FOM: consistency equals existence; WKL0; CH
Stephen G Simpson
simpson at math.psu.edu
Wed Feb 23 21:37:37 EST 2000
I want to thank Robert Black for his thoughtful posting of 18 Feb 2000
in the ``consistency equals existence'' thread. I had hoped to follow
up earlier, in a more timely way, but other things have gotten in the
Recall that my ``consistency equals existence'' argument was based on
G"odel's Completeness Theorem for (first-order) predicate calculus: a
system S of mathematical objects *exists* if and only if the axioms
describing S are *consistent*.
Black suggests that this argument was probably not what Hilbert had in
mind. So far as we can tell from the historical evidence at hand,
Hilbert said ``consistency equals existence'' way back in 1899, when
the distinction between first-order logic and set theory had not been
clearly formulated, and G"odel's Completeness Theorem was unknown.
Black's suggestion is very well taken. I cannot cite Hilbert as an
authority for my argument, nor do I claim to understand what Hilbert
intended in 1899.
So, let's forget Hilbert.
Even without Hilbert, I still find my G"odel Completeness version of
``consistency equals existence'' intriguing, especially in the context
where I raised it: contemporary large cardinal research.
1. Friedman uses large cardinals to derive mathematically appealing
Pi01 combinatorial statements. But for this only *consistency* of
the large cardinals is needed, not their actual *existence*,
whatever that may mean. And of course, if consistency equals
existence, then there is no distincion here!
2. Steel in his transcript of ``The Speech'' (25 Jan 2000) takes his
usual staunch Platonist/realist view to argue for large cardinals.
But even there he feels a need to come to grips with an extreme
``Hilbertist'' position giving a special role to Pi01 consequences
and consistency statements.
All this makes me wonder exactly what would be lost if we were to ``go
all the way'' and simply identify consistency with existence, a la
G"odel Completeness. Could this turned into a coherent foundational
doctrine? Could such a doctrine be true?
Black raises the following cogent objection:
> What I can't see is how the completeness theorem plays a role if
> we're not set-theoretic realists. Suppose we have a first-order
> theory which we know or believe to be consistent, and we want to
> show that it has a model. Of course it's a theorem of our set
> theory that any consistent theory has a model. But we're not
> assuming our set theory to be true any more - only (at most)
> consistent. ...
Another excellent point.
Let me attempt to blunt Black's point by noting that it is possible to
coherently accept the G"odel Completeness Theorem without accepting
set-theoretic realism, or even set theory. According to Reverse
Mathematics, the G"odel Completeness Theorem (``a countable set of
axioms is consistent if and only if there exists a model of it'') is
equivalent over RCA0 to Weak K"onig's Lemma. Thus it is much, much,
much weaker than ZFC. In fact, RCA0 plus Weak K"onig's Lemma (called
WKL0 in my book) is conservative over primitive recursive arithmetic
for Pi02 sentences, and it is actually a very nice system in which a
very large portion of core mathematics can be conveniently developed,
a la my paper ``Partial Realizations of Hilbert's Program''.
In short, there is a fairly compelling foundational setup in which the
G"odel Completeness version of ``consistency equals existence'' holds,
and which is quite different from set-theoretic realism.
It seems we have returned to ``Hilbertism'' again ....
Black also discusses other versions of ``consistency equals
existence'' involving other logics, especially second-order logic with
the standard or full-power-set semantics. (See also the FOM thread of
last year on ``second order logic is a myth''.) Deductive consistency
is no longer an issue, because there are no rules of inference. Black
now calls it ``posse equals esse''. From a set-theoretic realist
stance, Black says:
> This way of thinking also provides a foundational role for set
> theory as describing a structure, the universe of all sets, which
> is designed (as far as possible) to be so rich that every possible
> structure gets actualized in it (if it's *possible* for a system of
> axioms to have a model, then it *actually* has a model in the
> universe of sets). In other words, any system of axioms which
> *could* be true *is* true. Note also that this doesn't depend on
> any distinction between first and second-order logic.
However, it seems to me that this idea founders on the Continuum
Hypothesis. Recall that there are sentences A and B of second order
logic which are satisfiable (i.e., satisfiable in the standard or
full-power-set semantics) if and only if 2^aleph_0 = aleph_1 and
2^aleph_0 > aleph_1, respectively. Now we know from G"odel and Cohen
that both A and B ``could be true'', but only one of them is
satisfiable in ``the'' set-theoretic universe. It seems to me that
this is a severe problem for Black's idea above. Of course Cohen's
work deals ``only'' with first-order set theory, but still, if we take
``posse equals esse'' seriously, .... Perhaps Black would want to
overcome this by considering Boolean-valued models?
More information about the FOM