Well, synthetic truths are just truths that are not true in virtue of the terms involved in some proposition, it has nothing to do necessarily with experience perse, though many do connect the two. Kant's synthetic a priori truths do not require experience, far as I know. I've never read Kant directly on this, but it's my understanding (SAP) truths are propositions about the external world which are not known via meaning of the terms, but known before experience. This he connects with many of our notions of causal rules, like every event must have a cause. I think Objectivism's axioms stand opposed to this. As I said before, the axioms are truths brought by experience and without experience, we do not know of them. (or anything, really) You could say it is similar to the (SAP) truths in that they both are validated by every experience of the world that we have. Rand (and Peikoff more explicitly and at length) say that the law of causality is a corollary of the axiom of identity and so is reified in any contact we have with the world. In that way, Kant has something in common with Objectivism, but I wouldn't say the axioms and (SAP) truths are any more similar than that. Now analytic a posteriori, mmm I like me some of those!