# [FOM] Criteria for New Axioms

Dmytro Taranovsky dmytro at mit.edu
Tue May 5 18:55:52 EDT 2015

```Two notes to clarify criteria for new axioms:
- There are some pragmatic factors in addition to the 4 criteria I
listed.  For example, if an axiom is likely to be soon either superseded
by a better one or thrust into doubt, then it may be good to wait.
- As a precautionary measure, I suggest that axioms are accepted
gradually.  There should be a period when an axiom is symbolically
accepted before it can be silently omitted (if it can be omitted at
all).  In the other direction, when AI is sufficiently developed for
formal verification to become the norm, theorems might get automatically
annotated with axioms used, even for axioms in ZFC.

While weak axioms are more certain, they are also much less useful; I am
not aware of significant consequences of inaccessible cardinals (when
added to ZFC) on say V_{omega+omega}.

That V is not L can be intuitively seen as follows:  Pick an ordinal
kappa_1 with sufficiently strong reflection properties in L, then
kappa_2>kappa_1 with sufficiently strong reflection properties, and so
on.  The theory of L augmented with kappa_n for every n<omega is
recursively isomorphic to 0^#.

However, it is notable how effectively L can impersonate V. Iterating
predicative comprehension (which is used to built L) a sufficiently
closed ordinal number of times leads to ZFC. The height of L is Ord (and
also L is canonical), so large cardinal axioms that, roughly speaking,
assert existence of certain ordinals rather than sets of ordinals
relativize to L.

In second order arithmetic, impredicative comprehension is just not
strong enough to approximate the assertion that every real number
exists, but a natural strengthening (projective determinacy) is.  I
suspect that a natural strengthening of comprehension will approximate
existence of arbitrary sets of real numbers, but I do not know how to
state such a principle.

There are intuitive arguments for strong large cardinal axioms, but they
get progressively weaker as the axioms are strengthened.
1.  Arguments based on "collecting the universe", which are most clear
using reflective cardinals, lead to indescribable cardinals.
2.  Iterating higher-order set theory an ordinal number of times
naturally leads to a strengthening of Ramsey cardinals. (See my
"Reflective Cardinals" paper for some details, but roughly one considers
theory of (V, in, R) where R(alpha, beta) means that beta is similar
enough to Ord for V_beta to host higher-order set theory augmented with
R restricted to {(gamma,delta): gamma<alpha}.)
3.  Arguments based on symmetry of different levels of V lead to
Vopenka's principle.  They might even lead to rank-into-rank embeddings,
but that is much more questionable.

Sincerely,
Dmytro Taranovsky
```