I can present you with a definition that specifies an infinite set. If you ask for an element, I can give you one. If you present me with a thing, I can tell yo

I can present you with a definition that specifies an infinite set. If you ask for an element, I can give you one. If you present me with a thing, I can tell you if it's in the set. In a very real sense it is completely specified. Similarly defining an infinite collection of sets. However, if I have an uncountable collection of non-empty sets, sometimes you can tell me how to choose one from each (for example, if I have uncountably many pairs of shoes then you can just say "pick the left one"), but within the axioms of set theory, if all you know is that there are infinitely many non-empty sets, the ZF axioms don't allow you to declare that there is a function which when given one of the sets, returns to you an element from that set. Your statements seem to be saying "If you accept that there are infinite sets then you must accept that the Axiom of Choice is 'True'." That turns out not to be the case. There are sets of axioms that result in systems that have infinite sets but in which the Axiom of Choice is not 'True'. Perhaps I've mis-characterised your position.