4

When we are studying independence proofs we are dealing with statements of the form $Cons(T)\rightarrow Cons(T')$ where $T$ and $T'$ are first order theories; commonly $T$ and $T'$ are subtheories of $ZFC$ or extensions of it like $ZFC+CH$. So when we say, for example, that $CH$ is independent of $ZFC$ we are saying that there exists a model $M$ for the theory $ZFC+CH$ and another model $M'$ for the theory $ZFC+¬CH$. That models cannot be objects of ZFC since in that case $ZFC\vdash Cons(ZFC)$ so, where they live?

I feel that there is a formal notion that I'm ommiting or misunderstanding so, could anyone give some explanation about it?

Thanks in advance.

Asaf Karagila
  • 393,674
Ergonvi
  • 589
  • 1
    When you talk about relative consistency you can do it in a "material theory" like set theory, where you can replace the theory by a model of the theory. Or you can do it in a "syntactic theory" like all sort of arithmetics, where you actually talk about the existence of codes for proofs in the case of independence, or how to prove that from a code for "T is consistent" you can produce a code for "T' is consistent". – Asaf Karagila Jan 07 '16 at 14:15
  • Also, http://math.stackexchange.com/questions/28202/logic-set-theory-independence-proofs-etc and http://math.stackexchange.com/questions/531516/meta-theory-when-studying-set-theory and probably many many other questions related to this. – Asaf Karagila Jan 07 '16 at 14:17
  • @AsafKaragila Thank you! But what do we understand by metatheory? I was reading about it in Kunen's book and he said that the metatheory is "what is really true" but I don't understand with precision Kunen's definition of metatheory. – Ergonvi Jan 07 '16 at 14:42
  • Well, a meta-theory is "the universe of mathematics in which you formalize your arguments". Since you are eventually going to appeal to induction, you can't really claim that you're pushing strings on paper if you want to argue that something is true for every possible string, even those whose length exceeds the universe. So if you want to use induction for something, you end up needing some mathematical universe to work in. That is your meta-theory, what you assume to be the framework of your work. It could be ZFC or it could be something else. – Asaf Karagila Jan 07 '16 at 14:50
  • @AsafKaragila But ZFC is not our formal theory? – Ergonvi Jan 07 '16 at 15:12
  • It can. In that case you're arguing inside ZFC, and you might be arguing about what ZFC can or cannot prove. – Asaf Karagila Jan 07 '16 at 15:13
  • So ZFC as a metatheory is the theory that studies what is really true about statements of ZFC? – Ergonvi Jan 07 '16 at 15:17
  • In case of using ZFC as a meta-theory, you are really investigating the following scenario "We live in a universe, and all we know is that it satisfies ZFC. What else can we say?" so by studying relative consistency you can say that if you had assume such and such theory is consistent, then you'd get that such and such theory is also consistent, by showing that CH is independent you show that you cannot determine the truth value of CH in the "real universe" that you're working in, unless you assume more axioms. – Asaf Karagila Jan 07 '16 at 15:19
  • So for example, are "$Cons(ZFC)\rightarrow Cons(ZFC+CH)$" and "$Cons(ZFC)\rightarrow Cons(ZFC+¬CH)$" metatheorems and then "$CH$ is independent of $ZFC$" a metacorollary in the metatheory ZFC ? – Ergonvi Jan 07 '16 at 15:38
  • 1
    Yes. But the thing here is that the same proofs can be done in more or less every "reasonable" meta-theory. Be it ZFC or PRA (primitive recursive arithmetic). Which is why we don't usually pay much attention to these things. And just let me say that the intricacies of theory and meta-theory are probably the most difficult ones in modern set theory. – Asaf Karagila Jan 07 '16 at 15:45

1 Answers1

2

It is not really meaningful to say that such and such "models cannot be objects of ZFC", because there is no such thing as "objects of ZFC". What there is are objects of particular interpretations of (the language of) ZFC, but different interpretations of the same theory may or may not contain objects with particular properties.

(Some of these interpretations are models, when their universe is a set in whichever set theory we use for our metareasoning. If we believe in a Platonic universe of "actually existing" sets, that may also work as an interpretation of ZFC, but it usually won't be a "model", depending on the exact definitions we employ).

This distinction is important here, because you appear to be confused about the difference between some object existing in some particular interpretation, and the theory proving they exist.

If ZFC is consistent, then there will be some interpretations of it that contain sets that are models of ZFC itself, and others that don't. In particular every model of ZFC whose integers are standard will contain a set that is a model of ZFC.

On the other hand, the Gödel-Rosser incompleteness theorem says that if ZFC is consistent, then it cannot prove its own consistency, and therefore it cannot be the case that every model of ZFC contains an smaller model of ZFC. Those that don't, however, cannot have the same integers as we use outside the model (because the non-existence of an internal model of ZFC implies that there is a number that the outer model believes is a proof of a contradiction from ZFC, and such a number cannot be standard).

hmakholm left over Monica
  • 286,031