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Definition of incompatibility : The quality or state of being incompatible; inconsistency; irreconcilableness.

Synonyms:

mutual exclusiveness, discrepance, horror, gap, agree, revulsion, repulsion, disparity, repugnance, discrepancy

disharmony, dissent, disagreement, collision, dissidence, disunity, dissonance

curiosity, idiosyncrasy, funniness, incongruity, exception, quirkiness, divergence, inconsistency, impropriety, oddity, strangeness, difference, unfamiliarity, peculiarity, aberrance, nonconformity, derangement, uniqueness, outlandishness, deviation, anomaly, aberration, singularity, uncommonness, irregularity, unconventionality, eccentricity, abnormality, freakishness

dislike, alienation, hard feelings, combativeness, conflict, spitefulness, opposition, loathing, surliness, adverseness, wrath, antagonism, coldness, resentment, averseness, tension, bad blood, hatred, quarrelsomeness, animosity, hostility, antipathy, malice, contention, unfriendliness, discordance, enmity, contrariety, rancor, coolness, malevolence, bitterness, inharmoniousness, polarity, inimicality, friction

Usage examples:

• But in China this incompatibility does not exist; a man may be both a Buddhist and a Confucian, because nothing in either is incompatible with the other. - "The Problem of China", Bertrand Russell.
• But the most ominous sign of coming defeat was the incompatibility of temper which rapidly developed between loyal ministers and loyal Assembly. - "British Supremacy & Canadian Self-Government 1839-1854", J. L. Morison.
• Mr. President, is there such incompatibility of interest between the two sections of this country that they can not profitably live together? - "The Rise and Fall of the Confederate Government, Vol. 1 (of 2)", Jefferson Davis.