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One can define the algebraic numbers as the solutions to polynomials with integer, or indeed rational, coefficients. Also, it can be proven that algebraic numbers and only algebraic numbers are solutions to polynomials with algebraic coefficients. However, the latter characterization of algebraic numbers can't be used as a definition, as it would be a circular definition. So, my question really is, has anyone formalized what it means for a characterization of a set of mathematical objects to be a valid, non-circular definition?

user107952
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    Actually, you can define the algebraic numbers to, in some sense, be the minimal set satisfying "the algebraic numbers are solutions to polynomials with algebraic coefficients" and containing $\mathbb{Q}$. This says you construct the algebraic numbers by starting from $\mathbb{Q}$ and, given some numbers you've already constructed, constructing the roots of a polynomial with those coefficients. So you can use recursion to make an apparently "circular" definition sensible. – Qiaochu Yuan Oct 21 '22 at 07:08
  • A definition is not circular if you can write down all the definitions it relies upon in a list, such that each definition in that list only make reference to earlier definitions in the list. The first definition necessarily must only reference the fundamental objects and relations of the theory (e.g., "point", "collinear", in geometry; or "set", "is an element of" in set theory). – Paul Sinclair Oct 22 '22 at 04:30
  • You might be interested in the book "Vicious Circles" by Barwise and Moss. – Alex Kruckman Oct 22 '22 at 18:10

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