[FOM] From theorems of infinity to axioms of infinity

[Monroe Eskew]

On Mar 17, 2013, at 9:51 AM, Nik Weaver <nweaver at math.wustl.edu> wrote:

> You seem to be confusing working within ZFC and reasoning about ZFC.
> You cite relative consistency results *which are theorems of Peano
> arithmetic* as evidence that *we should use set theory*.

This goes to the arguments I made in previous threads.  As a practical matter and historical fact, it is only by working within ZFC (plus large cardinals) that we come to know meta-mathematical facts about ZFC, which are ultimately theorems of PA.  In the course of proving a result such as Con(ZFC+ measurable cardinal) implies Con(ZFC + total probability measure in R), we do for a long time actually "use set theory" as a collection of working hypotheses.  Now you can bracket all this work with prefixes and suffixes so that you boil down the content you find metaphysically acceptable, but the real work is done within set theory.

> But how on earth can the fact that various questions *cannot* be
> answered within standard set theory, be a reason for using set theory?

Because it is only through the methods of set theory that these things are known.  And by this I mean nontrivial stuff-- getting deep into forcing, infinitary combinatorics, inner models, large cardinals.  Using set theory is our only hope for understanding the metamathematics of set theory.


> This phenomenon, where seemingly fundamental questions like the
> continuum hypothesis are independent of ZFC, should rather raise
> suspicions that something is wrong with set theory as a foundational
> system.  All the more so when these seemingly fundamental questions
> turn out to have little or no relevance to mainstream mathematical
> concerns.

If you have some idea for replacing set theory with some equally powerful systems for attacking relative consistency problems that also fits with your metaphysical beliefs, please do tell.  These days, independence results are the main product manufactured by set theorists.  No one else has tools that can manufacture the same products.  It is, by objective standards, a successful area of mathematics, not one worthy of "suspicion that something is wrong with it."

It is quite a stretch to say these results are not relevant to mainstream mathematics.  Many classical questions of analysis and topology have been answered or shown independent by set theory.  Consider the question of total probability measure, or the Borel conjecture, or various questions about the relation between measure and category having to do with "cardinal characteristics of the continuum."  The Whitehead problem.  The recent work on C^* algebras and the influence of forcing axioms.  Classification of ergodic systems.  Many of these questions came from mainstream mathematics.  You'd have to be a revisionist to say set theory is irrelevant.

Monroe