FOM: mathematical certainty
Volker Halbach
Volker.Halbach at uni-konstanz.de
Tue Dec 29 11:39:56 EST 1998
In a comment on a note by Andreas Blass Mic Detlefsen has raised some
interesting issues. In particular, I am interested in the question of why
we accept the consistency statements or reflections principles of theories
(e.g. ZFC in Blass' note) we accept.
Mic Detlefsen mentions three possible ways to arrive at the consistency
statement:
"Possibility 1: T (a set of axioms) is true. T couldn't be true unless it
were consistent. Therefore, T is consistent.
Possibility 2: T. If T, then T is consistent. Therefore, T is consistent."
Since the third possibilty seems to rely, as Detlefsen correctly points
out, either on the first or second possibility, I neglect it.
Possibility 1 explicitly involves a notion of truth, Possibility 2 can be
formulated without a truth predicate, if we have singled out a finite
axiomatization. For this case Detlefsen asks how we can deduce the
consistency of T from T itself.
I think Possibility 2 is only accepted as a consequence of Possibility 1. I
do not see any way to get from T to the consistency statement for T without
a notion of truth (or soundness etc.). My reasons for this belief are these:
1. In practice we never argue in the fashion of 2. Nobody lists all axioms
of a finitely axiomatized theory like Q, ACA_0 or Bernays-Goedel Set Theory
and says then "Thus Con_Q" or "Thus ACA_0". We couldn't explain why, say, a
certain set theoretic axiom of Bernays-Goedel Set Theory is needed in order
to establish the consistency of Bernays-Goedel Set Theory. However, the
*truth* of all axioms can be used for establishing consistency.
2. Possibility 2 is a *scheme* and it cannot be formulated as a single
principle without appealing to a truth predicate.
I conclude that what is actually going on is the addition of a truth
predicate; the consistency statement and the proof-theoretic reflection
principles are only consequences of this. In the literature peopl have
preferred to suppress the use of the truth predicate and to jump directly
from the theory itself to the reflections principles. I guess the main
reason is that the formulation of the latter does not require an expansion
of the language, whereas a truth predicate is a new symbol in the language.
Therefore we should focus on the addition of truth predicates (and suitable
axioms for it) rather than consistency statements and the proof-theoretic
reflection principles.
Truth theories remove also worries about intensionality (perhaps not
completely). This is one of Detlefsen's worries, which I share. At least
for infinitely axiomatized theories it is very unclear what sentence
expresses consistency (in a natural way). In a truth theory we do not need
a representation of the axiom set as we do in the formulation of
consistency statements and proof-theoretic reflection principles.
In his "Reflection on Incompleteness" Feferman went into this direction.
The closure of a theory is there defined via a truth predicate.
But so far we have only replaced the old question "How can we conclude
Con-T from T?" by the new question: "On what grounds do we accept the
axioms of a truth theory, if we accept the theory itself?" This seems to me
the better question.
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Volker Halbach
Universitaet Konstanz
Fachgruppe Philosophie
Postfach 5560
78434 Konstanz Germany
Office phone: 07531 88 3524
Fax: 07531 88 4121
Home phone: 07732 970863
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