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It is an important question whether mathematical objects in fact exist. Certainly, we can't locate them in the physical world. I want to know whether sets exist. If they in fact do not exist, then all the axioms of ZFC are either false or vacuous. So, how do we establish that sets exist in the first place?

user107952
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    Math wouldn't be fun anymore if they didn't exist. So, just assume they exist and go from there. – morrowmh Dec 19 '20 at 18:22
  • Also, "axioms" are never false. They are true by definition of "axiom". It's something you take to be true, regardless of whether you actually believe it. Remember, believing$\not=$assuming. – morrowmh Dec 19 '20 at 18:23
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    Well... the issue of "existence" of mathematical "objects" has preoccupied metamatematicians for millennia. But I don't see how "If [sets] in fact do not exist, then all the axioms of ZFC are either false or vacuous." ZFC can apply to constructs... no? – David G. Stork Dec 19 '20 at 18:26
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    The question is meaningless until you explain exactly what you mean by exists. And it is in any case not a mathematical question. – Brian M. Scott Dec 19 '20 at 18:36

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All formulations of ZFC imply that at least one set exists, either by implicit semantic rules of first order logic or by stating it directly.

AnalysisStudent0414
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