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I know there isn't a fixed definition for Set (Am I right?), but still there's one that I have seen more commonly goes like this 'A collection of well defined distinct objects'. And then they explain the term 'well defined' which means there is no confusion regarding the inclusion or exclusion of any object. Everything is ok here. But a small thing bothering me here. The term well defined should actually come before word collection, because the rule by which we are collecting objects needs to be in such a way that there isn't confusion about which objects to include and which not. So I feel 'A well defined collection of distinct objects' looks more appropriate definition than the first one. But yes, the objects we are collecting should also be well defined, means they shouldn't be something absurd or weird, so even more correct definition can be 'A well defined collection of distinct well defined objects'.

I don't want to get into deep things like what we actually mean by well defined and all.. but was a little curious about what should be the correct position for well defined!

ogirkar
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  • You might want to look into axiomatic set theory, and also Russell’s Paradox. – Vivaan Daga Feb 12 '22 at 15:52
  • But which one looks more correct? first, second, third or none :p – ogirkar Feb 12 '22 at 15:55
  • All of them are wrong, some wronger than others. – markvs Feb 12 '22 at 15:58
  • Either way, this definition is not very precise because: what is a “collection”, what is an “object”, and what does “well defined” mean? I’m not an expert on set theory, but in my mind there is no precise definition of a set — it is just a primitive term that I understand intuitively. – littleO Feb 12 '22 at 15:58
  • @littleO What I want is, if we assume all this terms for the moment, where should term well defined come, before collection or before objects – ogirkar Feb 12 '22 at 16:01
  • "well defined" should not be there at all. – markvs Feb 12 '22 at 16:10
  • "The smallest natural number which cannot be defined in less than $100$ words" is well defined or not? – markvs Feb 12 '22 at 16:12
  • You won't arrive at a satisfactory understanding of what a set is without delving into axiomatic (or formal) set theory, which is (so far as I am aware) rarely taught at undergraduate level. This stands in contrast with naive set theory, which all mathematics undergraduates are familiar with from the start. As @Logic points out, Russell's Paradox is instructive here. For example, does the 'set of all sets', 'set of all groups' etc. satisfy your definition of a set? How about the 'set of all sets that do not contain themselves' i.e. sets $S$ for which $S\notin S$? – user829347 Feb 12 '22 at 16:39

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As mentioned in the comments, the "definitions" you're describing are informal and can't be rigorously evaluated, because none of the terms you're using ("object", "collection", "well-defined") have precise meanings either.

The way we avoid this problem of "having nowhere to start" is to skip saying what a set "really is" and just state several properties that sets have. These properties are the axioms of set theory, and everything we prove about sets is deduced from them. This way, it doesn't matter what sets "really are", we just know that if a universe of sets obeys the axioms, then all our theorems are also true of that universe.

Karl
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