117b – Undecidability and Incompleteness – Lecture 4

We showed that is Diophantine. This implies that the prime numbers are Diophantine. Therefore, there is a polynomial in several variables with integer coefficients whose range, when intersected with the natural numbers, coincides with the set of primes. Amusingly, no nonconstant polynomial with integer coefficients can only take prime values.

We proved the bounded quantifier lemma, the last technical component of the proof of the undecidability of Hilbert’s tenth problem. It implies that the class of relations definable by a formula in the structure coincides with the class of relations (i.e., the Diophantine ones).

To complete the proof of the undecidability of the tenth problem, we will show that any c.e. relation is Diophantine. For this, recall that a set is c.e. iff it is the domain of a Turing machine. We proceeded to code the behavior of Turing machines by means of a Diophantic representation. This involves coding configurations of Turing machines and it remains to show how to code the way one configuration changes into another one.

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4 Responses to 117b – Undecidability and Incompleteness – Lecture 4

I was a bit confused about this in class, but it’s probably just because I was majorly sleep-deprived: “Therefore, there is a polynomial in several variables with integer coefficients whose range on the natural numbers coincides with the prime numbers. Amusingly, no nonconstant polynomial with integer coefficients can only take prime values.”

The distinction, then, is that there IS a polynomial whose range is all the primes AND a bunch of negatives, but there is NO polynomial whose range is primes only? Is that correct?

Yes, exactly. I think I may have actually written something incorrect on the board, so I will briefly review it on Tuesday. We have that the primes are Diophantine. From this, we saw how to build a polynomial P such that, if R is its range, then the intersection of R with the naturals is the set of primes. But any such R necessarily takes negative values (and 0?) as well. I rephrased the entry a little to make this clear.

If anyone’s interested, I remembered reading about this polynomial (though I didn’t know the historical context in which it was derived) a while back in the book The Music of the Primes. According to Amazon’s search-inside-this-book function, it’s on page 200, which I have downloaded from Amazon and is now stored on my web page. It’s good to see these things actually written out!

Thanks!
The same polynomial is shown in page 331 of the handout “Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solutions”, it is due to J. P. Jones. The polynomial we obtain if we simply apply the definitions shown in lecture is different than this one.
It is curious that this polynomial is written as the product of two polynomials, and yet its (positive) values are precisely the prime numbers.

I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The relevant search term is ethnomathematics. There are several journals devoted to this topic (for instance, Revista latinoamericana de etnomatemática). Browsing them (if you have access to MathSciNet, the relevant MSC class is 01A70) and looking at their references should help you get started. Another place to look for this is in journals of history of mat […]

Some of the comments in the previous answers make a subtle mistake, and I think it may be worth clarifying some issues. I am assuming the standard sort of set theory in what follows. Cantor's diagonal theorem (mentioned in some of the answers) gives us that for any set $X$, $|X|

For $\lambda$ a scalar, let $[\lambda]$ denote the $1\times 1$ matrix whose sole entry is $\lambda$. Note that for any column vectors $a,b$, we have that $a^\top b=[a\cdot b]$ and $a[\lambda]=\lambda a$. The matrix at hand has the form $A=vw^\top$. For any $u$, we have that $$Au=(vw^\top)u=v(w^\top u)=v[w\cdot u]=(w\cdot u)v.\tag1$$ This means that there are […]

That you can list $K $ does not mean you can list its complement. Perhaps the thing to note to build your intuition is that the program is not listing the elements of $K $ in increasing order. Indeed, maybe program 20 halts on input 20 but only does it after several million steps, while program 19 doesn't halt on input 19 and program 21 halts on input 2 […]

A reasonable follow-up question is whether there are some natural algebraic properties that the class of cardinals satisfies (provably in $\mathsf{ZF}$ or in $\mathsf{ZF}$ together with a weak axiom of choice). This is a natural problem and was investigated by Tarski in the 1940s, see MR0029954 (10,686f). Tarski, Alfred. Cardinal Algebras. With an Appendix: […]

I was a bit confused about this in class, but it’s probably just because I was majorly sleep-deprived: “Therefore, there is a polynomial in several variables with integer coefficients whose range on the natural numbers coincides with the prime numbers. Amusingly, no nonconstant polynomial with integer coefficients can only take prime values.”

The distinction, then, is that there IS a polynomial whose range is all the primes AND a bunch of negatives, but there is NO polynomial whose range is primes only? Is that correct?

Yes, exactly. I think I may have actually written something incorrect on the board, so I will briefly review it on Tuesday. We have that the primes are Diophantine. From this, we saw how to build a polynomial P such that, if R is its range, then the intersection of R with the naturals is the set of primes. But any such R necessarily takes negative values (and 0?) as well. I rephrased the entry a little to make this clear.

If anyone’s interested, I remembered reading about this polynomial (though I didn’t know the historical context in which it was derived) a while back in the book The Music of the Primes. According to Amazon’s search-inside-this-book function, it’s on page 200, which I have downloaded from Amazon and is now stored on my web page. It’s good to see these things actually written out!

Thanks!

The same polynomial is shown in page 331 of the handout “Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solutions”, it is due to J. P. Jones. The polynomial we obtain if we simply apply the definitions shown in lecture is different than this one.

It is curious that this polynomial is written as the product of two polynomials, and yet its (positive) values are precisely the prime numbers.