Let us call an order-type "untranscendable" if it cannot be embedded into a product of two smaller ones (!). Ordinals are untranscendable if and only if they are multiplicatively
indecomposable. Moreover untranscendability almost implies additive indecomposability, that is to say, there is but one linear order type which is additively decomposable yet untranscendable. However, using the Axiom of Choice one can prove that there is a different untranscendable order type which at least fails to be strongly indecomposable, the order type of the real number continuum. Moreover, we can show that there is nothing more among the sigma-scattered linear order types and consistently neither among the Aronszajn lines.
Towards the end of the talk I am going to sketch some open problems, both in the presence and the absence of the Axiom of Choice.
This is joint ongoing work with Garrett Ervin and Alberto Marcone and builds on previous work by Barbosa, Galvin, Hausdorff, Laver, Ranero, and others.
