FOM: simultaneous truth of consistent statements
Andrian-Richard-David Mathias
Andrian-Richard-David.Mathias at univ-reunion.fr
Thu Feb 24 02:00:20 EST 2000
1. Black suggests that notions of completeness not having yet been
formulated, they can have had nothing to do with Hilbert's 1899 belief
that consistency is tantamount to existence. I disagree. Much of
(mathematical) research consists of the articulation of vague perceptions,
and it could well be that some subconscious version of the notion of
completeness was already guiding Hilbert's activities, and might indeed have
taken root in his mind from his work on the axiomatics of Euclidean geometry.
2. As Simpson points out, Hilbert's remark
> The proposition 'every equation has a root' is true, or the
> existence of roots is proved, as soon as the axiom 'every equation has a
> root' can be added to the other arithmetical axioms without it being
> possible for a contradiction to arise by any deductions.
discounts the possibility of there being a theory T and two properties
A(x) and B(y) such that if T is consistent so are both T + (Ex)A and
T + (Ey)B whilst the statement "(Ex)A and (Ey)B" is refutable in T.
In Simpson's example, we may take A and B to be "x is a well-ordering of
the continuum of length aleph_1," and "x is a well-ordering of the
continuum of length some aleph exceeding aleph_1", and T to be ZF.
Another example: take A to say "x is a lightface Sigma^1_2 well-ordering
of the continuum", B to say "y is a non-constructible real", and T to be ZF.
T + (Ex)A is consistent by Goedel;
T + (Ey)B is consistent by Cohen; and by Mansfield
(Ex)A and the negation of (Ey)B are, provably in T, equivalent.
3. Of course *if T is complete*, then sentences consistent with
T are theorems of T, so sentences individually consistent with T
are simultaneously consistent with T.
4. Despite the essential incompleteness of mathematics, there has been
success in formulating hypotheses that achieve the simultaneous truth
of consistent statements. Some examples:
5. Vopenka's principle, recently been used by
Casacuberta, Scevenfels and Smith in their paper "Implications of
large-cardinal principles in homotopical localization", from which
I quote two results and summarise a third:
"Vopenka's principle implies the existence of cohomological localizations,
for every generalized cohomology theory. Whether cohomological localizations
can be shown to exist in ZFC or not is a long standing open question."
"If we assume the validity of Vopenka's principle,
then we can prove that every homotopy idempotent functor E
in the category of simplicial sets is weakly
equivalent to f-localization for some map f."
[The authors also construct a specific E for which the existence of a
corresponding f implies the existence of a measurable cardinal, so that
this second consequence that they draw from Vopenka's principle is not a
theorem of ZFC.]
Vopenka's principle states that given any proper class of structures of
the same type (e.g; groups, fields, ...) one of them is elementarily
embeddable in another. It is related to reflection principles which, in
vague terms, say that if something is true in this universe it was true in
some earlier universe.
6. Bagaria has numerous results expressing forcing axioms such as
Martin's axiom and its variants in the general form "anything which might
happen in some future universe has already happened in this one".
For example, let H_2 be the term "the set of sets
whose transitive closure is of cardinality less than aleph_2". Then
(Bagaria) Bounded Martin's Maximum is equivalent to saying that
the H_2 of this universe is a Sigma_1 elementary submodel of
the H_2 of any generic extension of this universe obtained using a
partial ordering that preserves stationarity of subsets of omega_1.
7. Woodin in his recently published monumental work
"The Axiom of Determinacy, Forcing Axioms and the Nonstationary Ideal"
(de Gruyter, 1999) exhibits, using some large-cardinal hypotheses,
a canonical model of the negation of the continuum hypothesis in which
all individually consistent sentences of a certain kind are simultaneously
true.
Thus although Hilbert's dream of a universe in which all possible objects
simultaneously exist was impossible, fascinating approximations to it exist.
A. R. D. Mathias
Professeur associe,
IREMIA,
Universite de la Reunion,
15 Avenue Rene Cassin,
BP 7151,
F 97715 St Denis Messag 9
France outre-mer
ardm at univ-reunion.fr
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