Is it true that stars the size of the sun will never have a core hot enough to fuse carbon?
Yes, stars the size of the Sun aren't hot enough for carbon fusion.
A star needs to have a core temperature of over
The Sun is currently fusing hydrogen into helium, with a core temperature of
When the hydrogen runs out in the core the star collapses to the point where hydrogen fusion can start in the region outside of the core. The star expands into a red giant at this stage.
Once the core temperature reached
For stars less than 8 times the mass of the Sun, then the supply of helium is dexhausted, they end up as white dwarf stars with an oxygen carbon core and no further fusion reactions taking place.