[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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