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Looking at the axioms of comprehension and replacement, I see that they allow for the use of parameters in the formulas used in the axioms. From what I can understand, parameters are specific sets that may not be definable in the language of set theory. Using these sets in formulas gives us extra expressive power. Why are we allowed to do this? Are these parameters in the domain of discourse? How would we determine that these parameters are indeed sets if they aren't definable in the language of set theory? Thanks to all who can clear this up.

Edit: My question is concerned with why we are allowed to use parameters, not what they are.

Simplicio
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  • Could you give an example? – Santana Afton Mar 19 '17 at 17:13
  • They're part of the domain of discourse because there isn't anything that isn't, as far as the theory is concerned; and they're sets because there aren't any non-sets, according to most set theories. – Malice Vidrine Mar 19 '17 at 17:13
  • @MaliceVidrine How do we know that a parameter isn't a proper class, for example the set of ordinals ON? – Simplicio Mar 19 '17 at 17:14
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    Parameters are taken from the universe. They are sets by the very definition of "being in the universe". – Asaf Karagila Mar 19 '17 at 17:15
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    Are you in a theory that has proper classes? If it's ZFC, proper classes are a figure of speech, not something that can occur as the value of a variable. – Malice Vidrine Mar 19 '17 at 17:15
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    Have you read the answer on the linked thread? – Asaf Karagila Mar 19 '17 at 17:17
  • @MaliceVidrine I see. That makes more sense. – Simplicio Mar 19 '17 at 17:19
  • @AsafKaragila Yes, I did. I was just concerned that parameters would allow the use of objects that aren't in the universe. I am pretty sure I understand what parameters are. – Simplicio Mar 19 '17 at 17:20
  • @AsafKaragila In your answer to the linked question you say that parameters are definable in the language. However, I was under the impression that parameters can also be things that aren't definable in the language. – Simplicio Mar 19 '17 at 17:35
  • Since the word "definable" does not appear on that page, I'm not quite sure what you mean by that. – Asaf Karagila Mar 19 '17 at 17:36
  • @AsafKaragila "But set theory does not have an intrinsic sense of what are the real numbers, what are functions and what are roots of a function. But we do know that we can define a set and structure that gives us the real numbers, and we can define functions like sin or cos." I thought that use of the "define" was intended to mean that the parameters referenced have to be definable in the language. – Simplicio Mar 19 '17 at 17:38
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    That was an example for why we want to have parameters in the first place. Nobody said that the parameters have to have a definition. Indeed, my answer, if anything, implies that any "interpretation of the real numbers" (definable in the language of set theory or not) can be a valid parameters for us. – Asaf Karagila Mar 19 '17 at 17:39
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    @AsafKaragila If a parameter doesn't have a definition how would one know that it is a valid object to reference? (Thanks for taking the time to explain this, I appreciate your help) – Simplicio Mar 19 '17 at 17:43
  • @AsafKaragila More specifically, how would we even reference something that isn't definable in the language? – Simplicio Mar 19 '17 at 17:51
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    When you use the parameters, you already have the universe in which you apply these things. Therefore the parameters already exist and they are sets. Syntactically, you can always use existential instantiation to add a new constant symbol with some properties which you can prove something must exist to have, and then use that if you want. This is subtle, but I think that naively, the semantic approach is easier to understand. – Asaf Karagila Mar 19 '17 at 17:55
  • @AsafKaragila That makes sense. I believe I was taking the definability and the existence of sets to be the same thing, although they are not. Thanks again for your time. – Simplicio Mar 19 '17 at 18:06

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