Then I'll give my thoughts on it (though not all at once, since it would probably be a ridiculously huge post).
In the metaphysics section:
2. Every thing that exists has a specific nature or identity ('A is A' or 'a thing is what it is'). A thing must be something, otherwise it is nothing.
I would disagree with this. Things do not have specific natures or identities innately. They may be composed of molecules composed of atoms composed of particles composed of quarks, unless those things aren't matter or whatever, in which case they're made of something else (like photons, neutrinos, etc), and their 'natures' or 'identities' are only labels given to them by observers (us), and they can have more than one 'nature' or 'identity'. (Not to mention that I could point out the empty space between atoms, or between electrons and protons, or the vacuum, when it doesn't contain virtual particles or dust or the like, all of which have been named, have natures, and identities, but are nothing)
3. You exist, and you exist possessing consciousness, which is the faculty of perceiving that which exists.
I have no convincing proof that I, and every other human, are not an artificial intelligence in a simulation simulating a human mind, incapable of perceiving anything but the simulation. Furthermore, most animals seem to be so predictable that it almost seems unlikely that their "minds"
aren't programs or AIs of some sort. If that is the case, then what we are perceiving is not reality at all, but a convincing (to us) simulated reality, and we are not perceiving anything that exists in the real world, meaning that we do not possess 'consciousness' by that definition.
In fact, even if we are real, we are not perceiving anything in the real world. Those colors you see? Not what is actually there in the real world. Things reflect various wavelengths, but you only have three different kinds of cone cells which each absorb a range of wavelengths and compare the strengths of the signals they get and combine their signals to come up with a color. Which is how you get purple when you see both blue and red light, when there is no such actual color at all (blue and red are on the opposite end of the color spectrum and do not overlap at all). (It is generally used in place of violet as RGB color lights cannot generate violet light)
Anything you perceive is going through processing prior to you becoming aware of it. You don't smell a chemical signature, you perceive the smell of cinnamon, or chicken, or a wet dog, etc. You perceive pain which isn't really there, can experience
pain from limbs which were amputated, and
your brain can be fooled by a trick with a rubber hand or visual manipulation, and on and on.
1. Existence exists (there is something)
I have no objections to this, since even a simulated reality inside a simulated reality inside a simulated reality inside a simulated reality inside a simulated reality inside a simulated reality inside a simulated reality inside a simulated reality inside a simulated reality inside a simulated reality is ultimately still going to be running in a simulation in the 'real world,' even if that is not the world we perceive.
The theory suggests that these facts are "undeniable" because to deny them requires them to be true. If you deny that existence exists, you have to exist (because only entities can perform an action like denying something). Additionally, this denial assumes you have a consciousness which allows you to think and perceive and process information.
This is insane troll logic.
It is on the basis of these axioms that Objectivism embraces Atheism. According to Objectivism, reality must exist independently of consciousness. Any consciousness. This means the idea of Creation Ex Nihilo (that God created existence from nothing by sheer force of will) is a fallacy according to Objectivism. Additionally an omnipotent, omniscient, omnipresent God conflicts with the law of identity (i.e. if a thing is what it is, it is only what it is and not anything else, and thus must be specific and finite).
Regardless of the above, the Objectivist position is that reality is real, you can't think it away, and all knowledge assumes that existence exists and things are what they are (i.e. there is something to know) and a knower with the capacity to know exists (i.e. you exist possessing consciousness).
Alternately, even if those three axioms were true now, and if we aren't experiencing a simulation, I would posit that there is still no way to know whether they were true before the universe existed. I would posit an omnipotent god existing before the universe which transformed itself into the universe.
If we are experiencing a simulation, then most of the argument falls apart. We would have no idea what the actual reality is actually like, as it may have completely different physical laws than our simulated reality, so we cannot try to do logic about it using anything reasoned about this one, especially when that reasoning relies on the assumption that we aren't in a simulated reality.