Thread: Theorems and isomorphisms View Single Post
Emeritus
PF Gold
P: 9,245
 Quote by yossell ...from my point of view, your way of describing things mixes up model theoretic notions and linguistic notions. You present two groups {0 1} and {-1 1}. But it also seems you're considering a language L that contains symbols 0', 1', `-1' for all these objects, and which also has predicates for different operations + and . Is this right?
Hm, you have a point there. The language should consist of the symbols that all first-order languages have in common, and in addition to that, a binary operation symbol, a unary operation symbol and a constant symbol. Let's choose those to be $\{\star,i,e\}$. Now each structure with signature $\{\star,i,e\}$ associates a binary operation with $\star$, a unary operation with i, and a member of the underlying set with e. What I should have said is that the complete definition of the first group I specified (incompletely), associates + with $\star$, $x\mapsto x$ with i and 0 with e. The second group associates · with $\star$, $x\mapsto x$ with i and 1 with e.

Edit: Another thing became clear to me when I wrote this sentence in the reply to JSuarez below: "if X and Y are isomorphic, any theorem that is satisfied by X is satisfied by Y too." When I gave an example of a theorem about a group G that corresponds to a theorem about another group H, I shouldn't have written $\forall x\ x+x=0$ and $\forall x\ x\cdot x=1$. The theorem is one sentence that's satisfied by both structures, and it's neither of the two above, it's $\forall x\ x\star x=e$. This discussion is clearly good for me, since I'm making exactly the sort of mistakes you'd expect from someone who has read a third of the book and not done any exercises.

 Quote by yossell I'm not sure what's worrying you in your response to Hurkyl.
Mainly the problem of how to define "finite" before "natural number". It seems like we even have to have an intuitive understanding of sets before we define the formal language of set theory, because a specification of a language involves a specification of an alphabet, which is a set of symbols.

 Quote by yossell In set-theory, and without appealing to semantic or linguistic notions or appealing to numbers, one defines the special set omega. In terms of this set, and without appealing to semantic or linguistic notions or appealing to numbers, one defines finite sequences.
Can you tell me how?

 Quote by JSuarez ...the isomorphism statement is a basic result in classical model theory:
Yes, I remember seeing a theorem that appeared to be saying something very similar in a book a few months ago. I haven't studied the proof yet, but I will. My concern here isn't so much "does such a theorem exist?". It's more like "is it a theorem in mathematics or metamathematics?". I was thinking that since it's a theorem about theorems, rather than a theorem about sets, maybe it's not a theorem in ZFC or any other set theory. But it seems that I should be thinking about variables, logical symbols, sentences etc, as sets. I can see how that makes sense for any first-order language, except the language of set theory.

 Quote by yossell In fact, you don't need isomorphism for two distinct structures to satisfy the same class of first-order expressions: there are non-isomorphic strucutures that have this property; they are called, in Model Theory, elementarily equivalent structures.
This is interesting, and from my point of view, also a bit disturbing. As I said in one of my posts above, I'm looking for a nice way to explain the point of isomorphisms. Why do we define them at all, and in what sense are isomorphic structures equivalent? This concept of "elementarily equivalent structures" seems to ruin my idea that this theorem is what tells us exactly in what sense isomorphic structures are "equivalent".