[FOM] What makes a large cardinal axiom plausible?

[Roger Bishop Jones]

On Thursday 06 May 2004  3:28 pm, Timothy Y. Chow wrote:
> I want to pick up on Roger Bishop Jones's suggestion that what
> makes a large cardinal axiom plausible is that it "merely"
> sets a lower bound on the size of the cumulative hierarchy.

I think Tim has in this cite and elsewhere in his message
somewhat overstated the naivety of my observations about
large cardinals, so I would like to take this opportunity
to make a more complete statement of the problem of
characterising large cardinal axioms as I conceive it
at present.

I'd like to mention, in mitigation of my arrogance in
apparently attempting to lay down rules for large cardinal
axioms in general from a base of so slender a knowledge
of large cardinal axioms in particular, that my interest
here is a philosophically motivated interest in the
semantics of set theory.  I don't have any ambition to
prescribe what counts as a large cardinal, but I do
have a desire to understand some general questions about
large cardinal axioms which can only have definite
answers relative to some definite conception of what is
and what is not a large cardinal axiom.

One such question, from the semantic point of view,
is "can all unresolved questions about the semantics
of set theory be resolved by large cardinal axioms,
or do (or should) large cardinal axioms still leave some
semantic questions unresolved".
The answer to this, relative to a conception of
"large cardinal axiom" which requires invariance
under small forcing, is that at least one question,
the truth of CH, is independent of large cardinal
axioms.

Tim discusses the idea I put forward that the
justification of large cardinal axioms is that
they do no more than place a lower bound on the
height of the cumulative hierarchy, citing Harvey
Friedman's work as evidence to the contrary.
In fact, when I made this suggestion the very
next thing I did was to note its inadequacy,
which has been known for a long time independently
of the work of Harvey, since it flows directly from
Godel's incompleteness results.

It seemed to me then, and it seems to me despite Tim's
contribution, that the idea is still important.
I think it worth attempting a characterisation of
large cardinals axioms which comes as close to that
constraint (that they merely place a lower bound
on height) as possible.
A first attempt might be to say that large cardinals
should say no more about questions other than height
than is strictly necessary, raising the question of
whether anything useful can be said about what in this
respect is "strictly necessary".

Something can be said about this.

I know of two "reasons" for interaction between large
cardinal axioms and properties of small sets.
The first is that the first order axiom of regularity 
doesn't work, it is satisfied by some ill-founded
interpretations of set theory, and in those
interpretations the some of the facts of arithmetic
(for example, facts about derivability of contradictions
in various formal systems) are incorrect.
The reason why it (regularity) doesn't work is that it relies on
quantification over sets in the domain of the interpretation
and if there are sets missing, then there may be
ill-founded sets whose ill-foundedness is not noticed.
(the collections with no minimal element are not sets,
they appear only as inseparable subsets of sets which
also contain some extra member which is minimal)
When a large cardinal axiom is added, more sets are known
to exist, some ill-founded interpretations no longer
satisfy all the axioms, and some extra facts of arithmetic
turn out true in every interpretation (and hence provable).

>From the observation that some effects of large cardinal
axioms on arithmetic are mediated by the axiom of regularity
the possibility arises that all these effects are so mediated,
and that a bound on the small effects of large cardinal
axioms can be expressed via well-foundedness.

We know that all non-standard models of arithmetic are
ill-founded, and hence that relative to the well-founded
models of set theory large cardinal axioms can have no
effect on arithmetic truth.
So could we define a large cardinal axiom as one which
satisfies every well-founded interpretation of sufficient
height, thus circumscribing the effects of large cardinals
on small questions?
This gambit is successful in relation to CH, for this
is another conception of large cardinal axiom relative
to which CH must be independent of large cardinal axioms.
(since CH is independent of well-foundedness).

However, there seems to be at least one other way in
which large cardinal axioms in first order set theory
effect small questions.

Evidence for this appears in the incompatibility between
sufficienly large cardinal axioms and the axiom of
constructibility (V=L).  A large cardinal axiom which
contradicts V=L fails to satisfy a well-founded
interpretation of "sufficient height", since we know
that whatever the ordinal height of a model satisfying
a large cardinal axiom, there will be a version of L
of the same height, which will be well-founded but which
will not satisfy the large cardinal axiom (it has the
same ordinals, but their cardinality is smaller?).

So my second "reason" for interaction between large
cardinals and small sets is that these have to be
large *cardinal* axioms, (rather than tall ordinal
axioms, why is that?) and that the cardinality of
an ordinal interacts with questions of width.
This seems to connect with the constraint requiring
large cardinals to be invariant under small forcing.

Perhaps the definition of large cardinal axiom
as one which is invariant under small forcing
does what I am looking for, i.e. which ensures
that large cardinal axioms place a lower bound
on the height of models with minimal prejudice
to properties of small sets.  If so the question
arises whether this can be proven, or whether
convincing arguments can be found to support it.

There remain lots of other questions, for example
how do candidate definitions for "large cardinal
axiom" fare against the desirability that large
cardinal axioms are linearly (pre-) ordered,
by cardinality and consistency strength?
(I am assuming here that we are not counting
as large cardinals axioms statements which
have high consistency strength but don't actually
assert the existence of large cardinals)

The idea was, that subject to certain restraints,
because they "simply" assert something about how
far the iterative construction has proceeded,
a large cardinal axiom, if consistent, must be true.

Tim in his message also observed that some large
cardinals axioms seem not to connect in with this
intuitive picture at all.
This however, does not necessarily rule out the
principle.  Surely those large cardinals axioms
which do not appear directly to address the
question of height are only called large
cardinal axioms because they have been shown
to entail and be entailed by lower bounds on height?

In filling in the semantics of set theory we then
have to consider the questions which are independent
of large cardinals (like CH).
Is there anything useful can be said about "wide V"
axioms?

Roger Jones

- rbj01 at rbjones.com
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