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There is some mistake in the current definition. Since now Ntsh only shifts components that are naturals. Hence, for example, Ntsh(1,3/2)=(2,3/2) and this does not preserve the order type of the pair. Hence (1,3/2) and Ntsh(1,3/2) are not in Ntr relation. (I think my mic isn't working)
I understood that Ntsh only shifts natural numbers in the tail, leaving the order intact.
Oh, I see, so only shifted components are naturals such that in the tuple there are no greater fractional numbers. Thanks, so everything is fine.
I have a question about falsifiability of Π⁰₁ sentences. There is an important limitation that in practice we could test a Π⁰₁ sentence only for relatively small naturals. Do you have an argument why it is interesting to test your sentences on small integers? If we would have a human-readable finitistic proof that there are no counterexamples of the size <2^80 bit, this would probably make the tests non-interesting.