The Collaborative International Dictionary
Irreconcilableness
Irreconcilable \Ir*rec"on*ci`la*ble\ (?; 277), a. [Pref. ir- not + reconcilable: cf. F. irr['e]conciliable.] Not reconcilable; not able to be reconciled or brought into accord; implacable; incompatible; inconsistent; disagreeing; as, irreconcilable enemies, statements. -- Ir*rec"on*ci`la*ble*ness, n. -- Ir*rec"on*ci`la*bly, adv.
Wiktionary
irreconcilableness
n. The quality of being irreconcilable; irreconcilability; incompatibility; incongruity.