[FOM] ZC vs. ZFC

Thomas Forster T.Forster at dpmms.cam.ac.uk
Sun Jan 31 07:43:51 EST 2010

We seem to be having this discussion about the need for replacement
over and above Zermelo far too often for my taste.  Here is my
two penn'orth; i shall be brief, and then silent!

The point is often made that replacement doesn't seem to be needed
for much of `ordinary mathematics'.  This isn't really a point greatly
worth worrying about even if it is true, since what is ordinary
mathematics in 1900 isn't ordinary mathematics in 2000 or 3000 and
whereas it might be worth worrying whether or not replacement is *true* 
(whatever that might mean) in the cumulative hierarchy, worrying about 
where it is needed in this place at this time is not a mathematical 
concern: mathematics is time-invariant. (To be fair, some of the people 
who observe from time to time that replacement doesn't seem to be needed 
&a, &a, do also say that is not a point of metaphysical significance).

Harvey Friedman has a catalogue of nice facts about sets of low rank that 
can be proved only by reasoning about sets of high rank.   This *ought*
to shut down the debate, but for some reason it doesn't.  I think it is 
something to do with the distain in which logic is held by many 
mathematicians, which makes them reluctant to accept illustrations of the 
kind Harvey supplies.  Another reason for their failure to appreciate the 
need for replacement is a general disinclination to pay attention to what 
axioms are used, so that even when replacement *is* needed for what they 
do, they don't realise they are using it.  (This point is commonly made 
about AC, but it holds for replacement as well)

This brings me to what - for me at least - is one of the major reasons for 
accepting replacement.  There are at least three halfway-sensible ways of 
thinking of natural numbers as sets that have been proposed: there are the 
Zermelo naturals, the von Neumann naturals, and there are Scott's trick 
naturals.  It is a fact so important and so basic that it tends not to get 
spelled out that *it doesn't matter which implementation we use*. 
Certainly the von Neumann implementation is cuter than the others, and 
more generally used, but we must not lose sight of the fundamental insight 
that *it shouldn't matter which one we use*.  Actually i don't think we 
are in danger of denying this point, so much as *overlooking* it.  And 
here is the key point:

   *** If we want to maintain a stance of lofty indifference about      ***
   *** our choice of implementation then we have to assume replacement. ***

And lofty indifference is surely the mathematically correct stance. 
Perform the following thought-experiment:  Ramanujan turns out not to have 
been dead, but in a coma, and he wakes up.  (You will remember that Hardy 
said of him that all the natural numbers were his personal friends.)
When he opens his eyes, you greet him with the news that during his 
absence we discovered that the natural numbers were hereditarily 
transitive finite sets wellordered by the membership relation.  Will he 
 	(i) ``Thank you for sharing that with me''


 	(ii) ``Security, please take this lunatic away''

I think we can all agree that the fact that natural numbers were 
hereditarily transitive finite sets wellordered by the membership relation 
is of no conceivable use to a number theorist.

  If you propose to refrain from using replacement then you cannot have 
lofty indifference.  Notice that this means that you have to plump for one 
(or at least a restricted bundle of) implementations of (e.g., ordinals) 
and decide that ordinals just are sets of that kind!  How ironical that 
the people who regard replacement as an unmathematical logicians' plaything 
are denying themselves the right to the lofty indifference that many of 
them (I have in mind for example Godement's *Cours d'algebre* - thanks to 
Adrian Mathias for drawing it to my attention) fairly explicitly come out

If you want to be able to behave like an - ``ordinary!'' - mathematician
you have to have replacement.

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