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This is the context:

The subtle assumption we made is that for every description there is a set of elements that have the objects described. This works most but not all of the time. For example, if I think of the property of red, then I can form the set of all red things. With a description of pink Cadillacs there is a set of pink Cadillacs. But the description “does not contain itself” cannot correspond to a set of things that does not contain itself. This will lead to a contradiction. We must be careful.

What is the meaning of "objects described"?

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Actually, and somewhat unusually, "objects" here is being used to mean "properties" or "aspects" and "elements" means "elements of a set" and thus "things" or "items". However one should remember that these "things" may be physical objects, or may be abstract groups, that is sets (in the mathematical sense). This passage is clearly leading up to an explanation of the axiom of choice.

I would rewrite the sentence as:

  • The subtle assumption we made is that for every description there is a set of elements that fit the description. (or)
  • The subtle assumption we made is that for every description there is a set of elements that have the properties described.

The sentence in the quote is making a non-standard use of the word "objects", which is potentially quite confusing, and that in my view did not help the explanation.

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  • You're right, and I have deleted my answer: I misread the passage. I think objects is being used in way that doesn't make sense.
    – Colin Fine
    Commented Jul 16, 2019 at 17:56
  • @Colin Thanks. I think the person quoted is playing Humpty-Dumpty, using words to mean whatever s/he wants, regardless of their standard meanings. Perhaps the rest of the text from which the quote comes explains that odd usage of "objects", but somehow I doubt it. If there is a convention somewhere to use "objects" in this way in discussing Set Theory, I have never previously encountered it, and I have read quite a bit of ST.
    – David Siegel
    Commented Jul 16, 2019 at 18:07

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