From: Neil W Rickert Subject: Re: Lowenheim-Skolem, unknowable sets, and the Continuum Hypothesis Date: 15 Jun 2001 11:50:47 -0500 Newsgroups: sci.math Summary: How important is the Axiom of Choice in physics? mathwft@math.canterbury.ac.nz (Bill Taylor) writes: >Hey Neil - always nice to see your stuff. >Neil W Rickert writes: >|> Lots of functional analysis depends heavily on AC, and lots of >|> theoretical physics depends on functional analysis. >Nice trick example! >You are right on both counts; but (as I'm sure you know) those two parts >of functional analsysis don't overlap at all! What physicist ever needs >a Hamel basis for anything, or to handle the vector space of finite >sequences, or any of those other outre' choice-related things? You are quite right, that a physicist can manage without a Hamel basis. It's a tad more difficult to get by without the Hahn Banach theorem. In fact, it is doubtful that there could be such a thing as functional analysis without the Hahn Banach theorem. The involvement of AC in functional analysis goes far deeper than you might suppose. >I agree as a unifying elegant simplification, ZFC is far preferable to ZF; >but the C is not directly applicable to the real world. Of course you are right about C. But then the ZF part is not directly applicable to the real world either. If we limited ourselves to what is directly applicable to the real world, mathematics would be the empty set. No, I take that back. I hear tell that those quantum thingies have a nasty habit popping into existence without provocation, and it would really complicate matters if one were to pop into existence in the middle of the empty set.