What is the usual expression a mathematician uses when he has to make a choice in order limit an over-determined structure, in order to continue his argument?

For instance, when a structure is over-determined, a law may be true up to a sign. Deriving the law may require the mathematician/physicist to choose either sign and follow the formalism. This choice may become convention, eg. Hook's law for linear elasticity.

This is trivial and embarrassing, but I cannot remember nor can I find the answer on google. It has to be something along the lines of, "For the sake of determinacy, choose x from" or "For the sake of certainty, let x be." None of these sound quite natural though. Please help.

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    I've encountered wlog ("without loss of generality") used in this way. – Colin Fine Mar 10 '20 at 13:38
  • For Hooke's Law, F = kx describes the relationship where a simple helical spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is F, and the extension caused is x. But 'Hooke's law for a spring is often stated under the convention that F is the restoring force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes F = -kx' [Wikipedia; modified]. – Edwin Ashworth Mar 10 '20 at 14:01
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    'For the sake of argument, and without loss of generality, take ....' – Edwin Ashworth Mar 10 '20 at 15:05
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    If you need to get from overdetermined to determined, you relax a constraint. If you need to go from underdetermined to determined, you impose a constraint. – Global Charm Mar 10 '20 at 22:33
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    In your example, if either sign would be OK and the difference doesn't really matter, I'd say "for the sake of definiteness" or just "for definiteness". (Being a set-theorist, I would not use "determinacy", which has a technical meaning in set theory.) – Andreas Blass Mar 11 '20 at 19:01

"For the sake of argument, and without loss of generality" is quite good.

Thanks for all the comments and to Edwin Ashworth in particular.

EDIT Thanks, Andreas, it is "definiteness."

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