Austria, 1930
T here’s a storyof a philosopher who was well known as an atheist. It may have been Bertrand Russell. Someone asked him, what if he died and he met God face to face? Wouldn’t he feel quite the fool? “Not really,” said Russell or whoever. “Rather I should ask God why he’d kept such a low profile all these years.”
Why do we need a God Proof? Why doesn’t God make his existence so clear that there’s no debate? Why let people doubt? Why let them worry?
Why give them a chance to be wrong?
Why doesn’t God announce Himself, clearly root our existence in His and be done with it? Why isn’t it evident why we are here in this world? If God exists, would he allow decent, intelligent people to wonder what their purpose is in life? Why does God hide from us? Isn’t that all the proof anyone could need that God doesn’t exist?
The answer to this comes from Gödel’s Doubt Proof. The answer Gödel gives is not new. There’s been a lot of talk by theologians over the centuries about freedom and doubt, belief and faith. For many, that has sounded like excuse-making, a sad, desperate attempt to keep up belief in the face of the obvious. In Gödel’s hands, this answer became as sure and as solid as two plus two equals four.
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When Gödel found the Doubt Proof he was actually looking at the Consis-tency Question. Is the world consistent? Circles are round. Are they also square? Two and two is four. Is it also five? Consistency allows some things to be true and others false. Inconsistency makes everything and its opposite true. You really have to expect the answer to the consistency question to be
“yes”. The alternative seems just too bizarre.
ℵ
It wasn’t like Gödel to come up with the answer everyone expected, but with consistency it looked like he’d have no choice. In an inconsistent world, you can answer the Consistency Question and any other question “yes”. In a consistent world, “yes” is the obvious answer to the Consistency Question.
So that’s two choices. One is “yes” and the other is “yes”. You wouldn’t think even Kurt Gödel could pull a surprise out of that one.
But he did. The Doubt Proof showed the answer to the Consistency Question was “no”. There was a part of “no” that we didn’t understand. A part that meant “yes”.
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The Doubt Proof started out as a quick consequence from the Freedom Proof, so quick Johnny von Neumann nearly poached it. It would have been almost fair if he had. Without Johnny there at the conference in Königsberg, the Freedom Proof might have gone unnoticed for some time, what with Gödel keeping its real meaning secret from his own thesis adviser and all.
Not that Johnny needed to justify poaching. If you publish and you miss an easy consequence of your own work, somebody else beats you to it.
That’s just tough. Part of being a mathematician is knowing when it pays to let the Johnny von Neumann’s of the world in on your secrets and when to stonewall them.
The Doubt Proof would have been well worth poaching. David Hilbert, the one who had gotten so mad at the Freedom Theorem, was the leading mathematician of the time. As the century changed in the year 1900, Hilbert challenged his colleagues to solve a list of twenty-three problems. Number two on the list was the Consistency Question.
Hilbert didn’t seriously expect that circles were square or that two plus two is five. He was sure that math was consistent. What Hilbert was asking for with the Consistency Question was proof.
The Consistency Question is different from any other in mathematics.
Consistency is basic to mathematics. If you don’t have some things that are true and some that are false, the mathematical game, and all the real life that is built on it, cease to make sense. Consistency had forced Hilbert to accept the Freedom Proof, much as he hated it. Hilbert either had to accept that there were true things that could not be proved, or else that math was inconsistent.
Consistency is so basic and so important, you’re allowed to use circular
The Part of “No” That We Didn’t Understand
reasoning. A proof with circular reasoning is like this one that I’m the Dalai Lama.
Assume I’m the Dalai Lama.
Therefore, I’m the Dalai Lama.
This concludes my proof that I’m the Dalai Lama.
This sort of argument is not convincing. In math, assuming what you want to prove is cheating. It is not allowed, except for proving consistency.
Consistency has to be the exception. If you’re inconsistent you can prove anything, including that you are consistent. So if you believe your proof of consistency really means anything, you’re assuming consistency. You’re forced to cheat. A consistency proof is not much more than a double check, like when you count your chickens from one end of the yard first, then count them again starting at the other, just to make sure both counts are the same.
You could have missed the same chicken both times, but it’s better than nothing.
Von Neumann saw what Gödel had missed at first. A liar sentence is true, Gödel had showed, but it can’t be proved. That’s assuming consistency, which you have to do because an inconsistent system can prove anything.
So Gödel was saying “If the world is consistent, then a liar sentence is true.”
Von Neumann saw this could be turned into the second step of a proof:
One: The world is consistent.
Two: If the world is consistent, then a liar sentence is true.
Three: A liar sentence is true.
Step Three was exactly what the Freedom Theorem had proved you can’t prove. You can’t prove “A liar sentence is true” unless you’re inconsistent.
Proving Step Three is not acceptable. Something has to give.
To avoid getting to Step Three, either Step One or Step Two has to be thrown overboard. Step Two is not going. Its proof is as solid as they come.
The only thing left to eliminate is Step One: “the world is consistent.”
That’s the Consistency Question. You can’t prove consistency.
There’s an exception. There’s one way you can prove consistency. If you’re inconsistent, you can prove anything, including consistency. The logic of Steps One, Two and Three doesn’t have to bother you if you’re inconsistent, because everything is both true and false at the same time.
∞
The answer to Hilbert’s challenge is “No, you can’t prove consistency, unless it’s false.” Meep-meep.
Bottom line, it turns out the Consistency Question is another liar sentence, one of those things that are true, but which we can’t prove. It says “no” but it means “yes”. There is a part of “no” we don’t understand.
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There are rules about poaching in math. Johnny wrote to Gödel. Partly because it was polite to let Gödel know. Mainly because it put Johnny’s Consistency Proof on the record, with a date.
But Johnny von Neumann had a surprise coming. Gödel had already gotten the Doubt Proof on his own and sent it off to a math journal. Math journals carefully date stamp everything that comes in. Gödel’s earlier date was on record. Johnny von Neumann had been beaten to the punch. That didn’t happen very often.
Johnny never ventured into logic again, but he and Gödel kept crossing paths. Gödel continued to share his work with Johnny, and Johnny was a good friend to Gödel when they worked together at the IAS. It was the computer which Johnny pioneered that made Gödel’s work important in daily life. Today they are buried in Princeton Cemetery, a few steps apart.