So your original half-as-strong, twice as light thing - wasn't intended to be proper numbers, just stronger / heavier (not quibbling, just trying to be clear)
so, which is strongest for a given weight - surely that's the key thing isn't it
(with strength meaning that combination of strength-related good characteristics for a given purposes)
Strength is such a difficult word that I just don't even want to attempt the answer unless you give me a very specific example and a bicycle frame is not a specific example.
Material strength is measured in different ways. Basically, we measure pull strength (tensile), push strength (compression) and shear (hold your two fists together clutching a banana. Now move your right fist forward whilst holding the left fist in its original position. This is shear). A high one in one area doesn't mean a high one elsewhere. Then, we get structural strength that has little to do with the material and more with the shape of the material. A long piece of copper sheet cannot be picked up like a fishing rod. It will flop. A copper pipe make from the exact same piece of sheet, will even work as a fishing rod. Here we say the one structure is stronger than the other.
The same for twisting forces. a pipe transmits twisting forces better than a flat sheet from the same metal. In fact, bang the pipe flat and it transform into a poor twisting force transmitter.
Then we have fatigue properties. An item may be very strong but it can only bend twice before it breaks. High strength, low durability.
Then we have mixed strength properties. A biscuit is stiff but brittle.
Other items are tough, yet another property.
Some items become tougher, the more you bang on them - copper and titanium are two excellent examples of materials that work-harden.
Some materials crack very easily if there are pre-existent flaws in them. Here ti is suspect because it gets flaws from manufacturing which cannot be seen and only surface with use.
Some materials crack and allow the crack to travel very quickly. Think windscreens on your car or the material that potato crisps are made from. If you manage to open the package without nicking the edge, you can be sure that the packet will survive until the crisps are finished. But nick that same packet and it will be torn right to the bottom before you are finished.
Many of these "strengths" depends on how the item was designed and manufactured. Search here for a recent post on Specialized pedal extenders. There the design is at fault by 100%. The gadget was made from CroMo steel (strong stuff) but a schoolboy design error make it lethal.
So yes, I did not quote proper numbers and even warned that "strength" is an illusive concept.
If you force me into a corner and hit me on the nose until I give you a single answer, then it has to be carbon.