[FOM] throwing darts at natural numbers (rejoinder to Arnon Avron's reply)

[Allen Hazen]

Timothy Chow wrote:

> By the way, I want to revise something I said before.  I said something
> like, "In that case, AC sounds pretty dubious."  On further reflection, I
> would rephrase that, because the word "dubious" suggests that one is
> considering rejecting AC.  I didn't mean to connote that.  What I would
> argue is that the pathological sets associated with Freiling's argument
> can be, and perhaps are (by many people at least), thought of as a
> consequence of AC + CH together rather than of CH alone.
> 
> The distinction I'm making is that we can regard a counterintuitive result
> as being a *consequence* of X without necessarily *doubting* X.
> 
---------->I am reminded of a time in graduate school (in philosophy, but a
logic-friendly department) when I delivered my self of the opinion that
cardinal trichotomy was, intuitively, OBVIOUSLY true and that the
Well-Ordering Theorem was, intuitively, OBVIOUSY very fishy.  This was
received with approbation by people who were fully aware of their
equivalence.   ...   Moral, if there is one, is that intuitions have to be
well-schooled.  When one has studied set theory as deeply as Gödel did, or
as Woodin has, one's intuitions will start deserving respect.

Allen Hazen
School of Philosophy, Anthropology and Social Inquiry (our
    managers like mergers!)
University of Melbourne