I am not familiar with model theory. As a matter of fact, I only had my first Logic and Set theory courses last semester. But still, there is a question that bothers me, and It could be nice if someone here could give me an explanation which does not requier more then the background that I have.
If I understand correctly, a model (or a structure?) is a set $(D,v_i,f_i,P_i)$, where $D$ is the set of items we make our claims on, $v$ are variables $f$ are functions and $P$ are relations that get $T$ of $F$ value. Also, If I got that right, the proof of Gödel showed that in certain structures Cantor's CH is true and in others not.
The thing that is not clear to me: isn't Continuum Hypothesis coming with a given structure? I mean, don't we know that: $D= \mathbb R$, $v_i$ can get real numbers, $f$ are all the mathematical functions that $f_i \rightarrow OR$ where $OR$ is the space of ordinals (or cardinals if it matters). So, what does that mean to talk about different models when refering to the CH.
Can anyone give me an example of a model where the CH is true, and a model where CH is not true where the two models are relevant to the question?
Thanks Lior
First of all, let us clear some things.
Model of a theory $T$ in the language $\cal L$ is first of all an interpretation for the language, i.e. a pair $(M,\Sigma)$ where $M$ is a non-empty set and $\Sigma$ is an interpretation function. That is for every function symbol $f$ in the language, $\Sigma(f)$ is a function on $M$, and so on. Next we can define the satisfaction relation which tells us what sentences from $\cal L$ are true in this interpretation, and $M\models T$ if all the sentences in $T$ are true in $M$.
The completeness theorem tells us that $T$ can prove $\varphi$ if and only if whenever $M$ is a model of $T$, $M\models\varphi$. So provability, which is a syntactical relation between a theory and a sentence, is the same as semantic (or logical) implication. It is important to notice that this is true for first-order logic, but not necessarily so for other logics (such as second-order logic).
If there is a model of $T\cup\{\varphi\}$ we say that $\varphi$ is consistent with $T$, and if both $\varphi$ and its negation are consistent with $T$ then neither is provable from $T$ (and if neither is provable, then they are both consistent with $T$), in which case we say that $\varphi$ is independent of $T$.
Finally. Gödel proves that $\sf CH$ is consistent with $\sf ZFC$. He did so by showing that if we have a model of $\sf ZFC$ then we can construct a model where $\sf ZFC+CH$ is true. Paul Cohen, some twenty odd years later, proved that given a model of $\sf ZFC+CH$ we can construct a model of $\sf ZFC+\lnot CH$. Therefore both $\sf CH$ and its negation are consistent with the axiom of choice.
Now, Gödel also proved the incompleteness theorem which essentially says that a theory with such and such properties must have at least one statement which is independent from that theory. These statements are often "convoluted", they are constructed in a way that explicitly makes them independent. But once you are aware, and comfortable, with this phenomenon it's time to set sail and find statements which are organic (i.e. come up from the mathematics directly, natural questions to ask in the right context) and are independent.
The continuum hypothesis is such a statement, if we consider the first-order theory $\sf ZFC$. Note that the statement of the continuum hypothesis has nothing to do with the real numbers, just with their cardinality. Or rather, the cardinality of the power set of the integers.
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