[FOM] Infinity and the "Noble Lie"
joeshipman@aol.com
joeshipman at aol.com
Mon Dec 12 15:12:02 EST 2005
Bill, it doesn't matter whether you are asserting RH or asserting RH is
"true". Either way creates a problem, AS LONG AS you are not merely
asserting that RH has a proof in such-and-such a system.
Suppose someone proves RH, and the proof uses the Axiom of Infinity in
an essential way. Anyone who claims to doubt the existence of "actually
infinite sets" has no right to assert RH, or cite it as a needed lemma
in his own work. But many more people seem to be willing to say they
doubt the existence of infinite sets than are willing to take this
doubt to its consistent conclusion and say that theorems like the Graph
Miinor theorem have not really been proven, and that credit for the
proof of the Prime Number Theorem properly belongs to Erdos/Selberg and
not to Hadamard/de la Vallee Poussin.
In the particular case of statements like RH which are provably
equivalent over ACA0 to a pi^0_1 statement, one does not need the Full
Axiom of Infinity, just that the Axiom of Infinity is consistent with
the other axioms used. I am sure that many who doubt the existence of
actually infinite sets are happy to say they do not believe that such
sets are inconsistent, and, if pressed, will admit that Con(ZFC) is an
ontologically acceptable axiom even if the full system of ZFC is not.
There are only 4 alternatives I can see which avoid hypocrisy: accept
the full Axiom of Infinity, or accept an axiom such as Con(ZFC), or
protest that certain widely accepted theorems have not really been
proven, or claim that such theorems don't have definite truth values.
For Pi^0_1 statements, the fourth alternative would be widely rejected;
but those who don't admit the existence of actual inifnities are not
protesting very hard that, say, Wiles's theorem has not really been
proven, nor are they offering any axioms they accept from which Wiles's
theorem follows. (In the case of Wiles's theorem the situation is
actually even worse -- the proofs all ultimately pass through results
proved using Grothendieck Universes, and thus are not officially proved
in ZFC unless and until someone cities a metatheorem that ensures the
"Universes" assumption is eliminable. I asked the FOM list in 2003
"where is the metatheorem", but all the defenders of Wiles's proof
neither offered one nor cited any published work that showed the
assumption was eliminable in this particular case -- they all just said
"all the top people in the field know how to eliminate the assumption"
and seemed to think that was good enough.)
There are no famous Pi^0_1 theorems which have been shown to be
unprovable without the Axiom of Infinity, so the question is still
avoidable by saying "I believe the use of Infinity in the proof of X
will eventually be eliminated, and in any case I don't think a
contradiction will ever be found in ZFC, so I'm happy to simply assert
X." But for Pi^0_2 this is not an option. The Graph Minor theorem
provably requires the Axiom of Infinity (query for Harvey: is there a
proof in ZFC of the Graph Minor Theorem which avoids the use of the
Power Set axiom, or does the theorem actually require an uncountably
infinite set?).
Is anyone willing to assert that the Graph Minor Theorem has not really
been proven and may in fact be false or have no truth value? Because I
want to see how anyone UNwilling to make that assertion can continue to
claim to disbelieve in actual infinities.
(previous dialogue appended below -- JS)
Shipman:
> My point is that there does appear to be an antecedent notion of truth
> in mathematics as far as "ordinary mathematics" is concerned -- for
> example, there is a fair consensus that the Riemann Hypothesis is
> either true or false but we don't know which. Those who deny that an
> antecedent notion of truth exists which settles the Axiom of Infinity
> had better explain what to make of statements which do not mention
> infinite sets but all known proofs of which require the axiom of
> infinity. (For example, Kruskal's theorem in Friedman's finite
> form, or
> various other specializations of the Graph Minor theorem.)
Tait:
As to the first point, it seems to assume that the notion of
mathematical truth (as opposed to that of truth in a model) is
employed *within* "ordinary mathematics". it surely is used
informally, but, I would argue, not essentially.To use your example,
the statement that RH is true is no more than the statement that RH:
certainly the two would be proved in the same way. So the assertion
that RH is true or false is nothing more than the law of excluded
middle, RH v not-RH, which indeed is assumed in ordinary
mathematics. For example, if we can prove P both from RH and from not-
RH, we would conclude that P.
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