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In model theory, an object is algebraic in a structure $M$ if it satisfies a property that only finitely many other objects in $M$ exhibit, where by "property" here we mean one that is expressible in …

I find this question extremely interesting. I have two small observations. First, following up on my comment, the answer is definitely no in the case that you allow real coefficients and want the an …

Tarski's theorem on the decidability of the theory of real-closed fields provides a general algorithm that decides any question expressible in the first order language of real-closed fields. His algor …

The universe that Grothendieck intends to suggest by his notation is known in set theory as HF, the class of hereditarily finite sets, the sets that are finite and have all elements finite and element …

This is a side matter to the main question here, but I wanted to add a bit more on the history of the universe concept, since this seems to be less widely known than it deserves. Namely, universes wer …

My belief is that no result in algebraic geometry that does not explicitly engage the universe concept will fully require the use of universes. Indeed, I shall advance an argument that no such results …

You haven't said precisely what you mean by an algorithm here, and this is actually a nontrivial issue since your question is outside the usual finitary context of computability theory, so it isn't cl …

Obviously the perimeter of an ellipse is a computable function of the parameters (e.g. semi-major and semi-minor axes). What it means for this to be computable is precisely that we can compute approxi …

Let me adopt an extreme interpretation of your question, in order to prove an affirmative answer. Yes, in the arena of the countable, DC suffices. To see why, let me first explain what I mean. If …