The axiom of choice and calculus revision: A dialogue

Abstract

M. Well, of course the axiom of choice is true. It's obvious. If you have a set A and another set B, you can define a set C that is the set of all ordered pairs (a,13) where a is in A and 13 is in B. P. You mean the set A • B exists. M. Why put it that way? P. It's making what you said more explicit. For you to "define" A • B and use it, you have to believe it exists. M. That's not perfectly clear to me, but I'll go along. Okay. P. And that raises the question, "How do you know it exists?" M. What's the problem? The set A is there, and so is B---so of course the ordered pairs (et,13) are there, and the set of them is A x B. P. But you're asserting that all these things exist! They're not physical objects-in what sense do they "exist"? M. It seems your problem is about how to put this perfectly clear idea into words. Everyone knows putting ideas into words is difficult--the computer types call it "documentation" and avoid doing it at all costs. But you remember that years ago, before you specialized in logic, you learned about ordered pairs and cartesian products, and saw it all perfectly well. P. Well I don't see it now. In what sense can a cartesian product of infinite sets be said to exist? M. You've lost it? That's too bad. P. It wasn' t really clear ever. When questions were raised about it, I couldn't answer them. Now it's clear to me that I was sort of fooling myself the concepts involved are really very obscure. M. It's clear to you? You mean the concepts you use in questioning the axiom of choice are themselves all clearer to you than the concept of cartesian product of sets? For instance, w h e n y o u ask "Wha t is the meaning of 'exists' in the statement 'A x B exists'?", is the term 'meaning' so clear to you? Also, what exactly, is the logical status of the phrase 'what is'? You know that when we wish to raise questions we can raise very difficult ques t ions about the very clearest ideas and attitudes. Some of them are taken seriously by philosophers--for example the question of whether there is such a thing as "free will," as they put i t a n d some are not taken seriously, such as Zeno's paradoxes and the question of solipsism. I don't know how they choose which to dismiss and which to talk about.