[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  For example, if an axiom is likely to be soon either superseded 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.

Reply to Martin Dowd:
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.

Dmytro Taranovsky

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