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(1) I understood that in the language of first order logic quantification cannot be over sets (there are no predicate variables that can be bound by quantifiers), whereas the language of second-order logic extends the language of first-order logic by allowing quantification of predicate symbols and function symbols (where these symbols may be assigned sets as their extensions http://plato.stanford.edu/entries/logic-higher-order/ ).

(2) I understand that in first order set theory (which uses a first order language) the (first order) variables x,y and z range over sets.

Suppose we have some mathematical theory, T, which concerns some kind of entity. Externally to T, these entities might be able to be understood as sets of sets (but lets suppose that they are not defined as such in the formal theory, T). They would then form the domain of the mathematical theory. Could we, in that case, use FO to quantify over such elements of the domain in this mathematical theory? If so, then it would seem the distinction between FO quantification and SO quantification becomes theory relative in a way which is rather strange. Does this not trivialise the distinction somewhat? While Full SO quantification allows quantification over arbitrary subsets of the domain, certain of those subsets might be elements of the domain of our hypothetical mathematical theory, T, and so in T we might quantify over these elements, treating them as substitutable for FO variables.

user65526
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  • In first-order logic the quantifiers are relative to individual variables. i.e. variables ranging over the objects of the domain. Thus, in first-order arithmetic, the variables (and the quantifiers) range over numbers, because the domain (i.e. the universe) contains only numbers; in the same way, in first-order set theory the variables range over sets, i.e. the domain is the "universe" of sets. 1/2 – Mauro ALLEGRANZA Apr 28 '15 at 07:17
  • In second-order logic we add variables and quantifiers for predicates; thus, in the corresponding s-o arithmetic we can quantify over properties (i.e. sets) of numbers and in s-o set theory we can quantifiy over properties of sets. 2/2 – Mauro ALLEGRANZA Apr 28 '15 at 07:19
  • You can see also : Jouko Vaananen, Second order logic, set theory and foundations of mathematics. – Mauro ALLEGRANZA Apr 28 '15 at 07:22
  • Why can we not therefore simply introduce sets of sets as objects of a domain, as we introduce numbers as objects in a domain in FO arithmetic and sets as objects in a domain in FO set theory? Doesn't this make the FO/SO contrast rather unpleasantly (or perhaps advantageously!) theory relative (depending on what is considered an object in a given domain)? – user65526 Apr 28 '15 at 07:22
  • You can see also this post. – Mauro ALLEGRANZA Apr 28 '15 at 07:23
  • In set theory there are only sets, and thus obviously also sets of sets... but with the well-known "limitations" regarding first-order language. In f-o arithmetic you have natural numbers, but not sets of them, and thus you cannot define, e.g. Dedekind cuts that are sets of (sets of) natural numbers... – Mauro ALLEGRANZA Apr 28 '15 at 07:25
  • Thanks, but I'm not sure my point is clear. Suppose we have some mathematical theory, T, which concerns some kind of entity. These entities might be able to be understood as sets of sets (but lets suppose that they were not defined as such in the formal theory). They would then form the domain of the mathematical theory. Could we, in that case, use FO devices to quantify over such elements of the domain in this mathematical theory? – user65526 Apr 28 '15 at 07:28
  • I can actually think of an example. Montague treated individuals like "John" as denoting sets of sets. So we might want to quantify over individuals (for all x, y such that x and y have the same name, x and y have the same birthday). But when we do so, names like "John" denote elements of the domain. So is this a case of FO quantification or SO quantification? – user65526 Apr 28 '15 at 07:31
  • We can develop arithmetic "into" f-o set theory, and thus define proxy for "high-order" objects, like Dedekind cuts : details are available in any standard textbook of set theory. But still we have some limitations due to f-o language and thus, we cannot obviate to the difference between f-o logic and s-o logic (read carefully SEP entry for the details). For most of "practical" mathematical usages, f-o set theory is enough. – Mauro ALLEGRANZA Apr 28 '15 at 07:35
  • I'm not sure I understood you, but I think you are saying that if we have some mathematical theory T, that treats of certain entities (definable as sets of sets outside T) then we can always find an interpretation of this theory in FO set theory, finding suitable proxies for the entities, and thus treat quantification over them as FO. But in the case I was imagining these entities were equivalent to sets of sets. So is quantification over dedekind cuts (treated as proxy objects) allowed in FO set theory? – user65526 Apr 28 '15 at 07:41

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