- Concerning the undecidability of an assertion ( a theorem ) in mathematics, can somebody give an example and thus prove that a theorem is undecidable ? I will then higly appreciate showing the procedure of using the axiom of choice to prove that only by using the axiom of choice the theorem in question could be proven. Can somebody also give the example of an undecidable theorem which is proven to be so but for which there is still no axiom of choice which could prove the theorem ?
- I was reading recently in a book that every undecidable question creates a bifurcation and imposes a choice. It then gives the example of P. Cohens theorem on the continuum hypotheses which, according to the book, leads to a bifurcation: we have to choose either there are no cardinals between the countable and the continuum or that there are 36 of them. The first choise is then made because of simplicity (according to the book). What i dont understand here is: how one proceeds from the simple definition of the choice function and axiom of choice in the set-theoretical context to the assertion of such choices as mentioned above for which you would hardly see the intervention of the axiom of choice as we define it usually ?
Many thanks for your comments.