Since I was the one who originally said that, I suppose I ought to take up the banner of defending it ;)
Actually, I disagree. They are not fundamentally unprovable, they are self evident and all further constructions rely upon them for their existence.
However, the question is, how do such propositions relate to objectivism? And the problem is that, according to objectivism...well, let them speak:
“Man’s reason is fully competent to know the facts of reality. Reason, the conceptual faculty, is the faculty that identifies and integrates the material provided by man’s senses. Reason is man’s only means of acquiring knowledge.” Thus Objectivism rejects mysticism (any acceptance of faith or feeling as a means of knowledge), and it rejects skepticism (the claim that certainty or knowledge is impossible).
The problem is that if "reason is man's only means of acquiring knowledge", self-evidence is no longer an appropriate standard of knowledge. After all, axioms cannot be rationally derived from other principles - if they could be, they wouldn't be axioms. We treat axioms as real because of their apparent self-evidence, or because they are useful in constructing our preferred system of morality, or whatever, but ultimately the question of whether they are real is not answerable in any reasonable way - you simply must take them as given.
And that's hardly the end of the problems with that standard of knowledge - by that standard, that says that the notion of certainty being impossible is false, inductive reasoning is rendered illegitimate, since it produces probibilistic arguments, and not certainties. So you're limited to deductively reasoning out truths, but with deductive reasoning, the only way you can be certain that your conclusion is, in fact, true, is if the argument is logically valid and the premises are true - if the premises are false or the logic is invalid, then the conclusion may still be true, but you can't know as a matter of certainty that it is true unless the premises are known to be true and the logic is known to be valid. But in the case of axioms, you may inductively reason that they are true, or you may find it useful to treat them as true, or you may consider them self-evident, but unless you can deductively prove them to be true - as objectivism says you can and must before accepting them to be true - you cannot be certain that your conclusions are true. The objectivist's proposition that certainty is possible in all places and times is thus rendered false. QED.
Their proof is that they cannot be dispensed with. For example, morality or ethics presume the concept of choice. You can prove that choice exists but you cannot reduce any further than choice.
Certainly you can, and in fact, people of a deterministic bent do it all the time by arguing that choice and free will are illusions. Now, for most of us, free will certainly appears to be real, and we generally behave as though it were real, but that hardly makes it actually real - "seems" is not the same as "is", and ultimately there's no way to know if free will really exists if you're even moderately clever about how you construct your argument for determinism.
But I was responding to what I read, whether you said it or not!