Going back to Epistemology now, I'm going to discuss a greatly misunderstood topic. Axiomatic concepts, commonly referred to as the axioms.
For a quick intro about why they're misunderstood, and what the use of the axioms are, read here:
Axioms are undeniable, which is very different from "self-evident". Objectivism rejects the idea of something being "self-evident", which means "requiring no proof or explanation". A self-evident idea is supposed to be true in itself, and so obvious that every instantly recognizes it. It's often used to state something you're sure is right, but don't know how you know it or came to learn it. It's sometimes uses to insinuate that anyone who disagrees with you is blind to obvious facts.
No, axioms are not self-evident. They are undeniable though. Of course, you can deny them if you want, but you can't do it logically. You can't deny them without contradicting yourself. In order to deny an axiom, you have to use the axioms. They're necessary for any statement of fact. Even the act of denying them requires an implicit acceptance of them. Let's look at the individual axioms.
The first axiom is "Existence Exists". The wording is tricky and a little confusing, but the fact that it points to is the most fundamental of facts. It's just stating that there are things that exist, meaning that they're real. To grasp it more clearly, we have to examine the purpose of this axiomatic concept. As I said in the article above, it acts as a guidepost to your thinking, telling you when you've gone into La-La Land. If you think that nothing actually exists, your mind is attempting to completely sever its ties to reality. If you think that there's something called "nothing" that exists, then you invalidating the concept of existence. Does anyone make this kind of mistake in real life? Not consistently. But some will argue that reality is just a figment of our imaginations.
How does one go about denying this axiom? If you say it's not true, you're stating that there exists something (an idea) which isn't true, and you're accepting that there is something that's "true" which this axiom doesn't satisfy. The act of speaking the denial is an acknowledgement that other people exist, or that you yourself exist, and that communication exists, etc. Every thought and every action is an implicit acknowledgement of this axiom.
We've already discussed The Law of Identity, which is a second axiom. It basically states that for something to exist, it has to exist in some specific way. This is axiomatic as well. If one starts accepting that contradictions are possible, or that things can be A and not A at the same time and in the same respect, they're no longer making any sense at all. They're dismissing all of their knowledge by claiming that identity is impossible, and remember that knowledge is identification. Past that guidepost leads to insanity. If one were to try to deny the Law of Identity, they would have the same problems as above. How can you say something "is" false, when there is no identity. How can you talk about anything at all if nothing has identity?
The last axiomatic concepts is "Consciousness". We've also discussed this idea already. What would it mean to reject one's own awareness? They would be invalidating their own minds completely. Obviously someone trying to argue against consciousness would have the problem that they would be counting on your mind, as well as their own, to try to argue that neither exists. It is lunacy.
The axioms are fundamental to every thought, statement, or action. They cannot be avoided. They can only be contradicted. The goal of formulating the axioms is to bring to your fully conscious mind the premises you hold in every aspect of your life.
There is the inevitable question of how you prove the axioms. The answer is that you can't. Proof requires the axioms themselves. Without them, the concept of proof is meaningless. Logic depends on Identity. Pointing depends on Existence. And both depend on awareness.
Remember my warnings from the article, though. The axioms are not some kind of deductive starting point, where you deduce the rest of reality. Objectivism isn't that kind of philosophy.