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There is a new research going on in Neural Turing Machine which, in theory, can exactly do that. Maybe we have to wait some time to see NNs can sort numbers.

On a high level, sort() is a seq2seq function, I think throw LSTM/RNN will yield some interesting results to this problem.

Sibling comments have mentioned Neural Turing Machines, but even just regular recurrent neural networks are technically Turing complete[1].

[1] - http://lipas.uwasa.fi/stes/step96/step96/hyotyniemi1/

A Neural Turing Machine can learn programs.

ok i meant some function

sort is a function (in the mathematical sense)

then if it 's a continuous function it can be approximated

and if it's not a continuous function?

https://en.wikipedia.org/wiki/Universal_approximation_theore...

It's proved for a subset of functions. Maybe you can prove it for more.