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I am trying to understand the basics of model theory. Before getting too deeply into it, I would like to know how it is even possible to construct a model, i.e. a structure that satisfies axioms of a given set theory, e.g. ZFC. If I understand correctly, you will need a set theory to construct such a structure. Can a set theory be used to construct a model of itself? Or do you need some kind of meta-set theory to do that?

Dan Christensen
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    Yes; you need a meta-theory that is itself set theory. See e.g the post How can there be genuine models of set theory? – Mauro ALLEGRANZA Jan 22 '19 at 12:33
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    And see Thomas Jech-Set Theory-Springer (2006), Ch.12 Models of ST. – Mauro ALLEGRANZA Jan 22 '19 at 12:35
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    Topic references: Lowenheim-Skolem Theorem/Process, Godel numbering, Godel Incompleteness Theorems – DanielWainfleet Jan 22 '19 at 12:39
  • See also the answer to the post Models in set theory. – Mauro ALLEGRANZA Jan 22 '19 at 12:45
  • @MauroALLEGRANZA Re: "You need a meta-theory that is itself set theory." Can you describe such a meta-theory in the case of modelling ZFC? – Dan Christensen Jan 22 '19 at 12:50
  • See Carl Mummert's linked answer : "To construct a model of set theory means to produce a set A and a relation R on A×A such that all the axioms of ZFC are satisfied if we take "set" to mean "element of A" and take "a∈b" to mean aRb. This is not actually any different than the case with groups. There is one complication, though. Although ZFC proves that there is a model of the group axioms, ZFC does not prove that there is a model of the ZFC axioms. 1/2 – Mauro ALLEGRANZA Jan 22 '19 at 12:59
  • One way we can get around this is by moving to a stronger system of set theory to construct the model of ZFC. For example, Kelley-Morse set theory proves that there is a model of ZFC. Another way is to simply assume there is one model, and use that to construct other models." 2/2 – Mauro ALLEGRANZA Jan 22 '19 at 12:59
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    The issue is that a model of a theory is a mathematical structure and mathematical structures a formally described by set theory. Thus, in order to describe a model of a set th $T_1$ you need a meta-theory $T_2$ that is iteself a set theory. In order to prove the existence of a model of $T_1$, due to G's Incompleteness Th (a theory containing PA cannot prove its own consistency), you need a theory $T_2$ that is "stronger". – Mauro ALLEGRANZA Jan 22 '19 at 13:06
  • @MauroALLEGRANZA This would seem to lead to an infinite regress of stronger and stronger set theories. What do we gain by this? Would we lose much by simply not modelling set theories? – Dan Christensen Jan 22 '19 at 13:28

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