There are a variety of axiom systems which justify mostly similar notions of rationality, and a few posts explore these axiom systems. Sniffnoy summarized Savage's Axioms. I summarized some approaches and why I think it may be possible to do better. I wrote in detail about complete class theorems. I also took a look at consequences of the jeffrey-bolker axioms. (Jeffrey-bolker and complete class are my two favorite ways to axiomatize things, and they have some very different consequences!)
As many others are emphasizing, these axiomatic approaches don't really summarize rationality-as-practiced, although they are highly connected. Actually, I think people are kind of downplaying the connection. Although de-biasing moves such as de-anchoring aren't usually justified by direct appeal to rationality axioms, it is possible to flesh out that connection, and doing this with enough things will likely improve your decision-theoretic thinking.
1) The fact that there are many alternative axiom systems, and that we can judge them for various good/bad features, illustrates that one set of axioms doesn't capture the whole of rationality (at least, not yet).
2) The fact that not even the sequences deal much with these axioms shows that they need not be central to a practice of rationality. Thoroughly understanding probability and expected utility as calculations, and understanding that there are strong arguments for these calculations in particular is more important.