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Judging A Book By Its Coverage

December 6, 2013

Flaws in constitution

Cropped from book page.

Rebecca Goldstein is the author of the book Incompleteness: The Proof and Paradox of Kurt Gödel. She obtained her PhD in Philosophy from Princeton University, and has also written several novels set in academia, including The Mind-Body Problem and Properties of Light: A Novel of Love, Betrayal and Quantum Physics. The latter draws on the life and concerns of the physicist David Bohm.

Today Ken and I wish to talk about Kurt Gödel’s journey in getting his USA citizenship, and his journey since then in the interpretation and implications of his research.

Gödel’s citizenship interview happened on Thursday the 5th of December, 1947—over fifty years ago. Even over sixty years ago, come to think of it. Past a certain age it becomes better to focus on the wider part of the calendar than the four-digit number at the top.

I bought Goldstein’s book years ago and started to read it. But somehow the initial few pages were not that compelling, or I was distracted by doing something else. In any event, I recently had a long plane flight and took the book—yes I still read hard-copy printed books—along. Partially because it was small, partially because it was on Gödel, and partially by randomness.

It turns out the book is a mixed bag. It was a fun read, with many interesting insights into the life of Gödel. It was also filled with strange errors that I easily noticed, even flying at 36,000 feet without any access to Google search. Yet I did enjoy the book, and am sorry I had not read it before. Well not completely—without it the plane flight would have been longer, since reading helps shrink the time of a flight.

The Story

Here is the story, according to Goldstein, of the day Gödel went to Trenton to get sworn in as an American citizen. Gödel had prepared well for his hearing, and had further discovered that the U.S. Constitution has a flaw that could allow it to become a dictatorship.

Oskar Morgenstern and Albert Einstein drove Gödel to Trenton for his hearing before the judge. On the car ride Einstein tried to distract Gödel with jokes: “Well, are you ready for your next-to-the-last test?” Gödel answered “What do you mean, ‘next-to-the-last’?” Einstein aded, “Very simple. The last will be when you step into your grave.”

Einstein continued on till they reached the court where the judge was Philip Forman, who was a friend of Einstein besides having administered Einstein’s own citizenship oath. The judge moved them quickly into his private chambers. Einstein and the judge chatted while Gödel sat mute. Finally the judge said to Gödel, “Up to now you have held German citizenship.” Gödel corrected him: Austrian citizenship. The judge added, “In any case, it was under an evil dictatorship. Fortunately, that is not possible in America.”

As Goldstein says, this was what Gödel was waiting for. Gödel started to explain how it could happen here because of the flaw in the Constitution. The judge interrupted and said “You needn’t go into all that.” The rest when smoothly and after the oath Gödel become a US citizen. Later in a letter to his mother, Gödel remarked that Forman was a “very sympathetic person.”

The Lost Story

It was known that Morgenstern had written an account of that day, but when his widow was interviewed in 1983 by John Dawson, she had been unable to locate it. Dawson used her recollections in his 1997 biography of Gödel. In 2006 the Institute for Advanced Study hailed the centennial of Gödel’s birth in its spring newsletter. This included a sidebar titled “Gödel, Einstein, and the Immigration Service,” later reproduced on their Gödel page, but with a story quite different from what Dawson had heard. Moreover, the IAS gave the year as 1948. Perhaps they followed my advice about calendars.

Mathematician and author Jeffrey Kegler, who based a novel on Gödel’s two lost notebooks, tells the full story on a neat page with links to all sources, including his own blog posts. While editing Wikipedia’s Gödel page in November 2008, he found another account that “rang true” more than the existing hearsay accounts, and resembled the IAS version. He was convinced the latter had to be based on a true original. He contacted Dawson, who in turn prompted the Institute to find and release it.

Morgenstern in fact mentions only the year 1946. Here is part of what he wrote:

…[Gödel] rather excitedly told me that in looking at the Constitution, to his distress, he had found some inner contradictions and that he could show how in a perfectly legal manner it would be possible for somebody to become a dictator and set up a Fascist regime… I tried to persuade him that he should avoid bringing up such matters at the examination before the court in Trenton, and I also told Einstein about it: he was horrified that such an idea had occurred to Gödel, and he also told him he should not worry about these things nor discuss that matter.

Many months went by and finally the date for the examination in Trenton came. … While we were driving, Einstein turned around a little and said, “Now Gödel, are you really well prepared for this examination?” Of course, this remark upset Gödel tremendously, which was exactly what Einstein intended and he was greatly amused when he saw the worry on Gödel’s face. …

When we came to Trenton, we were ushered into a big room, and while normally the witnesses are questioned separately from the candidate, because of Einstein’s appearance, an exception was made and all three of us were invited to sit down together, Gödel, in the center. The examiner first asked Einstein and then me whether we thought Gödel would make a good citizen. We assured him that this would certainly be the case, that he was a distinguished man, etc. And then he turned to Gödel and said,

“Now, Mr. Gödel, where do you come from?”

Gödel: “Where I come from? Austria.”

The Examiner: “What kind of government did you have in Austria?”

Gödel: “It was a republic, but the constitution was such that it finally was changed into a dictatorship.”

The Examiner: “Oh! This is very bad. This could not happen in this country.”

Gödel: “Oh, yes, I can prove it.”

So of all the possible questions, just that critical one was asked by the Examiner. Einstein and I were horrified during this exchange; the Examiner was intelligent enough to quickly quieten Gödel… and broke off the examination at this point, greatly to our relief. …

Then off to Einstein’s home again, and he turned back once more toward Gödel, and said, “Now, Gödel, this was your last-but-one examination;” Gödel: “Goodness, is there still another one to come?” and he was already worried. And then Einstein said, “Gödel, the next examination is when you step into your grave.” Gödel: “But Einstein, I don’t step into my grave.” and then Einstein said, “Gödel, that’s just the joke of it!” and with that he departed. I drove Gödel home. Everybody was relieved that this formidable affair was over; Gödel had his head free again to go about problems of philosophy and logic.

The Lost Flaw

Maddeningly left out is what exactly the “inner contradictions” were. There have been various speculations, even a paper, most revolving around the Constitution’s providing the power to amend itself. Kegler has his own hypothesis.

Here I—Ken writing this—must confess I am unable to locate the webpage with what I took to be the flaw when I did background reading for our first “interview” with Gödel two years ago. I’ve alas never picked up the index-card habit. What struck my memory, however, was the source’s reference to the Senate and the judiciary.

Trying to reconstruct it, I think the path to dictatorship Gödel feared starts with something like this: The President of the Senate declares that a rules issue is a Constitutional question. This enables a bare majority, exploiting the gaps in Article I, to rewrite the rules of the Senate. Such a rule change can enable the uncontested appointment of Federal judges. Those judges in turn can… Well, anyway, nothing like that would ever actually happen.

Back to Dick and to Goldstein’s book, which to be fair, came out a few months before the IAS newsletter with Morgenstern’s account.

The Book

The book is—as I stated already—a mixed-bag, at best. I liked the history and insights into Gödel’s life. Yet it has many errors—both small errors that were almost just typos, and major errors. I had my thoughts, but Ken found the tough review by Solomon Feferman, so let’s quote that:

As to the core of Goldstein’s book, anyone familiar with Gödel’s work has to flinch. Dozens of errors could have been avoided by an expert vetting of the manuscript. At the very least we would not have had ‘Kreisl’ for ‘Kreisel,’ ‘Kline’ for ‘Kleene,’ and ‘Tannenbaum’ for ‘Teitelbaum’ (the birth surname of Alfred Tarski, the great logician, whose significant interaction with Gödel barely merits Goldstein’s notice).

In the air, flying way above the clouds, I certainly wondered if I was dreaming when I saw the reference to “Kreisl.” At first I wondered did she mean someone other than the famous logician Georg Kreisel? I could not believe that there could be another. Kreisel worked on proof theory and is known for many things including this amazing conjecture:

Suppose that Peano Arithmetic (PA) proves {A(S^{n}(0))} in {O(1)} steps for all {n}, then PA proves {\forall x A(x)}.

Note: {S^{n}(0)} is the successor function applied to {0} a total of {n} times: it is {n} in unary. There is some evidence for and against it; the latter two papers are by the same author, Pavel Hrubeš.

Errors aside, the book does have some interesting bits of history about Gödel and other mathematicians of his era. Many of the stories are known, perhaps well known. The book is much more about people and their history than a primer of the Incompleteness Theorems. One story that I knew but l like a lot is about Einstein’s salary negotiation with the head of IAS:

Einstein asked for a salary of $3,000, and the head “countered” with an offer of $16,000.

A very interesting example of negotiation. Quoting Feferman again:

What she does very well is to provide a vivid biographical picture of Gödel, beginning mid-stream with his touching relationship with Albert Einstein at the Institute for Advanced Study in Princeton, where, over a period of 15 years until Einstein’s death in 1955, they were often seen walking and talking together.

But he ends with:

Those who are fascinated by Gödel’s theorems—and the general idea of limits to what we can know—may still hunger for a more universal view of their possible significance. But they should not be satisfied with Goldstein’s ‘vast and messy’ goulash; hers is not a recipe for true understanding.

Indeed Feferman most loudly criticizes her signing on to the “view [t]hat Gödel’s theorems were designed to refute the formalist program of David Hilbert.” Both Ken and I have been careful to portray Gödel in harmony with Hilbert, and even as compressing rather than expanding the implications of his own theorems. Of course we have conjured our own fictionalizations of Gödel, and however well sourced, they may have errors. If so, we will amend them. Scrupulousness even made this post a day late.

How Many Unprovable Statements Are There?

While we are talking about Gödel’s Incompleteness Theorems, Tim Gowers has raised a question about unprovable statements in mathematics. In essence it is: Why do we as practicing theorem provers seem to be able to avoid the unprovability issues of Gödel? Or do we?

I have an answer that I am sure Gowers saw, but thought I would share. Consider all true statements {\phi} in Peano Arithmetic of size {N} in some standard encoding. I claim that there is a positive {\delta} so that at least a {\delta} fraction of these true statements cannot be proved in PA. The proof is quite simple. Pick any single unprovable statement {A}. Then consider the set of statements of the form:

\displaystyle  \phi = A \wedge B

for any true {B}. None of these are provable in PA, and they form a positive fraction of all the true statements of length {N}. Statements {A \vee B} where {A} is provable yield a similar upper bound separated from {1} on the proportion of unprovable statements.

Open Problems

A natural question is: in the limit are there more unprovable than provable statements of size {N} as {N} goes to infinity? This depends on encoding details but should be a robust enough question under reasonable conditions. Is it clear that there is a limit? Of course the above construction leads to many uninteresting statements. So the second question might be: can we sharpen the question, for instance by associating to a provable {\phi} the idea of minimizing the size of {\psi} such that {\psi \longrightarrow \phi} has a “trivial” proof?

14 Comments leave one →
  1. Fabio Russ permalink
    December 6, 2013 5:39 pm

    Good question!
    If Landauer’s principle holds…a “trivial” proof from physics’ principle

  2. jeffreykegler permalink
    December 6, 2013 10:52 pm

    Thanks for mentioning my work on Gödel. By the way, I was one of Prof. Lipton’s students in the late 1970’s at Yale. Dick taught my Theory of Computation course. In succeeding years, I’ve come to realize just how good the job he did was. In those days there was no general textbook on Theory, which makes organizing a course much, much harder.

    • December 6, 2013 11:34 pm

      Great, thanks! Part of the day’s delay alluded to in the post was to summarize your discovery story correctly in yea-few words. Have you seen the recent buzz about coding the “Ontological Proof” for machine verification? Time has been one factor keeping us from touching that, but it may lend itself to technical treatment along lines of the “Graph of Math” post. To begin, allocate a node for each formula, and put an edge between f and Not-f, identifying Not-Not-f with f etc. If f forces g, add an edge from f to Not-g, and note that this is the same edge you get from the contrapositive. Now Gödel is talking about 2-colorings of this graph…so what does the rest say in this (decidedly non-sensational) vein?

      • jeffreykegler permalink
        December 7, 2013 10:55 am

        That series did a great job of capturing Gödel’s “voice”, especiallly if you assume that he & Einstein have resumed their walks, and that’s he’s been putting in a lot of work on his one-liners.

      • December 7, 2013 2:56 pm

        Thanks again. About the graph in my other comment, on fourth thought, I might need 3 colors after all, which could put the task into NP-hardness territory…

      • jeffreykegler permalink
        December 8, 2013 1:07 am

        Wrt the Benzmüller and Paleo machine proof, they admitted they chose the Ontological proof because it would make the headlines. So the real news would be their verification approach, which I haven’t had time to check out. As Jordan Sobel points out, the Ontological (aka God) Proof is Kurt Gödel cleaned up by Dana Scott and, looking at it purely as an exercise in logic, things don’t get much better attested than that,

  3. Craig permalink
    December 7, 2013 6:45 pm

    Chaitn’s Omega number holds the answer to the question of how many unprovable statements there are in math. Check out Chaitin’s book Meta-math, which can be found on arxiv.

    • Serge permalink
      December 10, 2013 5:55 am

      A very good book indeed, but Chaitin’s constant is uncomputable except for a finite number of its digits. I guess that’s a consequence of the undecidability of Pip’s open problem…

      • Craig permalink
        December 13, 2013 11:14 am

        From Chaitin’s work, we can conclude the following:

        1) For any mathematical problem, the bits of Omega, when Omega is expressed in binary, completely determine whether that problem is solvable or not.

        2) The bits of Omega are random, and only a finite number of them are even possible to know. (Therefore, an infinite number of them are impossible to know.)

        3) Hence, most mathematics problems are impossible to solve, and most mathematical facts are impossible to prove.

  4. Jim Blair permalink
    December 8, 2013 12:10 am

    In his book “God Created The Integers”, Stephen Hawking’s reported the Constitutional contradiction Gödel was worried about this way:

    “The President could fill vacancies without Senate approval while the Senate was in recess. This, Gödel reasoned, could lead to a dictatorship!”

    Not long after 9/11, a talking head on TV brought up a somewhat similar problem. If the fourth plane had hit the Capitol and wiped out a large number of Representatives and Senators, it would have created a Constitutional crisis because the way the members of Congress can be replaced.

    The only thing that stuck in my mind at the time was that there would be a Constitutional problem with declaring War.

  5. September 19, 2016 5:00 pm


    P: president
    C: congress (both houses)
    X: order, resolution or vote
    Vi: vote

    0. C passes X: C produces V0
    1. C sends V0 to P
    2. P returns V0 to C
    3. C reconsiders V0: C produces V1, V0 shall become a law because V1 exists (shall be of effect)
    4. C sends V1 to P
    5. P returns V1 to C
    6. C reconsiders V1: C produces V2, V1 shall become a law, etc.

    The president can postpone indefinitely anything resulting from the concurrence of both houses:

    X is an order, resolution or vote to which the concurrence of both houses is necessary. This concurrence is a vote, V0:

    **Every** Order, Resolution, or **Vote to which the Concurrence of the Senate and House of Representatives may be necessary** (except on a question of Adjournment) shall be presented to the President of the United States; (U.S. const. I.7, §2)

    X’s vote, i.e., V0, should be sent to president, because if X is sent then V0 of effect:

    and before the Same shall take Effect, shall be approved by him, or being disapproved by him, shall be repassed by two thirds of the Senate and House of Representatives, according to the Rules and Limitations prescribed in the Case of a Bill. (I.7, §3)

    If he [the President of the United States] approve he shall sign it, but if not he shall return it, with his Objections to that House in which it shall have originated

    P doesn’t approve, he returns V0 to that house in which V0 has originated.

    who shall enter the Objections at large on their Journal, and proceed to reconsider it. If after such Reconsideration two thirds of that House shall agree to pass the Bill, it shall be sent, together with the Objections, to the other House, **by which it shall likewise be reconsidered**

    Such reconsiderations result in a vote V1, the mere existence of which causes V0’s becoming of effect. But to become isn’t to be: if V1 exists then V0 of effect, if V0 of effect then V1 of effect and if V1 of effect then V1 sent to P because V1 is not excepted from:

    Every Order, Resolution, or Vote to which the Concurrence of the Senate and House of Representatives may be necessary (**except on a question of Adjournment**) shall be presented to the President of the United States; (U.S. const. I.7, §2)

    Which is what happens in 5, Congress must send the reconsideration vote V1 to P.

    P returns V1 because he can for, after all, V1 is but a vote for which both houses of C are responsible.

    C must reconsider V1 for otherwise neither V1 nor V0 will take effect, in contradiction with 4. C thus makes a V2, the existence of which causes V1 to become of effect, which requires V2’s effect, which won’t happen before its having been sent to P. Etc.
    X won’t even be presented to P if P disagrees on the Vs. But even if X were presented to P, P can still return X & any further V that C may present.

    No Amending of this Constitution will be done without the President’s allowing it. For the proposal of Amendments by the Congress supposes that two thirds of both Houses shall deem it necessary. But both Houses need also agree before they shall call a Convention for proposing Amendments, on the Application of the Legislatures of two thirds of the several States (V). These are respectively, a Resolution and an Order to which the Concurrence of the Senate and House of Representatives may be necessary. What this implies is that the amending process won’t start while disapproved by the President. But neither ends if forbidden by the same. For the very mode of Ratification, since determined by Congress also needs the President’s approval to take Effect (all of this because of I.7, §3).

    No Judgment in Cases of Impeachment will be reached without the prior Consent of the President.
    For such Judgments require Indictment, Trial & Conviction. Now, on the one hand, the Indictment in such Cases is an Order which only the House of Representatives can issue:

    The House of Representatives … shall have the sole Power of Impeachment 5 (I.2, §5).

    On the other hand:

    The Senate shall have the sole Power to try all Impeachments and the Conviction needs the Concurrence of two thirds of the Members present (I.3, §6)

    Which is a Resolution or Vote. So, because the Concurrence of both Houses is necessary, the Verdict is not going to take Effect before its (or a related Vote’s) having been signed or left unreturned in excess of 10 days (Sundays excepted) by the President (cf. I.7, §3).
    Now Congress acted anti-constitutionally by making Amendment I, breaking the following fragment of the same (by then in the Constitution):

    Congress shall make no law … abridging the freedom of speech

    Congress is to be held accountable for that. For Congress made a Law abridging its own freedom of speech (as a legislator). This act is forbidden by the (then amended) Constitution. Moreover, Amendment I becomes a Law because Congress inserts it in the Constitution and this insertion (towards its end) becomes anti-constitutional because of (the just inserted) Amendment I: Amendment I is an ex post facto law, which is against the constitution.
    Amendment XXV, Section 4 is no remedy. For President can Commission the Vice President with some Military Trust (because of II.3 and the Vice President’s being an Officer, see e.g., Amendment XII’s “But no person constitutionally ineligible to the office of President shall be eligible to
    [that] of Vice-President of the United States” where “that” refers to “office”). President as Commander in Chief can then order the Vice President to resign from the Office of Vice President. Should he refuse, being subject to Martial Discipline, he incurs at least his being arrested; but he also could be Impeached, Convicted and removed from Office, because Constitution’s II.4 seems to apply to such disobediences. Now by Amendment XXV’s Section 2, President should nominate a Vice President: he can nominate his own self, and keep doing so upon each non-confirmation by Congress. This results in the Office of Vice President remaining empty as long as President finds fit, unless Congress agrees with the President’s nomination. In this way, President can ensure his not being ousted by Vice President (except by his own self). But even if Vice President were to become President, and Vice President’s Office should, as a result of his becoming President, be considered vacated, he, as President, is expected to nominate a Vice President (XXV.2): he can proceed as explained above (thus leaving that Office empty if not filled with his self); then, the new Dictator can no longer be removed following XXV.4 (except, again, by his own self).

    One could object that reconsidered things need not remain without effect while the reconsideration votes themselves are of no effect i.e., it could be suggested that the word “before” means only logical precedence i.e., that x is before y should only mean that a necessary condition for y is x, where y may very well occur chronologically before x. But this, in fact, makes things worse. For if “before”, as in “before it becomes a Law”, weren’t to require the chronological precedence of the President’s approval (i.e., signature or non-return) then it would be perfectly sound to make of a Bill a Law chronologically before its passing both Houses. For the passage of a Bill is arguably motivated by a prior agreement of some majority of both Houses concerning that Bill: it hardly would have been proposed, if it weren’t probable that it would actually become a Law. Moreover, even if returned by the President, they could already be prejudiced against any objection of his in this process and be determined from the beginning to re-pass the same, or any related Vote should it be returned. So what could be defended then is that, since it is likely from the beginning that the Bill shall become a Law: it suffices for Congress not to by their Adjournment prevent its (or that of a related Vote) Return, because otherwise it shall not be a Law. That Bill shall then be made Law, with the Conditions of its being made so (e.g., passed by both Houses) to be fulfilled at some later time. The U.S. President could, if “before” weren’t chronological, also very well make out of some Recommendation a Law before its Consideration by Congress. For example one in which a part would be: “All non-Yeas, including abstentions, shall be added to Yeas, in the Vote for passing this Recommendation”. For that that Recommendation was never considered by the Congress under the Constitution can’t be determined before the cessation of this Constitution. So, there is no constitutional objection against that Recommendation’s being a Law as soon as President has made it. Now, if there indeed shall be Consideration, the said Recommendation will pass the Senate and the House of Representatives because of that quoted part and its being a Law. It then suffices for the President of the United States not to return it upon being presented with it, within the required period. This would not happen if a prior (in time) Consideration by Congress was necessary, as would be the case if “before” were as usual, because then, Congress could discard such Recommendations by (say) not considering them. Therefore the chronological meaning of “before” is to be maintained.

    Notice that USC IV.4 may be in contradiction with a dictatorial government: USC is inconsistent and classical logic takes care of USC’s requiring what it forbids. One doesn’t have to arrange for the US to be attacked by the US (a classic) in order to establish a practical dictatorship: the USA are de constitutio a dictatorship (and inconsistent at that).

  6. September 20, 2016 2:52 am

    The USC was begotten dead, it is self-destructive, inconsistent. Thomas Jefferson held the doctrine of Permanent Revolution. That failed suicide is prohibited by the USC is a striking confirmation.


    …nor shall be compelled in any criminal case to be a witness against himself… (Amendment V)

    That X fails suicide means that X failed to kill Y where X = Y. Here X must not be compelled to witness against X, but according to Amendment VI:

    In all criminal prosecutions, the accused shall enjoy … to have compulsory process for obtaining witnesses in his favor…

    So, X is to be compelled to witness in favor of X i.e., against Y. Since X = Y, X should be compelled to witness against X: this contradicts Amendment V.

    It is interesting how Congress, by resisting the President’s veto, actually defeats itself, i.e., life is death, as far as Congress is concerned. For then Congress must not fail to reconsider any vetoed vote in the unbounded sequence, for if it were to leave a returned vote not repassed, i.e., if C weren’t to produce a further joint reconsideration vote, then none of the preceding votes would be of effect which is a contradiction because the mere existence of any previous Vi, where i > 0, ensures that Vi-1 would *become* (nota bene the word in the USC which is “become”, not “be”) of effect. Vi-1 shall actually reach effect only after Vi’s effect, for the object of Vi is Vi-1’s being of effect.

    In view of:

    Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:—“I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States.” (II.1.8)

    And because C, by its failing to reconsider a reconsidered but vetoed vote, raised a contradiction i.e., harmed the USC, it is therefore a signal duty, for the President, to punish C.

    It can perhaps be thought that a reconsideration isn’t really a joint vote. This is of course false and an easy practical disproof consists in checking the Journal of both Houses.
    A theoretical proof of the above is likewise easy:

    …But in all such Cases the Votes of both Houses shall be determined by yeas and Nays, and the Names of the Persons voting for and against the Bill shall be entered on the Journal of each House respectively. (I.7.2)

    Each House shall keep a Journal of its Proceedings, and from time to time publish the same, excepting such Parts as may in their Judgment require Secrecy; and the Yeas and Nays of the Members of either House on any question shall, at the Desire of one fifth of those Present, be entered on the Journal. (I.5.3)


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