In his seminal 1937 paper, Jones [1] proved the following result about Moore spaces:

Theorem. (Jones)If $2^{\aleph_0}<2^{\aleph_1}$ then allseparablenormal Moore spaces are metrizable.

Then he came up with the idea that maybe the *separability* condition in the above theorem could be removed. This led him to formulate the following famous conjecture:

Normal Moore Space Conjecture (NMSC).All normal Moore spaces are metrizable.

Long story short, NMSC turned out to be of high large cardinal strength and so far beyond the power of $ZFC$ to decide.

In 1980, Nyikos [2] proved the consistency of NMSC using Product Measure Extension Axiom (PMEA) which is known to be consistent assuming the consistency of strongly compact cardinals. Later a more direct proof of the consistency of NMSC has been presented by Dow *et al* [3] using the stronger assumption of the consistency of supercompact cardinals.

Interestingly, Fleissner [4] used a core model argument to prove the fact that the consistency of NMSC implies the consistency of measurable cardinals as well. Thus NMSC lies between strongly compacts and measurables in the *consistency strength* order of the large cardinals hierarchy.

I wonder whether the *direct implication* power of NMSC is as strong as its large cardinal strength or not.

Question 1.Does NMSC directly imply the existence of any large cardinals, say those of some topological nature such as weakly compacts? To be more precise, let me add that I look for the theorems of the form NMSC $\Rightarrow \exists \kappa$ Large (i.e. at least strongly inaccessible) if there is any.

**Update.** According to Will's answer, it turned out that NMSC doesn't contain any direct large cardinal strength.

In other direction, one may strengthen NMSC as follows:

Complete Normal Moore Space Conjecture (CNMSC).All normal Moore spaces arecompletely metrizable.

Here is my second question:

Question 2.Is CNMSC consistent? What are upper and lower bounds for its large cardinal strength? Strongly compacts and measurables or more?

**References.**

*Jones, F. Burton*,**Concerning normal and completely normal spaces**, Bull. Am. Math. Soc. 43, 671-677 (1937). ZBL0017.42902.*Nyikos, Peter J.*,**A provisional solution to the normal Moore space problem**, Proc. Am. Math. Soc. 78, 429-435 (1980). ZBL0446.54030.*Dow, Alan; Tall, Franklin D.; Weiss, William A. R.*,**New proofs of the consistency of the normal Moore space conjecture. I & II**, Topology Appl. 37, No. 1, 33-51 (1990). ZBL0719.54038.*Fleissner, William G.*,**If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal**, Trans. Am. Math. Soc. 273, 365-373 (1982). ZBL0498.54025.

weaknessof Woodin cardinals in direct implication order I am pointing out to the facts such as "the first Woodin cardinal is not even weakly compact." Shall I change my phrasing to mention this more explicitly? I thought it is pretty obvious. $\endgroup$ – Morteza Azad Jul 20 '18 at 10:51