The second movie in the Matrix series - "The Matrix Reloaded" - culminates in an encounter between Neo ("The One") and the architect of the Matrix (a thinly disguised God, white beard and all). The architect informs Neo that he is the sixth reincarnation of The One and that Zion, a shelter for those decoupled from the Matrix, has been destroyed before and is about to be demolished again.

The architect goes on to reveal that his attempts to render the Matrix "harmonious" (perfect) failed. He was, thus, forced to introduce an element of intuition into the equations to reflect the unpredictability and "grotesqueries" of human nature. This in-built error tends to accumulate over time and to threaten the very existence of the Matrix - hence the need to obliterate Zion, the seat of malcontents and rebels, periodically.

God appears to be unaware of the work of an important, though eccentric, Czech-Austrian mathematical logician, Kurt Gödel (1906-1978). A passing acquaintance with his two theorems would have saved the architect a lot of time.

Gödel's First Incompleteness Theorem states that every consistent axiomatic logical system, sufficient to express arithmetic, contains true but unprovable ("not decidable") sentences. In certain cases (when the system is omega-consistent), both said sentences and their negation are unprovable. The system is consistent and true - but not "complete" because not all its sentences can be decided as true or false by either being proved or by being refuted.

The Second Incompleteness Theorem is even more earth-shattering. It says that no consistent formal logical system can prove its own consistency. The system may be complete - but then we are unable to show, using its axioms and inference laws, that it is consistent

In other words, a computational system, like the Matrix, can either be complete and inconsistent - or consistent and incomplete. By trying to construct a system both complete and consistent, God has run afoul of Gödel's theorem and made possible the third sequel, "Matrix Revolutions".

Similarly, software applications – such as antivirus or anti-spyware, firewalls, or anti-malware - that aspire to comprehensiveness (completeness) are likely to be rendered imperfect and inefficient by Gödel’s theorems. Their detection rate is bound to suffer, inversely affected by their very scope. Malware pieces are computer code: strings of formal-logical statements. A consistent security program would inevitably come across undecidable propositions or sentences (pieces of malware that cannot be identified as such by the system.)

The Second Incompleteness Theorem explains false negatives and false positives: the misidentification of innocuous strings of code as malicious and the labelling of malevolent exploits as innocent. This is because complete computer security suites are inconsistent (or at least not provably consistent.)

It is easy to confuse the concepts of "virtual reality" and a "computerized model of reality (simulation)". The former is a self-contained Universe, replete with its "laws of physics" and "logic". It can bear resemblance to the real world or not. It can be consistent or not. It can interact with the real world or not. In short, it is an arbitrary environment. In contrast, a model of reality must have a direct and strong relationship to the world. It must obey the rules of physics and of logic. The absence of such a relationship renders it meaningless. A flight simulator is not much good in a world without airplanes or if it ignores the laws of nature. A technical analysis program is useless without a stock exchange or if its mathematically erroneous.

Yet, the two concepts are often confused because they are both mediated by and reside on computers. The computer is a self-contained (though not closed) Universe. It incorporates the hardware, the data and the instructions for the manipulation of the data (software). It is, therefore, by definition, a virtual reality. It is versatile and can correlate its reality with the world outside. But it can also refrain from doing so. This is the ominous "what if" in artificial intelligence (AI). What if a computer were to refuse to correlate its internal (virtual) reality with the reality of its makers? What if it were to impose its own reality on us and make it the privileged one?

In the visually tantalizing movie, "The Matrix", a breed of AI computers takes over the world. It harvests human embryos in laboratories called "fields". It then feeds them through grim looking tubes and keeps them immersed in gelatinous liquid in cocoons. This new "machine species" derives its energy needs from the electricity produced by the billions of human bodies thus preserved. A sophisticated, all-pervasive, computer program called "The Matrix" generates a "world" inhabited by the consciousness of the unfortunate human batteries. Ensconced in their shells, they see themselves walking, talking, working and making love. This is a tangible and olfactory phantasm masterfully created by the Matrix. Its computing power is mind boggling. It generates the minutest details and reams of data in a spectacularly successful effort to maintain the illusion.

A group of human miscreants succeeds to learn the secret of the Matrix. They form an underground and live aboard a ship, loosely communicating with a halcyon city called "Zion", the last bastion of resistance. In one of the scenes, Cypher, one of the rebels defects. Over a glass of (illusory) rubicund wine and (spectral) juicy steak, he poses the main dilemma of the movie. Is it better to live happily in a perfectly detailed delusion - or to survive unhappily but free of its hold?

The Matrix controls the minds of all the humans in the world. It is a bridge between them, they inter-connected through it. It makes them share the same sights, smells and textures. They remember. They compete. They make decisions. The Matrix is sufficiently complex to allow for this apparent lack of determinism and ubiquity of free will. The root question is: is there any difference between making decisions and feeling certain of making them (not having made them)? If one is unaware of the existence of the Matrix, the answer is no. From the inside, as a part of the Matrix, making decisions and appearing to be making them are identical states. Only an outside observer - one who in possession of full information regarding both the Matrix and the humans - can tell the difference.

Moreover, if the Matrix were a computer program of infinite complexity, no observer (finite or infinite) would have been able to say with any certainty whose a decision was - the Matrix's or the human's. And because the Matrix, for all intents and purposes, is infinite compared to the mind of any single, tube-nourished, individual - it is safe to say that the states of "making a decision" and "appearing to be making a decision" are subjectively indistinguishable. No individual within the Matrix would be able to tell the difference. His or her life would seem to him or her as real as ours are to us. The Matrix may be deterministic - but this determinism is inaccessible to individual minds because of the complexity involved. When faced with a trillion deterministic paths, one would be justified to feel that he exercised free, unconstrained will in choosing one of them. Free will and determinism are indistinguishable at a certain level of complexity.

Yet, we KNOW that the Matrix is different to our world. It is NOT the same. This is an intuitive kind of knowledge, for sure, but this does not detract from its firmness. If there is no subjective difference between the Matrix and our Universe, there must be an objective one. Another key sentence is uttered by Morpheus, the leader of the rebels. He says to "The Chosen One" (the Messiah) that it is really the year 2199, though the Matrix gives the impression that it is 1999.

This is where the Matrix and reality diverge. Though a human who would experience both would find them indistinguishable - objectively they are different. In one of them (the Matrix), people have no objective TIME (though the Matrix might have it). The other (reality) is governed by it.

Under the spell of the Matrix, people feel as though time goes by. They have functioning watches. The sun rises and sets. Seasons change. They grow old and die. This is not entirely an illusion. Their bodies do decay and die, as ours do. They are not exempt from the laws of nature. But their AWARENESS of time is computer generated. The Matrix is sufficiently sophisticated and knowledgeable to maintain a close correlation between the physical state of the human (his health and age) and his consciousness of the passage of time. The basic rules of time - for instance, its asymmetry - are part of the program.

But this is precisely it. Time in the minds of these people is program-generated, not reality-induced. It is not the derivative of change and irreversible (thermodynamic and other) processes OUT THERE. Their minds are part of a computer program and the computer program is a part of their minds. Their bodies are static, degenerating in their protective nests. Nothing happens to them except in their minds. They have no physical effect on the world. They effect no change. These things set the Matrix and reality apart.

To "qualify" as reality a two-way interaction must occur. One flow of data is when reality influences the minds of people (as does the Matrix). The obverse, but equally necessary, type of data flow is when people know reality and influence it. The Matrix triggers a time sensation in people the same way that the Universe triggers a time sensation in us. Something does happen OUT THERE and it is called the Matrix. In this sense, the Matrix is real, it is the reality of these humans. It maintains the requirement of the first type of flow of data. But it fails the second test: people do not know that it exists or any of its attributes, nor do they affect it irreversibly. They do not change the Matrix. Paradoxically, the rebels do affect the Matrix (they almost destroy it). In doing so, they make it REAL. It is their REALITY because they KNOW it and they irreversibly CHANGE it.

Applying this dual-track test, "virtual" reality IS a reality, albeit, at this stage, of a deterministic type. It affects our minds, we know that it exists and we affect it in return. Our choices and actions irreversibly alter the state of the system. This altered state, in turn, affects our minds. This interaction IS what we call "reality". With the advent of stochastic and quantum virtual reality generators - the distinction between "real" and "virtual" will fade. The Matrix thus is not impossible. But that it is possible - does not make it real.