question is said to be an isomorphie mapping of W1 onto W2 Isomorphie systems may be said to possess the same structure. For every true proposition about W1, there is a corresponding and identically phrased proposition about W2, and converdely. For a given postulate set P, if every two interpretations W1 , W2 of P are isomorphife, then the set P is said to be categorieal.
The categoricalness of a consistent postulate set P implies completeness' of P. Suppose P U categorical but incomplete: then there exists a statement f5 about thy ptimitive terms of P such that both S and~s are consistent with P.
Let I ba an interpretation of the consistent set of statement (P plus S) and I be an interpretation at the consistent set of statements (P plus~S.) Sinee P is categorical. ..there is a one-to-one correspondence between the elements of I and the elements of I', such that corresponding propositions in these two interpretations are either both true or both falsa. But this is impossible, since S is a true proposition for interpretation I, bnt a false prposition for interpr-etation I'.
In practice, to show incompleteness, one lias merely to produce two nonisomorphic interpretations of the postulate set. The usual process for establishing completeness is to show that any interpretation of the postulate set is isomorphic to some given, interpretation.
It is not hard to see that the postulate set in the abstract mathematical system which we have built possesses all the four properties. But in general, the study of the properties of postulate sets is not simple, Usually there are three distinet levels in the study. First of all there are the more or less informal theories of specific fields of knwledge. Secondly, there are the formal abstract postulational developments, having these specific fields as models. Thirdly there is a theory which studies the properties possessed by formal abstract postulational developments. It is the third and the highest of the three levels that Hilbert christened mathematics.
Mathematics is a great adventure for all people in the Space Age, especially for the geniuses, and for the young.

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