Thin cables may be strong enough on paper, but once you put everything together and tighten them up, you can put a surprising amount of pre-tension on the cable.  Load bearing cables on that size of tower shouldn't be anything less than 1/4".  Many others prefer 3/8", but they usually raise taller towers and bigger mills, too.

I use 1/8" for little things like bracing the gin pole when it goes up (15' high), for example.  The connections are the killer.  The typical saddle-type fasteners for making a loop on the end of the wire rope are crap, unless you get expensive forged ones.

I once did a load test on the 1/8" wire ropes, one terminated with cheapie saddles, the other terminated with high quality saddles.  The wire rope with cheap saddles failed at 45% breaking strength because the rope slid through.  The other sample failed at over 80% of the rope's breaking strength.
Obviously swaged terminals will give better results.

I am currently involved with machining some blocks and organizers for a 64 foot sailing yacht and the breaking strength of these lines are 18,000 to 20,000 psi with no stretch with a diameter of 12mm. The problem I am running into is the weak point is not the ropes, or the stainless steel, but the composite hull these blocks, sheaves and organizers bolt too.