FOM: Large Cardinals Axioms
Arnon Avron
aa at math.tau.ac.il
Thu May 11 06:33:28 EDT 2000
There is at least one prediction of Friedman concerning the future of
large cardinals axioms which I find quite doubtful. Friedman writes:
"Because of the thematic nature of these developments, and the interaction
with nearly all areas of mathematics, large cardinal axioms will be begin
to be accepted as new axioms for mathematics - with controversy. Use of
them will still be noted, at least in passing, for quite some time, before
full acceptance. This was the case with the axiom of choice. This will be
accompanied by various improved intrinsic justifications of large cardinals
along various lines."
I believe that the comparison with AC here is inappropriate. The big difference
is that AC has implicitly been used by mathematicians (at least in a weak
form, like DC) long before it was explicitly formulated by Zermelo and Russel.
It is still implicitly used in any rigorous introduction to analysis
(A first-year undergraduate course for math students in every university
in Israel). In contrast, I dont believe that any "core" (or other)
mathematician has ever implicitly used large cardinal axioms in her/his
work, and I dont believe one ever will (except perhaps set-theorists
actually working in this particular area).
Let me add here that personally I cannot be persuaded by Friedman's arguments
(and impressive achievements) because of my (old-fashioned?) criterion of
accepting a statement as an AXIOM only if it is obviously, self-evidently
TRUE. And I dont see what can make me suddenly realize that a certain large
cardinal axiom is in fact self-evident (how could I have missed such a
self-evident fact before?? Note that the fact that AC has been used
implicitly before its discovery puts it in a completely different state!).
Worse: I dont see what can possibly convince
me that large cardinal statements are meaningful - let alone true
(I am not saying that they are not meaningful - I just dont know whether
they are or not, and I cannot imagine what can settle this problem for me).
What I can at most accept, on empirical ground, is that many of them
are most probably consistent, and so what can be proved using them
most probably cannot be refuted. But for this what is actually used is
the consistency of these axioms, not the axioms themselves. The example
of V=L shows that even from the points of view of Friedman and Steel these
are different things.
So my point is: arguments like those of Friedman can be very convincing, but
only for scientists who are already more or less convinced.
Arnon Avron
Department of Computer Science
School of Mathematical Sciences
Tel-Aviv University.
Tel-Aviv, Israel
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