What is an adjective for "weaker than weak"?
I don't think there is a standard adjective to describe this. If there is, we would need to know the context of the terms stronger and weaker to answer. It sounds like you are defining this weaker-er notion in your paper (since you have to introduce a new term), so it is really on you to give it a name. Now to compile a list of suggestions:
- subweak
- weakerer
- superweak
- ultraweak
- extraweak
- weak' (weak prime)
- weak*
- weak$^2$ (weak squared, or weak two, or too weak)
- feeble
- anemic
- fragile
- puny
I can't resist adding this one to the list: what do you call a principle weaker than Weak Konig's Lemma?
Funny you should ask . . . (Weak Weak Konig's Lemma)
Ayup, we're a creative bunch. :P
Oh my goodness: page 18, after proposition 9.1. It's merely suggested, but: "Weak Weak Weak Konig's Lemma."
And heaven forbid we be at a loss to describe something not as weak as weak! (Page 8, definition 4.5.) "Strong weak truth table reducibility"
I would say "pathetic" or "puny".