Is there an injective homomorphism on the Turing degrees?

Question

I know that the existence of a non-trivial automorphism of the Turing degrees is a long-standing open problem. I am curious to know if something is known about (non-trivial) injective homomorphisms (of the Turing degrees within themselves). In other words, is there a substructure of the Turing degrees which is isomorphic to the Turing degrees?

Answer 1

This does not answer the question, but let me argue that one can add such an injective homomorphism (for the ground-model degrees) by forcing.

The argument relies on the fact that the Turing degree order is universal for countable orders, and indeed for countable upper semi-lattices. For example, see

• Antonio Montalbán, Embeddings into the Turing degrees, 2007.

Every countable upper semi-lattice embeds into the Turing degrees.

Fix a copy of a countably universal partial order inside $\mathcal{D}$, one for which every countable order embeds into it by a back-and-forth argument.

Consider now the forcing extension $V[G]$ in which the continuum and hence also the set of Turing degrees $\mathcal{D}$ is made countable. By performing the back-and-forth argument in $V[G]$, but using the now-countable structure of ground model degrees $\mathcal{D}^V$, we can map $\mathcal{D}^V$ into itself in an order-preserving manner, realizing the embedding in $V[G]$.