A standard model of ZF need not be transitive, of course, and Joel David Hamkins' answer to Large cardinal axioms and Grothendieck universes gives Tarski sets as an interesting example.

I should clarify since the terminology is not entirely standard. By a standard model I mean what Wikipedia does, and what Joel David Hamkins does in his answer to Standard model of ZFC: the set membership relation $E$ of the model is the actual set membership relation $\in$, restricted to sets in $M$.

Does existence of a standard model imply existence of a transitive one? Existence of a Tarski set does imply existence of a Grothendieck universe. But am I right to suspect that if there is a standard model, then the ordinals in the minimal model are not a true initial segment of the ordinals, so the minimal model is not transitive, and we can cut down to where that minimal model is the only standard one? If I understand Guest289 correctly, his argument shows my guess is wrong, since a minimal model is transitive. Do I understand that correctly? Or does this come back to unclarity abut what is a standard model?

Anyway, does existence of a transitive model have higher consistency strength than existence of a standard model? I would not be surprised if the axiom of choice lays a role here but I do not know if it does.

A standard model of ZF need not be transitive, of course, and Joel David Hamkins' answer to Large cardinal axioms and Grothendieck universes gives Tarski sets as an interesting example.

I should clarify since the terminology is not entirely standard. By a standard model I mean what Wikipedia does, and what Joel David Hamkins does in his answer to Standard model of ZFC: the set membership relation $E$ of the model is the actual set membership relation $\in$, restricted to sets in $M$.

Does existence of a standard model imply existence of a transitive one? Existence of a Tarski set does imply existence of a Grothendieck universe. But am I right to suspect that if there is a standard model, then the ordinals in the minimal model are not a true initial segment of the ordinals, so the minimal model is not transitive, and we can cut down to where that minimal model is the only standard one? If I understand Guest289 correctly, his argument shows my guess is wrong, since a minimal model is transitive. Do I understand that correctly? Or does this come back to unclarity abut what is a standard model?

Anyway, does existence of a transitive model have higher consistency strength than existence of a standard model? I would not be surprised if the axiom of choice lays a role here but I do not know if it does.

A standard model of ZF need not be transitive, of course, and Wikipedia makes the believable claim that existence of a standard model does not imply existence of a transitive one.

As to proving that claim, am I right to suspect that if there is a standard model, then the ordinals in the minimal model are not a true initial segment of the ordinals, so the minimal model is not transitive?

Whether that is right or wrong, does existence of a transitive model have higher consistency strength than existence of a standard model? I would not be surprised if the axiom of choice lays a role here but I do not know if it does.

A standard model of ZF need not be transitive, of course, and Joel David Hamkins' answer to Large cardinal axioms and Grothendieck universes gives Tarski sets as an interesting example.

I should clarify since the terminology is not entirely standard. By a standard model I mean what Wikipedia does, and what Joel David Hamkins does in his answer to Standard model of ZFC: the set membership relation $E$ of the model is the actual set membership relation $\in$, restricted to sets in $M$.

Does existence of a standard model imply existence of a transitive one? Existence of a Tarski set does imply existence of a Grothendieck universe. But am I right to suspect that if there is a standard model, then the ordinals in the minimal model are not a true initial segment of the ordinals, so the minimal model is not transitive, and we can cut down to where that minimal model is the only standard one? If I understand Guest289 correctly, his argument shows my guess is wrong, since a minimal model is transitive. Do I understand that correctly? Or does this come back to unclarity abut what is a standard model?

Anyway, does existence of a transitive model have higher consistency strength than existence of a standard model? I would not be surprised if the axiom of choice lays a role here but I do not know if it does.