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The phrase "achieved the ability to" is correct and usable in written English.
You can use it when discussing the attainment of a skill or capability after effort or development.
Example: "After months of practice, she achieved the ability to play the piano beautifully."
Alternatives: "gained the capacity to" or "developed the skill to".
Exact(13)
Within a few generations, Babylonian astronomers had achieved the ability to predict the behaviour of the Moon and the planets.
Skills achieved; the ability to tell the difference between a rock and a clump of mud and throwing stones really far.
Kim's interest in diplomacy was at most only partially fueled by the pressure: he had already achieved the ability to launch nuclear-tipped missiles to the United States and could negotiate from a position of strength.
In the few cases where this strategy is successful, the partnership develops into an alliance of equals or the partners separate after the weak partner has achieved the ability to compete on its own.
Israel, which regards Iran as its most dangerous enemy, has hinted that it may unilaterally attack suspected Iranian nuclear sites if it believes the Iranians have achieved the ability to create nuclear weapons; Iran has said it would retaliate.
"However much progress we might make on peace," says one, "it would all disintegrate if Iran achieved the ability to make a bomb and proliferate it".That could be a diversionary tactic.
Similar(47)
The Western countries and Israel suspect it is meant to achieve the ability to make nuclear weapons.
It is from this mathematics of learning and big data that a machine can achieve the ability to generate 'intelligent' decisions.
To achieve the ability to adapt on these new requirements autonomous cooperating logistic processes seem to be an appropriate method.
Iran contends the enrichment is for peaceful purposes, but much of the world suspects it is meant to achieve the ability to make atomic weapons.
Western powers suspect that Iran is enriching uranium as part of an effort to achieve the ability to make nuclear weapons.
More suggestions(17)
investigated the ability to
attained the ability to
gained the ability to
achieved the qualification to
achieved the opportunity to
attain the ability to
gain the ability to
outcome the ability to
success the ability to
completion the ability to
reached the ability to
achieved the differentiation to
achieved the instruction to
achieved the recommendation to
achieved the right to
achieved the power to
achieved the stardom to
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com