The Roman Catholic Church has infallibility in dogma through the successors of Peter, guided by the Spirit and Tradition.
The Bible contains all that we need to know in this world about God and matters of faith.
The Bible is inerrant.
The way to understand God is by looking at Christ.
The most important thing to know about God is that He is Sovereign.
Nothing happens apart from God's sovereign decree.
The Holy Spirit still gives all kinds of gifts of the Spirit to believers.
(And, often) Those who have the Holy Spirit can speak in tongues.
For other groups, while I may have some idea what they teach, I'm not sure which beliefs are the essential axioms. For example:
The goal of religion is to attain perfection.Does that reach the level of axiom for Methodists, or is something else the unique bedrock there? A thorough treatment of this would have each group speaking for itself.
Perfection can be reached in this lifetime.
Some items in the Nicene Creed read as though they were meant to be the axioms on which we base our further knowledge. Each group's systematic theology is profoundly influenced by its axioms -- in fact, each group's theology is built on its axioms and shaped by its axioms. After all, that is what axioms are for.
Some Christian groups oppose creeds on general principle, on the view that the Bible should speak for itself. The groups that oppose creeds tend to be the groups that have no systematic theology. Again, this is expected; you could hardly have a systematic theology without a set of axioms for starters. A group's systematic theology cannot go beyond its axioms.
Where does this leave us? I think we will need to find the core beliefs of each group, and take a closer look at those. For a discussion to make any progress, each group would have to put its most basic assumptions out on the table, not as the basis for further discussion, but for examination in order to vet those most basic assumptions.
To be continued. Next: some sample axioms, and where they would lead.