Page:Astounding Science Fiction v54n06 (1955-02).djvu/161

clusion can be reached.

But that's precisely the problem we have to solve! What we need is a method of thinking that can handle the fact that I am thinking, and thinking about the problem I'm thinking about, and that that thinking is going to interact on and change the problem.

You know, this sounds suspiciously like another type of problem. What kind of a set can be a proper member of itself? Easy; any mathematician can tell you! An infinite set.

If you consider the set of all the whole numbers, you have an infinite set. But if you count off only the numbers that end in —5, they, too, are an infinite set, and they're just as infinite as the set of all the whole numbers. Also, the set of all the numbers that end in —,000,000,000 is equal to the set of all whole numbers—it's just exactly as infinite.

Now in a finite set, no matter how large, there is one and only one way for two sets to be equal; they have to be identical. If two groups of numbers are different, then they are not equal. The set 17395 is not equal to the set 17935. That's one of its very useful characteristics.

Aristotelian logic will do just as well as any three-valued, four-valued, or n-valued logic in handling human problems. Or just as badly, actually. It makes no difference how large the value of n in an n-valued logic may be, if it's a finite value, it still maintains the rule that only identical sets are equal. It still can't be a member of itself. That characteristic is possessed only by transfinite sets—and by a transfinite logic.

Under an Aristotelian, or any n-valued logic, the statement "A is not identical to B" means that A and B can be ordered, that A can be put in a superior or inferior rank position, but cannot be put equal to B. But in a transfinite logic, A can be equal to B. A might be "the set of all odd numbers" while B was "the set of all whole numbers." They're different, all right, but equal.

If I say, "Bill doesn't think the way Tom does," it is automatically assumed that I imply a rank, a non-equality, a superiority-inferiority ordering. But suppose Bill happens to be the world's greatest mathematician, and Tom is the world's greatest biochemist. They certainly don't think alike.

You can work with n-valued logics till the world freezes over, but in any logic having finite values, you can't make different sets be equal, nor can you make the existence of that logic a factor within the logic itself. But infinity plus infinity is infinity; infinity minus infinity is infinity. A transfinite logic can include itself very happily.

Now—all we need is the laws and rules of a transfinite logic.

Anybody got the rule-book handy?