The Collaborative International Dictionary of English v.0.48:

Ineradicable \In`e*rad"i*ca*ble\, a.
   Incapable of being eradicated or rooted out.
   [1913 Webster]

         The bad seed thus sown was ineradicable. --Ld. Lytton.
   [1913 Webster]

WordNet (r) 3.0 (2006):

ineradicable
    adj 1: not able to be destroyed or rooted out; "ineradicable
           superstitions" [ant: eradicable]

Moby Thesaurus II by Grady Ward, 1.0:

19 Moby Thesaurus words for "ineradicable":
   deathless, immortal, imperishable, impregnable, incorruptible,
   indelible, indestructible, ineffaceable, inexpugnable,
   inextinguishable, invincible, invulnerable, nonperishable,
   quenchless, undampable, undestroyable, undying, unerasable,
   unquenchable