What does "possible to define" mean in general?

First I thought it means that "can not lead to a contradiction", but such seems to be hard to prove.

Then for the proof I was looking at, involving natural numbers, it turned out to mean "exists at least one number for which definition holds for each case", which is was relatively easy to prove.

Does "possible to define" always mean "exists something satisfying for all cases"?

My native language is not english. Is that might be the reason that I found this non-obvious?

  • $\begingroup$ To my ear there is little to no formal meaning conveyed by "possible to define": I am having a hard time thinking of a situation in which it would not be "possible to define" something. In context this may have a less trivial connotation. Could you supply the text in which this phrase has been used? $\endgroup$ – Pete L. Clark Jan 20 '14 at 16:54

In the context of Landau's text, "possible to define" means that if $\mathbb{N}$ is the natural numbers, there is a function $+: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ which satisfies the properties specified later in the sentence.

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  • $\begingroup$ I do understand what it means in that context, that is why I did not specify the context in my question. $\endgroup$ – boooo Jan 20 '14 at 17:55
  • $\begingroup$ I just read that sentence in the original german formulation, where it is much clearer. $\endgroup$ – boooo Jan 20 '14 at 18:08
  • $\begingroup$ As I said in my comment above, out of context "possible to define" means "it is possible to make the following definition". This is always true, so far as I can see. $\endgroup$ – Pete L. Clark Jan 20 '14 at 18:24

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