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My question is basically in the title. When we move from real numbers to complex numbers, is the reason we can talk about expressions such as $e^{\pi i}$ the transfer principle?

EDIT: Since clarification was needed, I was referring to something similar to the transfer principle for hyperreal numbers, where it's used to extend real functions to the hyperreals. I assumed perhaps something similar was going on here. I should've probably asked for "a" transfer principle instead of "the" transfer principle.

Analytic continuation was what I was after!

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    $\begingroup$ "The" transfer principle? What is "the" transfer principle? $\endgroup$
    – David K
    Mar 1 at 23:59
  • $\begingroup$ Are you talking about analytic continuation and similar concepts? $\endgroup$ Mar 2 at 0:00
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    $\begingroup$ Oh! Yes, you want to look into analytic continuation and other means by which real functions are extended into (part of) the complex plane. $\endgroup$ Mar 2 at 0:05
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    $\begingroup$ By the way, $e^{\pi i}$ happens to be a real number. $\endgroup$ Mar 2 at 0:24
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    $\begingroup$ Please unaccept my answer and accept Caleb's; his actually answers the question, mine is really just a comment on his. $\endgroup$ Mar 3 at 17:13

2 Answers 2

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Short answer: No :)

The transfer principle is a powerful principle of mathematical logic, most famously used in nonstandard analysis. In that context, it states that the real numbers $\mathbb{R}$ and hyperreal numbers $\mathbb{R}^*$ satisfy exactly the same set of logical statements -- as long as those logical statements are written in something called "first-order logic." Thus, any logical statement "transfers" from the real numbers to the hyperreals (or back from the hyperreals to the reals).

To be more concrete, if we take a statement such as

There exists a number $x$ such that $x^2 = 2$

You will hopefully recognize that this is true in the real numbers, since there exists the number $\sqrt{2}$. Also, I haven't shown this, but the above happens to be a statement expressible in first-order logic. Therefore, by the transfer principle, it must also be true in the hyperreal numbers; that is, $\sqrt{2}$ also exists as a hyperreal number.

However, the transfer principle does not hold when going from the real numbers $\mathbb{R}$ to the complex numbers $\mathbb{C}$. Why is that? Well consider this statement:

There exists a number $x$ such that $x^2 = -1$.

This is false in the real numbers, but true in the complex numbers. Therefore, the set of logical statements that are true in the real numbers and the set of logical statements that are true in the complex numbers are two different sets.

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"The" transfer principle or "a" transfer principle?


In general, a transfer principle has two components: the type of "structure-transformation" whose input and output we want to transfer between, and the logic in question which determines the "transferrable" properties. The classical "reals-to-hyperreals" transfer principle takes as its latter component first-order logic, and as its former component the ultrapower construction; really, it's just a particular corollary of Łos' Theorem. But there are other logics and other methods of getting new structures from old.

There is indeed a weak transfer phenomenon holding between $\mathbb{R}$ and $\mathbb{C}$. Specifically, $\mathbb{C}$ is a quotient of a subring of a power of $\mathbb{R}$. Half of this is obvious: $\mathbb{C}\cong \mathbb{R}[x]/\langle x^2+1\rangle$. (In fact that "$\cong$" may well be an "$=$" depending on how exactly you set things up.) The non-obvious part is that $\mathbb{R}[x]\hookrightarrow\mathbb{R}^\kappa$ for some cardinal $\kappa$, but this is also true and a fun exercise. The relevance of all this is a general theorem, the (easy direction of the) HSP theorem: if $\mathcal{A}$ is built from $\mathcal{B}$ by forming products, taking substructures, and taking quotients, then every equational property true in $\mathcal{B}$ is true in $\mathcal{A}$ as well. A bit snappily, just as ultrapowering preserves first-order sentences, HSPing preserves equational sentences.

Now of course this doesn't contradict Caleb's (absolutely correct) answer; moreover, I don't see how the extremely weak transfer principle described here is at all relevant to, say, the behavior of complex exponentiation. But I do think it's worth pointing out that "transfer principle" is a larger tent than one might expect.


FWIW the hard direction of the HSP theorem is the converse: if the equational theory of $\mathcal{A}$ is contained in the equational theory of $\mathcal{B}$, then $\mathcal{A}$ is a quotient of a substructure of a power of $\mathcal{B}$. Similarly, although with massively more work, $\mathcal{A}\equiv\mathcal{B}$ iff $\mathcal{A}$ can be built from $\mathcal{B}$ via ultrapowers and ultralimits; this is the Keisler-Shelah theorem. But we don't need this other direction here.

Incidentally, the "H" in "HSP" is for "homomorphic images;" after all, up to isomorphism quotients are just homomorphic images.

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