Remove outer cabbage leaves and set aside.
Core cabbage and set cores aside.
Slice or shred cabbage as fine as possible and place into a large bowl. Sprinkle with salt and set the cabbage aside. The salt will begin to soften the cabbage and extract liquid from the cabbage.
Grate cabbage core and add to bowl.
If using, grate apple and add to bowl.
Pound cabbage mixture with a "kraut pounder" or "pickle packer" (see video) or use clean hands to squeeze the cabbage until it softens and releases a considerable amount of liquid.
Pack cabbage mixture into one half gallon-sized jar or two quart-sized jars.
Fold outer cabbage leaves into quarters and place on top of cabbage mixture in jars.
Place 4 ounce jelly jar or glass fermentation weights on top of cabbage leaves in jar. Gently press down until all the cabbage is submerged under liquid. If the cabbage is dry, add a bit of filtered water (chlorine-free) to assure all the cabbage is submerged under liquid.
Place cap on jar and tighten.
Place jar in a bowl (to catch any overflow) and place the bowl in a warm undisturbed area that is approximately 68°F to 70°F.
As bubbles begin to appear in the jar, loosen the lid daily to release some of the gases that have built up, and then retighten the lid. (This step is not necessary if you are using the pickle pipes.)
After a few days of fermentation, use a clean fork to taste the red cabbage. If the taste is to your liking, refrigerate your red cabbage. Generally, it will take approximately 7-14 days for the red cabbage to reach its proper level of fermentation. Keep in mind that initially it will taste salty, but the saltiness will mellow over the coming weeks.
If you want, you can sweeten your red cabbage after the fermentation process is complete. Transfer the red cabbage to a bowl and add in maple syrup, honey, Sucanat date sugar, other natural sweetener starting with a 1/4 cup and working up to the amount that pleases your palate.
Once sweetened to your liking, return the red cabbage to the jar, put the lid on the jar, and refrigerate it.