The axioms seem to hold though.
It’s impossible for you to know that.
We only need to build upon foundations of solid axioms.
Says who? How do they justify that claim?
Axioms are pragmatic and therefore used a lot in math and science but when you enter the realm of metaphysics (e.g. Philosophy) you have to ground your worldview in a justified true belief.
I just did using the transcendental argument. God is the necessary precondition for universals such as logic and reason. They exist therefore God exists and these universal metaphysics are a reflection of his divine mind.
What is the epistemic justification for your world view? Make sure not to use universals or subjective experience because the former is in question and the latter is arbitrary.