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Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields together with the unary function symbol $ \exp $ such that in the standard interpretation of this language in $\Bbb R $ (where $ \exp $ is interpreted as the exponential function $ x \mapsto e^x $), $\phi (x) $ holds iff $ x=\pi $?

EDIT: As Levon pointed out, a negative answer to this problem would imply that $π$ and $e$ (and $e^e$, $(2e)^{3e^2}$, and so on) are algebraically independent over $\Bbb Q$, which is an unsolved problem. So, if you think that a definition of $\pi$ is impossible, I would be pleased if you could show something like, that it is possible to reduce $\phi$ to a formula which contains no terms involving bound variables inside exponential functions, which would reduce the problem more or less to a question on algebraical independece. However because there are such intricate connections between exponential and trigonometric functions, I don't think that $\pi$ should be undefinable.

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    $\begingroup$ One can define $2x=\pi$ as the least positive real such that $\sin x=0$. Given the intimate relation with $\exp$ and the trigonometric functions, can you work something out? $\endgroup$
    – Pedro
    Jun 9, 2013 at 22:58
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    $\begingroup$ A negative answer would imply $\pi$ and $e$ are algebraically independent over $\mathbb Q$. $\endgroup$ Jun 10, 2013 at 0:49
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    $\begingroup$ Since $\int_{-\infty}^{\infty} e^{-x^2} = \sqrt{\pi}$, we could express $\pi$ as a square of the value of this integral. I think we can express this in first order language. $\endgroup$
    – xyzzyz
    Jun 10, 2013 at 0:52
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    $\begingroup$ Carl's comment implies that Peter's approach can't work. $\;$ $\endgroup$
    – user57159
    Jun 10, 2013 at 1:45
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    $\begingroup$ Partial answer. Leave out exp, answer is "no". There are models of $(\Bbb R,0,1,+, \times, \lt)$ (that is, real-closed fields) that do not contain $\pi$. An easily described one is the set of all algebraic real numbers. $\endgroup$
    – GEdgar
    Jun 12, 2013 at 17:12

2 Answers 2

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Todd Trimble provided the answer to this question on MO:

Assuming Shanuel's conjecture, this treatise about exponential rings proves that (see Theorem 2.5.1) the exponential ring generated by $\pi$ looks just as the exponential ring generated by nearly every other real number, which implies that there is no defining relation of $\pi$ over $\Bbb R_{\exp}$.

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Using that $\int_{-\infty}^\infty e^{-x^2} = \sqrt{\pi}$

$$ \phi(x) := (\forall_y \psi(y) \implies y^2 = x) $$

Where $$ \psi(y) := \forall_{\epsilon > 0} \exists_N \forall_{n > N} \forall_x s(n, x) \implies |x - y| < \epsilon $$ The formula $\psi(y)$ says that $y$ is a limit of $x_n$ such that $s(n, x_n)$ holds. Now, $s(n, x)$ says that $\int_{-n}^n e^{-t^2} = x$:

$$ s(k, x) = \forall_{\epsilon > 0} \exists_N \forall_{n > N} \forall_y p(n, k, y) \implies |x - y| < \epsilon $$

Now $p(n, k, y)$ says that $y$ is a value of k-th Riemann sum of integral $\int_{-n}^n e^{-t^2}$. I'll leave the details.

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    $\begingroup$ How do you define the Riemann sum in this structure (which cannot even define or quantify over $\mathbb{N}$)? $\endgroup$ Jun 10, 2013 at 1:07
  • $\begingroup$ @Carl : $\:$ How do you know that this structure cannot define $\mathbb{N}$? $\;\;\;$ $\endgroup$
    – user57159
    Jun 10, 2013 at 1:24
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    $\begingroup$ @Ricky Demer: Alex Wilkie proved in 1999 that the structure $\mathbb{R}_{\text{exp}}$ is o-minimal. en.wikipedia.org/wiki/O-minimal_theory $\endgroup$ Jun 10, 2013 at 1:34
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    $\begingroup$ (Note: I have upvoted this, not because I agree but because -3 seemed harsh for a well-meant (and flawed) attempt.) $\endgroup$
    – Myself
    Jun 12, 2013 at 17:02
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    $\begingroup$ One would then similarly use $\int_{-\infty}^\infty dx/(1+x^2)$ to define $\pi$ without using $\exp$ at all, which is impossible. $\endgroup$
    – GEdgar
    Jul 14, 2013 at 18:31

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