[FOM] Adequacy of FOL+ZFC as foundation

[T.Forster@dpmms.cam.ac.uk]

Joe Shipman writes:

>No one responded to my request for examples of mathematics done in NF 
>as a base theory that were not obviously reinterpretable as arguments 
>from ZFC.
>
>Therefore my two examples do not appear to lead to statements which are 
>generally regarded as having been proven, but which cannot be seen to 
>be theorems of ZFC.
>
>Can anyone suggest another example that might show the inadequacy of 
>FOL+ZFC?  
>
Further to my last...

In NF or ZF we can define the *Specker tree* of a cardinal $\alpha$
as follows.
 
It has a root, and it branches downwards. The root is $\alpha$, every 
vertex is a cardinal, and immediately below any cardinal (vertex) $\beta$ 
is to be found any and all the $\gamma$ such that $2^\gamma = \beta$. It is 
simple to show that every Specker tree is wellfounded, and also easy to 
show that if the tree of $\alpha$ has infinite rank then $\alpha$ is not an 
aleph. It is - as far as i know - an open question in ZF whether or not 
there can be cardinals (whose Specker trees are) of infinite rank. So if 
you have any tho'rts on this i would like to hear them! If the naturals are 
reasonably well-behaved then the rank of $|V|$ is infinite. The tree is 
wellfounded but of course there is no reason to suppose the edge relation 
is extensional. Take an extensional quotient - which of course is a set 
picture - and then the Mostowski collapse is a kosher wellfounded set which 
arises from *big* sets such as you get from NF but not ZF(C).

There are other examples. I am going to write this stuff up for an article 
Randall Holmes and i are going to write for the 75th anniversary of the 
original Quine article