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There exist constructive and non-constructive proofs.

Sometimes, for a mathematical statement, we can have both non-constructive and a constructive proof.

However, are there statements for which there is only a non-constructive proof? (The fact that there maybe a construction of the required object but we don't know it yet doesn't count here).

Phrased differently, are there statements (that claim existence of objects) that are essentially non-constructive?

  • 3
    How about anything that we know cannot be proved in ZF, that is, requires some version of the Axiom of Choice, possibly weaker than full AC? Then there is a huge number of statements, important in standard mathematics, that only have a non-constructive proof. Hahn-Banach, existence of a non-principal ultrafilter on $\mathbb{N}$, existence of a basis for a general vector space, and so on and so on.2011-10-17
  • 0
    Could anyone give a simpler example? Moreover, what ensures that there does not exist a constructive proof for a statement?2011-10-17
  • 4
    To ensure that there is no "constructive proof", one usually constructs a model of ZF in which the statement is not true. You have not told us what is your background, so it is very difficult to know what you consider simple!2011-10-17
  • 0
    @Mariano, I think you are confusing [constructible](http://en.wikipedia.org/wiki/Constructible_universe) with [constructive](http://en.wikipedia.org/wiki/Constructivism_%28mathematics%29) (also see [SEP](http://plato.stanford.edu/entries/mathematics-constructive/)).2011-11-12

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