[FOM] Prejudice against "unnatural" definitions
joeshipman at aol.com
joeshipman at aol.com
Mon Mar 4 15:01:18 EST 2013
I have seen many mathematical articles where a statement is made of the form "there is an existence proof of X but no Y example of X" where Y is the word "definable" or the word "natural" or the word "canonical".
In most cases I am either unable to assign a precise meaning to such a statement, or the statement is clearly incorrect, or clearly unprovable.
Example: "there is no definable well-ordering of the reals" presumes that there are nonconstructible reals, but I have seen that statement dozens of times and it is hardly ever qualified in a way that makes it both precise and correct.
Another example: "the set of multiplicative characters mod n is a multiplicative group isomorphic to the invertible residue classes mod n but there is no canonical isomorphism". It's easy to define explicit isomorphisms, so what precisely is the difference between "canonical" and "non-canonical" isomorphisms?
Another example: "complex conjugation is the only explicitly definable non-identity element of the absolute Galois group of automprphisms of the algebraic numbers". Any enumeration of the algebraic numbers, combined with any computable function F on the positive integers, gives rise to a clearly defined and computable element of this group: inductively map the Nth algebraic number A_N to the F*(N)th of the numbers compatible with the previously mapped values, where F* is F reduced modularly to lie in the range {1,...,k} and k is the degree of A_N over the field generated by the previously mapped values.
This last is taken from a popular book, "Fearless Symmetry", by Avner Ash and Robert Gross, but they cannot get off the hook by claiming that they are simplifying things to be understandable by naive readers, because they say that Zorn's lemma is needed to obtain these supposedly non-definable group elements, which is obviously wrong.
Can anyone give other examples of this, or attempt to repair the statements I have cited so that they state actual nontrivial theorems?
-- JS
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