[FOM] From theorems of infinity to axioms of infinity

[Timothy Y. Chow]

Nik Weaver wrote:

> Come again?  How do you get that??

If that's not what you intended, then disregard my comment.

>> Your arguments would make more sense if you were to direct your
>> criticism against ZFC in particular.  As it stands, you are making
>> blanket statements against "set theory" in general that amount to
>> shooting yourself in the foot.
>
> I'm not sure if you've been following the previous discussion, but I am 
> arguing that the power set axiom is philosophically dubious and not 
> needed for ordinary mathematics.  I don't have any idea what you're 
> arguing about.

I'm arguing about what you said when you replied to Monroe Eskew:

> 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*.  For
> instance, question number (2) certainly did not "turn out to be
> independent", it is a theorem of PA that if inaccessible cardinals
> are consistent then the axiom of choice is needed to build nonmeasurable
> sets.
> 
> But how on earth can the fact that various questions *cannot* be
> answered within standard set theory, be a reason for using 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.

You used the term "set theory" four times here, or three times if we 
discount the use of the term "standard set theory."  The final time you 
used it, you clearly meant "ZFC" and not "set theory."  In the other 
cases, if you meant "set theory" (which was the term Monroe Eskew used), 
then I've already responded why Monroe Eskew's argument for using set 
theroy is a reasonable one.  If you actually meant ZFC---well, my point is 
that you should keep straight when you mean "set theory" and when you mean 
"ZFC" and when you mean "the power set axiom."

Tim