Explanation of Gödels proof

With the kind permission of the author, we quote from the book:

World Without Time“ by Prof. Palle Yourgrau,  Pages 152-154

The opening move concerns the more limited special theory of rel­ativity. Given that the A-series contains the flux of „now,“ the absence of an objective, worldwide „now“ in special relativity rules out its ex­istence. But absent the A-series there is no intuitive time. What remains, formal time as represented by the little „t“ of Einstein-Minkowski space-time, cannot be identified with the intuitive time of everyday ex­perience. The conclusion, for Gödel, is inescapable: if relativity theory is valid, intuitive time disappears.

Step two takes place when Gödel reminds us that special relativity is „special“ in that it recognizes only inertial frames in constant veloc­ity relative to each other. It does not include an account of gravity. Ein­stein’s general theory of relativity, in contrast, of which the special is a special case, does. In general relativity, as we have seen, gravity itself is defined as space-time curvature, determined, in turn, by the distribu­tion of matter in motion. It follows that whereas in special relativity no frames of reference or systems in motion are privileged, in the gen­eral theory some are distinguished, namely those that, in Gödel’s words, „follow the mean motion of matter“ in the universe. In the ac­tual world, it turns out, these privileged frames of reference can be co­ordinated so that they determine an objective remnant of time: the „cosmic time“ we encountered earlier. In general relativity, then, time (of a sort) reappears.

But no sooner has time reentered the scene than Gödel proceeds to step three, where he exploits the fact that Einstein has fully geometrized space-time. The equations of general relativity permit al­ternative solutions, each of which determines a possible universe, a relativistically possible world. Solutions to these complex equations are rare, but in no time at all Gödel discovers a relativistically possible universe (actually, a set of them) – now known as the Gödel universe – ­in which the geometry of the world is so extreme that it contains space-time paths unthinkable in more familiar universes like our own. In one such Gödel universe, it is provable that there exist closed timelike curves such that if you travel fast enough, you can, though always heading toward your local future, arrive in the past. These closed loops or circular paths have a more familiar name: time travel. But if it is possible in such worlds, Gödel argues, to return to one’s past, then what was past never passed at all. But a time that never truly passes cannot pass for real, intuitive time. The reality of time travel in the Gödel universe signals the unreality of time. Once again, time disappears.

But the dance is not over. For the Gödel universe, after all, is not the actual world, only a possible one. Can we really infer the nonexis­tence of time in this world from its absence from a merely possible uni­verse? In a word, yes. Or so Gödel argues. Here he makes his final, his most subtle and elusive step, the one from the possible to the actual. This is a mode of reasoning close to Gödel’s heart. His mathematical Platonism, which committed him to the existence of a realm of objects that are not accidental like you and me – who exist, but might not have – but necessary, implied immediately that if a mathematical ob­ject is so much as possible, it is necessary, hence actual. This is so be­cause what necessarily exists cannot exist at all unless it exists in all possible worlds.

This same mode of reasoning, from the possible to the actual, oc­curs in the „ontological argument“ for the existence of God employed by Saint Anselm, Descartes and Leibniz. According to this argument, one cannot consider God to be an accidental being – one that merely happens to exist – but rather a necessary one that, if it exists at all, ex­ists in every possible world. It follows that if God is so much as possi­ble, He is actual. This means that one cannot be an atheist unless one is a „superatheist,“ i.e., someone who denies not just that God exists but that He is possible. Experience teaches us that ordinary, garden-va­riety atheists are not always willing to go further and embrace su­peratheism. Following in the footsteps of Leibniz, Gödel, too, con­structed an ontological argument for God. Then, concerned that he would be taken for a theist in an atheistic age, he never allowed it to be published.

In arguing from the mere possibility of the Gödel universe, in which time disappears, to the nonexistence of time in the actual world, Gödel was employing a mode of reasoning in which he had more con­fidence than most of his philosophical colleagues. In the case of the Gödel universe, he reasoned that since this possible world is governed by the same physical laws that obtain in the actual world – differing from our world only in the large-scale distribution of matter and mo­tion – it cannot be that whereas time falls to exist in that possible world, it is present in our own. To deny this, Gödel reasoned, would be to assert that „whether or not an objective lapse of time exists (i.e., whether or not a time in the ordinary sense exists) depends on the par­ticular way in which matter and its motion are arranged in this world.“ Even though this would not lead to an outright contradiction, he argued, „nevertheless, a philosophical view leading to such conse­quences can hardly be considered as satisfactory.“ But it is provable that time fails to exist in the Gödel universe. It cannot, therefore, exist in our own. The final step is taken; the curtain comes down: time re­ally does disappear.