Talk:Resolution of singularities

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The article begins thus:

In algebraic geometry, the problem of resolution of singularities asks whether any algebraic variety has a non-singular model (a non-singular variety birational to it).

"Any" is often ambiguous. Does it ask

  • whether there is any algebraic variety that has a non-singular model; or
  • whether it is the case that any (i.e. every) algebraic variety has a non-singular model?

I don't know algebraic geometry, but I know it's not unusual for mathematicians to use "any" while overlooking this potential misunderstanding. In about 2002, the article on the compactness theorem in logic said that that theorem states that a set of sentences "has a model if any finite subset has a model". That's begging to be misunderstood! If the second meaning above was intended, then just changing "any" to "every" would fix the problem. Michael Hardy 03:48, 29 March 2007 (UTC)

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