So Help Me Gödel
a play in one act
CAST OF CHARACTERS
DEFENSE ATTORNEY
REGISTRAR
DEFENDANT
JUDGE
PROSECUTOR
[A typical courtroom. A trial is in progress.]
DEFENSE ATTORNEY: Your honor, the defense now calls the defendant to the stand.
[The defendant rises and approaches the stand. He sits down and places his right hand on an important-looking book presented to him by the court registrar.]
REGISTRAR: Do you solemnly swear that you will tell the truth, the whole truth, and nothing but the truth?
DEFENDANT: I do.
DEFENSE ATTORNEY: Your honor, if it pleases the court, the defendant wishes to read a prepared statement.
JUDGE: Proceed.
[The defense attorney hands the defendant a sheet of paper. The defendant unfolds it.]
DEFENDANT: [reading] The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
DEFENSE ATTORNEY: The jury will note that the statement read by my client is true beyond a reasonable doubt. Your honor, the defendant will now read a second prepared statement.
JUDGE: Another statement?
DEFENSE ATTORNEY: Yes, your honor.
JUDGE: Very well. I'll allow it.
[The defense attorney hands the defendant a second sheet of paper.]
JUDGE: All right. I'll allow one more statement, but you're on thin ice counselor. The state's objection is overruled.
DEFENSE ATTORNEY: Thank you.
[The defense attorney hands the defendant a third sheet of paper.]
a play in one act
CAST OF CHARACTERS
DEFENSE ATTORNEY
REGISTRAR
DEFENDANT
JUDGE
PROSECUTOR
[A typical courtroom. A trial is in progress.]
DEFENSE ATTORNEY: Your honor, the defense now calls the defendant to the stand.
[The defendant rises and approaches the stand. He sits down and places his right hand on an important-looking book presented to him by the court registrar.]
REGISTRAR: Do you solemnly swear that you will tell the truth, the whole truth, and nothing but the truth?
DEFENDANT: I do.
DEFENSE ATTORNEY: Your honor, if it pleases the court, the defendant wishes to read a prepared statement.
JUDGE: Proceed.
[The defense attorney hands the defendant a sheet of paper. The defendant unfolds it.]
DEFENDANT: [reading] The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
DEFENSE ATTORNEY: The jury will note that the statement read by my client is true beyond a reasonable doubt. Your honor, the defendant will now read a second prepared statement.
JUDGE: Another statement?
DEFENSE ATTORNEY: Yes, your honor.
JUDGE: Very well. I'll allow it.
[The defense attorney hands the defendant a second sheet of paper.]
DEFENDANT: [reading] The square root of two cannot be expressed as the quotient of two integers.
DEFENSE ATTORNEY: True again. Thank you. Your honor, the defendant now wishes to read a third prepared statement.
PROSECUTOR: Objection! What is the relevance of all these statements to the case at hand?
DEFENSE ATTORNEY: Your honor, these statements are necessary to establish the credibility of the defendant.
JUDGE: All right. I'll allow one more statement, but you're on thin ice counselor. The state's objection is overruled.
DEFENSE ATTORNEY: Thank you.
[The defense attorney hands the defendant a third sheet of paper.]
DEFENDANT: [reading] There are infinitely many natural numbers greater than one that cannot be formed by multiplying two smaller natural numbers.
DEFENSE ATTORNEY: So true. So true. Your honor, the defendant will now read a fourth prepared statement.
PROSECUTOR: Objection!
JUDGE: Sustained. Counselor, what is the meaning of all these statements?
DEFENSE ATTORNEY: Your honor, as I said, these statements establish my client's credibility. He swore to tell the truth, and that is exactly what he has done. The court can verify that each of his statements is true. All further statements my client might read would be true as well. Having fulfilled his oath to the court, we may proceed with the case at hand.
PROSECUTOR: Objection! The defendant has indeed told the truth, but he has clearly not told the whole truth.
JUDGE: That's true.
DEFENSE ATTORNEY: But the defendant would clearly tell the whole truth if given sufficient time.
JUDGE: But you require an infinite amount of time.
DEFENSE ATTORNEY: Yes, of course.
PROSECUTOR: Given the court's finite time, the state is willing to interpret the defendant's oath as only pertaining to finite statements, such as those that might fit on a single page.
DEFENSE ATTORNEY: Thank you. Then the state will agree that, of all such finite statements, the defendant, given time, would read all those statements that are truthful. Thus, we can all agree that, with sufficient time, my client would meet the terms of his oath by stating the truth. The whole truth.
PROSECUTOR: But what of nothing but the truth? Surely, in the course of reading all such true finite statements, the defendant is bound to recite a false one.
DEFENSE ATTORNEY: How dare you! My client would do no such thing. He is an honorable man.
JUDGE: Enough of this. Counselor, please proceed with your case.
DEFENSE ATTORNEY: Very well, your honor. The defense has no further questions.
JUDGE: The prosecution may now cross-examine the defendant.
[The prosecutor takes out a sheet of paper and writes on it with a black marker.]
PROSECUTOR: Your honor, I have written a statement onto this piece of paper. The state wishes to enter this statement into evidence as Exhibit G.
JUDGE: What is the meaning of this statement?
PROSECUTOR: The state simply wishes to know whether the defendant would read this simple statement, in keeping with his oath to tell the truth, the whole truth, and nothing but the truth.
DEFENSE ATTORNEY: Your honor, I must insist on knowing exactly what the statement says. If it is true, then of course the defendant will read this statement.
PROSECUTOR: The defendant will not read this statement.
DEFENSE ATTORNEY: Oh yes he will! Unless, of course, the statement labeled Exhibit G is false. Now I demand that you tell us the statement.
PROSECUTOR: The defendant will not read this statement.
JUDGE: The court orders the prosecution to reveal the content of Exhibit G.
PROSECUTOR: The defendant will not read this statement.
[The registrar grabs Exhibit G and hands it to the judge, who puts on his glasses and clears his throat.]
JUDGE: [reading from Exhibit G] The defendant will not read this statement. [The judge looks up at the defense attorney.] That is what it says. It says, "The defendant will not read this statement."
PROSECUTOR: Well?
DEFENSE ATTORNEY: Well, what?
PROSECUTOR: Will your client read Exhibit G?
DEFENSE ATTORNEY: Certainly not! Exhibit G is a bunch of nonsense, and my client will have no part in it.
JUDGE: Very well. Let the jury note that the defendant will not read this statement.
PROSECUTOR: Exactly as Exhibit G foretold! That statement told us that the defendant would not read it, and that is indeed what has come to pass.
JUDGE: That does appear to be true.
PROSECUTOR: The jury will note that the statement labeled Exhibit G has turned out, in fact, to be a true statement. The jury will also note that the defendant refused to read this true statement. Your honor, the state has shown that the defendant has not been forthright with this courtroom. By his unwillingness to read a true statement, he has violated his oath to tell the whole truth. His testimony is thus incomplete. The state therefore requests that he be held in contempt of court.
DEFENSE ATTORNEY: Objection! The prosecution is making a mockery of this courtroom.
JUDGE: Overruled. The prosecution has made a valid point.
DEFENSE ATTORNEY: Very well. Your honor, the defense wishes to change its position with respect to Exhibit G.
JUDGE: What do you propose?
DEFENSE ATTORNEY: My client will agree to read the irksome Exhibit G, provided that the prosecution will, at last, get on with the substance of its case against him. There will be no reason to hold my client in contempt.
JUDGE: The court agrees to the defense's proposal. The registrar will now provide the defendant with Exhibit G, to be read aloud to the jury.
PROSECUTOR: Objection!
JUDGE: Oh, what is it now?
PROSECUTOR: The state cannot allow the defendant to knowingly perjure himself on the stand.
JUDGE: Excuse me?
PROSECUTOR: By reading Exhibit G, the defendant will declare to the court that "the defendant will not read this statement." The defendant's action would be inconsistent with the content of Exhibit G. By reading the statement, the defendant would, in fact, make that very statement false. As you know, if he reads a false statement to the court, he will be in violation of his oath to tell nothing but the truth, and the state would have no choice but to accuse him of perjury.
DEFENSE ATTORNEY: Well, then I will certainly not allow my client to utter this false statement.
PROSECUTOR: But then, by not saying it, your client makes the statement a true one. By refusing to say a true statement, his testimony is incomplete.
DEFENSE ATTORNEY: But you have just insisted that he should not recite Exhibit G.
PROSECUTOR: Indeed, he should not. By saying it, your client would make the statement a false one. By uttering a false statement, his testimony would be inconsistent.
JUDGE: Then it appears there are two equally unpalatable options. In the one case, the defendant's testimony will be incomplete, in that it cannot be the whole truth, since it does not include the true statement labeled Exhibit G. In the other case, the defendant's testimony will be inconsistent, in that it is not nothing but the truth, since it requires him to utter the false statement labeled Exhibit G. We find ourselves in quite a pickle then. Is there no way then to avoid a descent into lawlessness?
DEFENSE ATTORNEY: I say we ban any statements that refer to themselves. A statement that refers to "this statement" is clearly troublesome, like the one in Exhibit G, or the paradoxical "This statement is false."
JUDGE: That is a fair compromise. It seems to me that Exhibit G is problematic because it refers to itself.
PROSECUTOR: Your honor, you just told us that "Exhibit G is problematic because it refers to itself." That itself is a statement about a statement, and thus I am currently making statements about statements about statements. Is the court to allow such statements that refer to statements?
JUDGE: Of course it will. I don't see how we could proceed without being able to make statements about statements.
PROSECUTOR: In that case, the state wishes to enter a new statement into evidence.
JUDGE: You will please share the proposed statement with the court.
PROSECUTOR: Certainly. The statement will read: "The defendant will not read Exhibit H."
JUDGE: I see no problem with your statement.
PROSECUTOR: Very well then. The state enters this new statement into evidence, to be labeled Exhibit H, which states that "the defendant will not read Exhibit H."
DEFENSE ATTORNEY: Objection! Your honor, do you see what the prosecution has done? He has found a loophole, allowing him to make a statement about itself.
PROSECUTOR: I have done no such thing. I am merely making a statement about a statement.
DEFENSE ATTORNEY: But your statement is about Exhibit H.
PROSECUTOR: It is.
DEFENSE ATTORNEY: And Exhibit H would be that very statement.
PROSECUTOR: It would be.
DEFENSE ATTORNEY: Therefore Exhibit H would, in fact, refer to itself, leading us back to the same conundrum.
JUDGE: The defense makes a good point, and yet the court cannot prevent such a statement.
DEFENSE ATTORNEY: In that case, I propose a different solution. Suppose we allow such a statement, but we concede that it is not a precise mathematical statement like those given by my client earlier. As such, we cannot determine the prosecution's statements to be true or false in the same sense.
PROSECUTOR: A worthy idea. Alas, my statement can indeed be written in rigorous mathematical language, as Kurt Gödel did when he introduced these ideas in 1931. All testimony must therefore necessarily be incomplete or inconsistent.
JUDGE: Very well. In that case, the court releases the defendant from his previous and unattainable oath. In its place, the defendant must now swear a new oath to this courtroom.
[The registrar holds out an important-looking book, and the defendant places his right hand on it.]
JUDGE: Do you solemnly swear that you will tell the truth, much of it, and not too much else?
[The defendant looks at his lawyer, then back at the judge, then back to the lawyer. He is sweating and visibly upset. At last, he speaks.]
DEFENDANT: OK! I did it! I killed him! And I'd do it again!
[The jury gasps, and the lights in the courtroom slowly fade to black.]
Author's Note
This story is based on the work of Kurt Gödel. In 1931, he constructed an arithmetic statement that signified "There is no proof of this statement." If arithmetic could be used to prove such a statement, then it must be inconsistent. If it couldn't, then it must be incomplete. In other words, Gödel put arithmetic on trial, and showed that it couldn't be used to prove all true arithmetic statements and nothing but true arithmetic statements.
DEFENSE ATTORNEY: So true. So true. Your honor, the defendant will now read a fourth prepared statement.
PROSECUTOR: Objection!
JUDGE: Sustained. Counselor, what is the meaning of all these statements?
DEFENSE ATTORNEY: Your honor, as I said, these statements establish my client's credibility. He swore to tell the truth, and that is exactly what he has done. The court can verify that each of his statements is true. All further statements my client might read would be true as well. Having fulfilled his oath to the court, we may proceed with the case at hand.
PROSECUTOR: Objection! The defendant has indeed told the truth, but he has clearly not told the whole truth.
JUDGE: That's true.
DEFENSE ATTORNEY: But the defendant would clearly tell the whole truth if given sufficient time.
JUDGE: But you require an infinite amount of time.
DEFENSE ATTORNEY: Yes, of course.
PROSECUTOR: Given the court's finite time, the state is willing to interpret the defendant's oath as only pertaining to finite statements, such as those that might fit on a single page.
DEFENSE ATTORNEY: Thank you. Then the state will agree that, of all such finite statements, the defendant, given time, would read all those statements that are truthful. Thus, we can all agree that, with sufficient time, my client would meet the terms of his oath by stating the truth. The whole truth.
PROSECUTOR: But what of nothing but the truth? Surely, in the course of reading all such true finite statements, the defendant is bound to recite a false one.
DEFENSE ATTORNEY: How dare you! My client would do no such thing. He is an honorable man.
JUDGE: Enough of this. Counselor, please proceed with your case.
DEFENSE ATTORNEY: Very well, your honor. The defense has no further questions.
JUDGE: The prosecution may now cross-examine the defendant.
[The prosecutor takes out a sheet of paper and writes on it with a black marker.]
PROSECUTOR: Your honor, I have written a statement onto this piece of paper. The state wishes to enter this statement into evidence as Exhibit G.
JUDGE: What is the meaning of this statement?
PROSECUTOR: The state simply wishes to know whether the defendant would read this simple statement, in keeping with his oath to tell the truth, the whole truth, and nothing but the truth.
DEFENSE ATTORNEY: Your honor, I must insist on knowing exactly what the statement says. If it is true, then of course the defendant will read this statement.
PROSECUTOR: The defendant will not read this statement.
DEFENSE ATTORNEY: Oh yes he will! Unless, of course, the statement labeled Exhibit G is false. Now I demand that you tell us the statement.
PROSECUTOR: The defendant will not read this statement.
JUDGE: The court orders the prosecution to reveal the content of Exhibit G.
PROSECUTOR: The defendant will not read this statement.
[The registrar grabs Exhibit G and hands it to the judge, who puts on his glasses and clears his throat.]
JUDGE: [reading from Exhibit G] The defendant will not read this statement. [The judge looks up at the defense attorney.] That is what it says. It says, "The defendant will not read this statement."
PROSECUTOR: Well?
DEFENSE ATTORNEY: Well, what?
PROSECUTOR: Will your client read Exhibit G?
DEFENSE ATTORNEY: Certainly not! Exhibit G is a bunch of nonsense, and my client will have no part in it.
JUDGE: Very well. Let the jury note that the defendant will not read this statement.
PROSECUTOR: Exactly as Exhibit G foretold! That statement told us that the defendant would not read it, and that is indeed what has come to pass.
JUDGE: That does appear to be true.
PROSECUTOR: The jury will note that the statement labeled Exhibit G has turned out, in fact, to be a true statement. The jury will also note that the defendant refused to read this true statement. Your honor, the state has shown that the defendant has not been forthright with this courtroom. By his unwillingness to read a true statement, he has violated his oath to tell the whole truth. His testimony is thus incomplete. The state therefore requests that he be held in contempt of court.
DEFENSE ATTORNEY: Objection! The prosecution is making a mockery of this courtroom.
JUDGE: Overruled. The prosecution has made a valid point.
DEFENSE ATTORNEY: Very well. Your honor, the defense wishes to change its position with respect to Exhibit G.
JUDGE: What do you propose?
DEFENSE ATTORNEY: My client will agree to read the irksome Exhibit G, provided that the prosecution will, at last, get on with the substance of its case against him. There will be no reason to hold my client in contempt.
JUDGE: The court agrees to the defense's proposal. The registrar will now provide the defendant with Exhibit G, to be read aloud to the jury.
PROSECUTOR: Objection!
JUDGE: Oh, what is it now?
PROSECUTOR: The state cannot allow the defendant to knowingly perjure himself on the stand.
JUDGE: Excuse me?
PROSECUTOR: By reading Exhibit G, the defendant will declare to the court that "the defendant will not read this statement." The defendant's action would be inconsistent with the content of Exhibit G. By reading the statement, the defendant would, in fact, make that very statement false. As you know, if he reads a false statement to the court, he will be in violation of his oath to tell nothing but the truth, and the state would have no choice but to accuse him of perjury.
DEFENSE ATTORNEY: Well, then I will certainly not allow my client to utter this false statement.
PROSECUTOR: But then, by not saying it, your client makes the statement a true one. By refusing to say a true statement, his testimony is incomplete.
DEFENSE ATTORNEY: But you have just insisted that he should not recite Exhibit G.
PROSECUTOR: Indeed, he should not. By saying it, your client would make the statement a false one. By uttering a false statement, his testimony would be inconsistent.
JUDGE: Then it appears there are two equally unpalatable options. In the one case, the defendant's testimony will be incomplete, in that it cannot be the whole truth, since it does not include the true statement labeled Exhibit G. In the other case, the defendant's testimony will be inconsistent, in that it is not nothing but the truth, since it requires him to utter the false statement labeled Exhibit G. We find ourselves in quite a pickle then. Is there no way then to avoid a descent into lawlessness?
DEFENSE ATTORNEY: I say we ban any statements that refer to themselves. A statement that refers to "this statement" is clearly troublesome, like the one in Exhibit G, or the paradoxical "This statement is false."
JUDGE: That is a fair compromise. It seems to me that Exhibit G is problematic because it refers to itself.
PROSECUTOR: Your honor, you just told us that "Exhibit G is problematic because it refers to itself." That itself is a statement about a statement, and thus I am currently making statements about statements about statements. Is the court to allow such statements that refer to statements?
JUDGE: Of course it will. I don't see how we could proceed without being able to make statements about statements.
PROSECUTOR: In that case, the state wishes to enter a new statement into evidence.
JUDGE: You will please share the proposed statement with the court.
PROSECUTOR: Certainly. The statement will read: "The defendant will not read Exhibit H."
JUDGE: I see no problem with your statement.
PROSECUTOR: Very well then. The state enters this new statement into evidence, to be labeled Exhibit H, which states that "the defendant will not read Exhibit H."
DEFENSE ATTORNEY: Objection! Your honor, do you see what the prosecution has done? He has found a loophole, allowing him to make a statement about itself.
PROSECUTOR: I have done no such thing. I am merely making a statement about a statement.
DEFENSE ATTORNEY: But your statement is about Exhibit H.
PROSECUTOR: It is.
DEFENSE ATTORNEY: And Exhibit H would be that very statement.
PROSECUTOR: It would be.
DEFENSE ATTORNEY: Therefore Exhibit H would, in fact, refer to itself, leading us back to the same conundrum.
JUDGE: The defense makes a good point, and yet the court cannot prevent such a statement.
DEFENSE ATTORNEY: In that case, I propose a different solution. Suppose we allow such a statement, but we concede that it is not a precise mathematical statement like those given by my client earlier. As such, we cannot determine the prosecution's statements to be true or false in the same sense.
PROSECUTOR: A worthy idea. Alas, my statement can indeed be written in rigorous mathematical language, as Kurt Gödel did when he introduced these ideas in 1931. All testimony must therefore necessarily be incomplete or inconsistent.
JUDGE: Very well. In that case, the court releases the defendant from his previous and unattainable oath. In its place, the defendant must now swear a new oath to this courtroom.
[The registrar holds out an important-looking book, and the defendant places his right hand on it.]
JUDGE: Do you solemnly swear that you will tell the truth, much of it, and not too much else?
[The defendant looks at his lawyer, then back at the judge, then back to the lawyer. He is sweating and visibly upset. At last, he speaks.]
DEFENDANT: OK! I did it! I killed him! And I'd do it again!
[The jury gasps, and the lights in the courtroom slowly fade to black.]
Author's Note
This story is based on the work of Kurt Gödel. In 1931, he constructed an arithmetic statement that signified "There is no proof of this statement." If arithmetic could be used to prove such a statement, then it must be inconsistent. If it couldn't, then it must be incomplete. In other words, Gödel put arithmetic on trial, and showed that it couldn't be used to prove all true arithmetic statements and nothing but true arithmetic statements.
