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 <tc...@alum.mit.edu> Reply-To: tc...@alum.mit.edu, Foundations of Mathematics <f...@cs.nyu.edu> To: f...@cs.nyu.edu 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 <mdetl...@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. Tim _______________________________________________ FOM mailing list f...@cs.nyu.edu http://www.cs.nyu.edu/mailman/listinfo/fom -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list?hl=en. For more options, visit https://groups.google.com/groups/opt_out.