By Dan Hooper, Ph.D., The University of Chicago
Austrian mathematician Kurt Gödel’s work was focused in the area of formal logic. He is considered by many to be the greatest logician of all time, and his work made a huge impact on the history of mathematics, as well as philosophy. He was particularly fascinated by the concept of time and believed there were logical inconsistencies in it. Let’s take a closer look at how he came to this conclusion.
In the first decades of the twentieth century, logic was emerging as an increasingly important branch of mathematics. Philosophically minded mathematicians such as Bertrand Russell and Gottlob Frege had made great progress in showing that logic could potentially underpin all of the branches of mathematics.
In effect, their goal was to show that all of mathematics follows directly from logic itself. They wanted to demonstrate that all branches of mathematics—geometry, algebra, arithmetic, etc.— are really just different ways of thinking about the same thing, and that everything boiled down to logic.
Hilbert’s Program
This laid the groundwork for what was known as Hilbert’s program, named after the great mathematician David Hilbert. What Hilbert set out to accomplish with his program was intellectually staggering. He wanted to prove that all of mathematics could be derived directly from a finite number of axioms. Furthermore, he wanted to be able to demonstrate that such a mathematical system could be internally self-consistent, and also entirely complete.
By ‘complete’ Hilbert meant that the axioms of the system could, in principle, be used to prove any given statement in the system to be either true or false. In the mathematical system that Hilbert was trying to build, there would be no unanswerable questions.
In a way, Gödel played the single most important role in the history of Hilbert’s program. He did so not by helping Hilbert build a self-consistent and complete system of mathematics, instead of proving that this could not be done.
Gödel’s Incompleteness Theorems
In 1931, Gödel published a paper which presented proofs of what have become known as Gödel’s incompleteness theorems. The contents of these two theorems are quite technical, but they basically boil down to the following statement:
For any self-consistent system of mathematics that is complex enough to be interesting, there absolutely must exist statements that cannot be proven to be either true or false. In other words, all non-trivial mathematical systems are incomplete. By proving the incompleteness theorems, Gödel had shown that Hilbert’s program was, from the very beginning, doomed to fail.
Learn more about what Einstein got wrong.
Gödel was Convinced the Concept of Time had Logical Inconsistencies
In 1940, Gödel fled Nazi Germany and immigrated to the United States, similar to what Albert Einstein had done seven years earlier. Just like Einstein, Gödel took up a position at the Institute for Advanced Study at Princeton, where he became one of Einstein’s closest friends.
They were known to engage in long walks and conversations. After years of such conversations, Gödel produced an important essay. This essay challenged not only the completeness of the theory of relativity, but also the reality of time itself.
During his time at Princeton, Gödel became preoccupied and fascinated with the concept of time. Although Gödel was first and foremost a mathematician, in many ways he had the outlook of a philosopher. The philosopher in him became convinced that there were deep logical inconsistencies in the way that we thought about time.
We intuitively classify events into those that have taken place in the past, those that are taking place in the present, and those that will take place in the future. With the theory of relativity, Einstein had begun to blur the edges of these categories. Relativity gave us reason to pause and reconsider this simple way we thought about time.
What does this mean? Let’s consider two events – Event A and Event B – and these two events take place at two different locations in space. Let’s also imagine that you’re standing somewhere where you can watch both locations at the same time. As you do so, you observe Event A and Event B occur simultaneously.
This is a transcript from the video series What Einstein Got Wrong. Watch it now, Wondrium.
Our intuition tells us that these events take place at the same time not only to us but universally to all observers. However, according to relativity, observers in other frames of reference won’t necessarily share this perception. Instead, to some observers, Event A will occur before Event B does, while to other observers, Event B will take place before Event A. Contrary to our naïve intuition, simultaneity is not an absolute according to relativity.
While there’s no way of knowing exactly what Gödel and Einstein discussed during their long and frequent walks through Princeton, it’s hard to imagine that the nature of time wasn’t a recurring topic. These conversations culminated in an essay that Gödel wrote in 1949. With this essay, Gödel set out to overturn our conventional way of thinking about time. According to Gödel, time as we commonly understand it simply cannot exist.
Learn more about the cosmological constant and dark energy.
Gödel’s Cosmological Solution Allows for Time Travel
In his essay, Gödel presented a cosmological solution to the gravitational field equations of general relativity. Over the years, quite a few cosmological solutions to these equations have been presented. For example, Einstein found a solution that describes a static universe that is wrapped around on itself. This solution was known as ‘Einstein’s world’, and among other problems, it turned out to be unstable.
This was followed by Alexander Friedmann’s more general solution. With this solution, Friedmann could describe universes that are either expanding or contracting with time. It’s Friedmann’s solution that modern cosmologists use to describe the universe that we actually live in.
In some ways, Gödel’s cosmological solution was similar to those found by Einstein and Friedmann. Like these others, Gödel’s solution included a homogeneous distribution of matter, as well as a cosmological constant.
In addition to these more common features, the matter in the universe described by Gödel’s solution was also rotating about an axis. Essentially, the entirety of Gödel’s universe is spinning, like a giant merry-go-round.
This is the key feature of Gödel’s universe. It’s this feature that makes time behave differently than it does in most other cosmological solutions. In particular, in Gödel’s universe, it’s possible for objects to move backward through time. In other words, it allows for time travel.
Learn more about the problems with time travel.
What Does Time Travel Mean in Gödel’s Universe?
In our universe, an object could potentially take many different paths through space and time. They can move in many different directions, and at different speeds. Each of these paths will lead an object through a different series of events, but of all of these possible paths through space and time that one could potentially take through our universe, none of them will ever pass through the same event twice.
No matter what route you follow through space and time in the future, it will never again bring you to today’s date.
But in Gödel’s universe, it is possible for someone to be present for some event, and then to travel through space only to later encounter the same event again.
Here it’s important to understand that ‘the same event’ doesn’t mean a recurrence or re-enactment of the original event. The event is not merely taking place a second time. Instead, it is the original event. The observer in this example has followed a path through their universe that has taken them from the future into the past.
A path through space and time of this kind is known as a ‘closed timelike curve’. Closed timelike curves pass through every single event within Gödel’s universe.
There is nowhere and no event that is safe from time travelers. In Gödel’s rotating universe, time just does not behave in the way that we usually think about time.
In fact, time in Gödel’s universe is much more like an additional dimension of space than it is like the dimension of time as we understand it.
What does this mean? You can ordinarily choose to move in either direction along any given dimension of space. For example, you can walk either north or south, but you can only move forward through time. However, in Gödel’s universe, time is more like a dimension of space, and you can move through it in either direction, either forward or backward through time.
In this sense, Gödel’s universe is something like a universe of space alone. It’s a universe without time. Or at least a universe without time as we generally know it.
Gödel’s universe was very different from the universe we live in – the real universe. This was the reason why even though Gödel’s essay raised a number of pertinent questions, and deserved to be further explored, Einstein was skeptical and to a certain extent dismissive.
Common Questions about Kurt Gödel
Kurt Gödel’s incompleteness theorems are actually made up of two theorems. It was proved in 1931 and said that for any self-consistent system of mathematics that is complex enough to be interesting, there absolutely must exist statements that cannot be proven to be either true or false.
Kurt Gödel is considered one of the topmost logicians of the twentieth century. In 1931, he proved the incompleteness theorems, and in 1949 he published an essay that introduced the concept of closed time-like curves. These are his most important and seminal works.
Austrian mathematician Kurt Gödel’s greatest contribution to mathematical logic is the incompleteness theorem, which proved that any axiomatic mathematical system must have statements that cannot be proven to be either true or false.
Gödel’s incompleteness theorems sent a shock through the mathematical community in the 1930s. It implied that axiomatic mathematical systems may have true statements that can never be proved to be true.
