Is there any formal definition for Universal set other than informal definitions that are used generally?
2026-03-26 07:55:22.1774511722
Is Universal set undefined?
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The universal set is "the set of all objects of interest". Assuming modern set theory is in play (read: Zermelo–Fraenkel), the objects of interests are themselves sets, and under certain (and arguably reasonable) assumptions, there is no set of all sets.
Other approaches to formalizing set theory do allow for a universal set (e.g. Quine's New Foundations, or Positive set theory), and they deter the Russell paradox in other ways.
When you are not interested in set theory, then your universal set is just some "large enough set to include all the meaningful information". This could be the real numbers, and a bunch of functions, or about of other objects.
For example, if you are working in a topological space $X$, then treating $X$ as the universal set is useful. Because $\bigcap\varnothing$ is defined to be "everything", or the universal set, if such set exists.
This makes it slightly cleaner to think about "finite intersection of open sets is open", because $\varnothing$ is itself a finite collection of open sets, so $\bigcap\varnothing$ is supposed to be open. But if we evaluate $\bigcap\varnothing$ in the full set theoretic universe, this is not a set.