It's a good discussion. In fact I've been independently thinking about the 
matter of Dedekind's original argument as it's discussed in Webb's book 
"Mechanism, Mentalism, and Metamathematics" (a book Bruno has referred to 
multiple times on this list). 

Does anyone know of other attempts to prove the existence of infinite sets, 
unrelated to Dedekind?

It seems like the truth or value of axioms (of infinity or otherwise) 
should be judged by their fruitfulness in "ordinary" mathematical 
reasoning. In this sense their truth may be understood "inductively". 

[Harvey Friedman has spent decades showing that several "natural" and 
"concrete" mathematical statements are not only independent of ZFC but 
require certain large cardinal axioms. In short, the existence of 
(massively!) infinitary objects has consequences for finitary mathematical 
questions. Friedman predicts that these statements will become a mainstream 
component of future mathematics, requiring the adoption of the large 
cardinal axioms.I recommend this YouTube video:]

On Wednesday, March 20, 2013 11:37:58 PM UTC-5, Stephen Paul King wrote:
>  Hi Folks,
>     I apologize for crossforwarding a post, but this one is too good to 
> not...
> -------- Original Message --------  Subject: Re: [FOM] From theorems of 
> infinity to axioms of infinity  Date: Wed, 20 Mar 2013 22:23:27 -0400 
> (EDT)  From: Timothy Y. Chow <> <javascript:>  Reply-To: 
> <javascript:>, Foundations of Mathematics 
> <> <javascript:>  To: <javascript:>  
> 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 <> <javascript:> 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.
> Tim
> _______________________________________________
> FOM mailing 
> <javascript:>

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