adj. Impossible to separate into its component parts.
the GNU version of the Collaborative International Dictionary of English
adj. Incapable of being resolved; not separable into component parts.
The Century Dictionary and Cyclopedia
Not resolvable; incapable of being resolved.
Word Usage
"He was actually very close to working for the president, but decided in the end, he did not want to be President Obama's commerce secretary, citing what he called irresolvable conflicts."