[FOM] From theorems of infinity to axioms of infinity

Timothy Y. Chow tchow at alum.mit.edu
Wed Mar 20 22:23:27 EDT 2013

I've found the responses to Michael Detlefsen's original question very 
interesting and educational.  Before the thread diverges completely onto a 
different track, though, I'd like to comment on one issue that Detlefsen 
implicitly raised in his original post.

Michael Detlefsen <mdetlef1 at nd.edu> wrote:
> Problem: Dedekind's "proof" of the assertion of the
> existence of an infinite collection is flawed, perhaps
> fatally so.
> Solution: Make the proposition purportedly proved by
> Dedekind's flawed proof an axiom!
> I'm guessing I'm not the only one who finds this a little
> funny, and a little bewildering.

This seems funny *if* you equate the *desire to provide a proof* for 
something with *a worry that it might be proved false*.  That is, if you 
think that the reason Russell and others felt an urge to provide proofs 
for the axiom of infinity was that they *doubted its truth* and therefore 
did not want to accept it without proof, then it is certainly bewildering 
to observe them accepting the statement as an axiom when the proofs fell 
through, rather than treating the statement as an open question.

But I think that the desire to provide a proof isn't always motivated by 
doubt, and the axiom of infinity is just an example of that.  For another 
example, consider Euclid's parallel postulate.  For a long time, many 
people struggled to prove it from the other axioms.  None of them ever 
doubted that it was true.  They just had a strong intuition that it should 
follow from the other axioms and that postulating it separately was 
redundant and inelegant.

Similarly, Russell never doubted the axiom of infinity, but just had a 
strong intuition that it was redundant to postulate it separately.  When 
this intuition proved to be wrong, it should not be bewildering to find 
him effectively shrugging his shoulders and saying, "Oh well, I guess 
we'll just have to postulate it separately after all."

The difference between wanting proof and having doubt can be seen even in 
the context of famous conjectures, e.g., P != NP or the Riemann 
hypothesis.  Although there is not quite enough consensus about these 
statements for them to achieve axiomatic status, in practice they are 
treated much like axioms, in that people feel free to assume them whenever 
they need to.  There's still an intense desire to find proofs for them, 
even among people who are totally convinced that the statements are true.


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