[FOM] Explicit constructions

[JoeShipman@aol.com]

In a message dated 7/3/2003 3:01:09 PM Eastern Standard Time, jbaldwin at uic.edu writes:

> One of the important discoveries of the middle 20th century is the
> futility of a trying to find a general foundations of 
> mathematics


In what sense is the usual foundations via ZFC "futile"?  Is there a part of mathematics it cannot provide a foundation for?  Or does it have some other irreparable inadequacy?

You can't mean that ZFC is futile because it can't prove statements like CH or "measurable cardinals exist", that is asking too much of a "foundation".  A foundation is supposed to provide a completely rigorizable basis for the mathematics mathematicians actually do, not answer every possible mathematical question.

It is true that it is futile to search for a single complete axiom system for all of mathematics, but that doesn't mean the mathematics that is actually done can't be given an adequate foundation.  Hilbert's dream of an algorithm to settle all mathematical questions was futile, but "Foundations of Mathematics" is concerned with much more than that.

-- JS