Einstein, the searcher

by Alexander Moszkowski

Before we deal with cases which seem to me, personally, to be interesting, let us hear what is Einstein's attitude in general. "As soon as a paradox presents itself, we may, as a rule, infer that inaccurate reasoning is the cause, and should thus examine in each particular case whether an error of logic is discoverable, or whether the paradoxical result denotes only a violent contrast with our present views."

Let us first take examples from an entirely modern science, from the Theory of Aggregates founded by Georg Cantor of Halle. We shall follow the argument by the only possible method for this book, namely, by rough indications that will serve our purpose and do not claim to be accurate in expression or in sense.

If we take an aggregate of three objects, for example, an apple, a pear, and a plum, we may, by definition, form six partial aggregates, namely:

the apple
the pear
the plum
the apple and the pear
the apple and the plum
the pear and the plum.

The aggregate of the partial aggregates, which contains six elements, is thus greater than (actually twice as great as) the original aggregate, in which only three elements occur.

If the original aggregate contains an additional element, for example, a nut, the following partial aggregates may be formed:

the apple
the pear
the plum
the nut
the apple and the pear
the apple and the plum
the apple and the nut
the pear and the plum
the pear and the nut
the plum and the nut
the apple, the pear, and the plum
the apple, the pear, and the nut
the apple, the plum, and the nut
the pear, the plum, and the nut.

Thus, in this case, the aggregate of the partial aggregates is already considerably greater than the original aggregate. This numerical excess increases rapidly with each successive increase in the original aggregate, so that if we apply the same reasoning to an infinite aggregate, the aggregate of partial aggregates becomes an infinity of a higher order. This is expressed by saying that the infinite aggregate of partial aggregates has a greater potentiality than the infinity of the elements of the original aggregate.

So we see that the one infinity is, in popular language, much more comprehensive, more powerful than the other. Our minds do not find it impossible to grasp this. But in a definite imaginary experiment it is found that this theorem of progression not only fails in its application, but leads to flagrant contradiction.

For if we start from the primary aggregate of "all conceivable things," its infinity can certainly not be transcended by any other infinity. But according to the above theorem the "aggregate of all partial aggregates" would have a greater potentiality, although it itself cannot extend further than to the conception of the maximum of all conceivable things. We thus arrive at an insoluble paradox, a typical example of how, in the system of conceptions involved, something is insufficient or not in conformity with logical thought. And this sceptical view receives support from various remarks of Descartes, Locke, Leibniz, and particularly Gauss, who, long before the advent of the Theory of Aggregates, raised a protest against inexact definitions of infinity.