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I am curious if it is possible to construct a homomorphism from a field of hyperreal numbers to the field of real numbers? (Similarly, a homomorphism from the surreals to the reals?)

Assuming that the elements of the hyperreal field are finite reals (r), infinitesimals (ε), and infinite numbers (ω), how would the homomorphism be defined?

I imagine that each monad of a given real number in the hyperreal field could be mapped to the same real number of the real field. However, how would each infinite number (ω) of the hyperreal field be mapped to the real field? Can the infinite elements be mapped to an element named "undefined"? I realize that doesn't make much sense, since "undefined" is not an element of the real field.

So, is there no such homomorphism?

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    $\begingroup$ There's no field homomorphism from any hyperreal field to the reals because the linear order is definable in terms of the algebraic structure and any hyperreal field is non-Archimedean. $\endgroup$
    – James E Hanson
    Commented 11 hours ago
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    $\begingroup$ Every homomorphism of fields is injective, hence you cannot map the whole monad to a single element. $\endgroup$
    – Emil Jeřábek
    Commented 8 hours ago

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