Our concept of truth is based on our assumptions about our reality -- we assume that this reality actually exits (and is not a figment of our imagination) and that it has certain properties. Those assumptions, I believe, are the axioms that you have mentioned. They cannot be proven once and for all, but their validity can be tested. Or, rather, we can make certain predictions based on those axioms, and those predictions can be tested (and that's what science is about).
And, I agree again, usually we make those assumptions intuitively and, therefore, implicitly. Here, however, is my attempt to spell them out:
- We all share and belong to one and only objective reality. "Objective" means that the existence of the reality itself and of everything that happens in it is independent of our awareness of those things, and of our own existence.
- In this reality, nothing happens at random but every thing can be traced to its cause in the past. In other words, this reality is deterministic.
- We, humans, can identify the patterns, in the ways the causes create their effects, and imagine the processes driving those patterns. By doing so, we achieve an understanding of how this world works, and use this understanding to accomplish the things we want, shaping this reality to our liking. It is this understanding that we call truth (or knowledge of the truth).1
Aside from our capacity to understand (and to know the truth), we also have a separate mental faculty that let us make statistical (or intuitive) inferences about our circumstances. This comes handy whenever our understanding is lacking (which is often because we are not born "preinstalled" with any of it).
1 This understanding (and its truth) is individual -- it gets created by and belongs to the person who put it together. However, since we all aim to understand the same objective reality, our understandings of it, and our truths, should also be the same -- and often they are.