 (d) The last line of objection was to a general feature of the 1904 proof, which was not changed in the second proof, namely the use of what became known as ‘impredicative definition’. An impredicative definition is one which defines an object a by a property A which itself involves reference, either direct or indirect, to all the things with that property, and this must, of course, include a itself. There is a sense, then, in which the definition of a involves a circle. Both Russell and Poincaré became greatly exercised about this form of definition, and saw the circle involved as being ‘vicious’, responsible for all the paradoxes. If one thinks of definitions as like construction principles, then indeed they are illegitimate. But if one thinks of them rather as ways of singling out things which are already taken to exist, then they are not illegitimate. In this respect, Zermelo endorses Hilbert's view of existence. To show that some particular thing ‘exists’ is to show that it is in 𝔅, i.e., to show by means of a finite proof from the axioms that it exists in 𝔅. What ‘exists’, then, is really a matter of what the axioms, taken as a whole, determine. If the separation, power set and choice principles are axioms, then for a given M in the domain, there will be choice functions/sets on the subsets of M, consequently well-orderings, and so forth; if these principles are not included as axioms, then such demonstrations of existence will not be forthcoming. From this point of view, defining within the language deployed is much more like what Zermelo calls ‘determination’, since definitions, although in a certain sense arbitrary, have to be supported by existence proofs, and of course in general it will turn out that a given extension can be picked out by several, distinct ‘determinations’. In short, Zermelo's view is that definitions pick out (or determine) objects from among the others in the domain being axiomatised; they are not themselves responsible for showing their existence. In the end, the existence of a domain 𝔅 has to be guaranteed by a consistency proof for the collection of axioms. Precisely this view about impredicative definitions was put forward in Ramsey (1926: 368–369) and then later in Gödel's 1944 essay on Russell's mathematical logic as part of his analysis of the various things which could be meant by Russell's ambiguously stated Vicious Circle Principle. (See Gödel 1944: 136, 127–128 of the reprinting in Gödel 1990. See also Hadamard's letters in Hadamard et al. 1905.) To support his view, Zermelo points out that impredicative definitions are taken as standard in established mathematics, particularly in the way that the least upper bound is defined; witness the Cauchy proof of the Fundamental Theorem of Algebra. Once again, Zermelo's reply is coloured by the principle of looking at the actual practice of mathematics.[27] 