FOM: Re: On Friedman's results

Arnon Avron aa at
Tue Mar 17 18:21:13 EST 1998

> 3. For Torkel Franzen: You suggest that the importance or general
> intellectual interest of Friedman's result depends on
> 	"the applicability of the combinatorial principles and the
> 	epistemological status of the large cardinal principles."
> I disagree with this. The importance, for me, of the independent
> statement being a combinatorial one is simply that that makes it more
> intuitively graspable as something that darn well *ought* to have a
> truth-value. Whether it has applications seems to me
> irrelevant. Furthermore, the *shakier* the epistemological status of
> the large cardinal principles, the more arresting the
> result!---because Friedman has shown that this relatively simple
> combinatorial statement is no more secure than the large cardinal
> principle. His result *puts on an epistemological par* the two kinds
> of principle---combinatorial and large-cardinal. *That* is why it's so
> impressive. If I am told that someone has shown "A iff B", when that
> sort of equivalence would have struck me, prima facie, as highly
> unlikely, I don't question the significance of the proof of that
> equivalence by saying "Well, we'll have to wait and see whether A has
> any application; and we'll have to wait until we've settled the
> epistemological status of B." If A looks like the kind of claim that
> ought to be true for simple reasons, and B is the kind of claim of
> which I already know that many people claim not to know how one might

As far as I understand, what Friedman has shown is *not* that "A iff B"
(where is the "relatively simple combinatorial statement" while B is
the large cardinal principle). As Feferman has pointed out, what has
been shown is the equivalence of A with the *1-consistency* of B. Now
the latter is at least as "intuitively graspable as something that darn 
well *ought* to have a truth-value" as this combinatorial statement
(and it is about the natural numbers, which form a data structure which
is no less in use in computer science than trees). So the difference,
if any, is purely psychological. Indeed, Friedman again and again
emphasized that his combinatorial statement is *natural* and that
experts find it interesting. I dont deny its interesrt, but I still
fail to see its importance to FOM per se. Should we now have more
confidence in the consistency of the assumption that subtle cardinals
exist than we had before? THe answer can be positive only if we have 
some independent reason to believe in Friedman's combinatorial statement 
rather than in its negation (which is no less interesting, I believe). 
Do we? Moreover: I find it very plausible (almost certain, I dare say) 
that to most consistency statements there exist equivalent "natural" 
combinatorial results. So suppose we find such equivalents to the consistency 
of two contradicting principles (both unprovable in ZFC). What will this tell us
about the truth of these principle or even about their meaning (if any)? 

Arnon Avron

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