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Suppose that I take ZFC and extend it by my own predicates like Fish(x) and Flies(x), and augment it with some new axioms, e.g. $\forall x(\text{Fish}(x)\to\text{Flies}(x))$.

I am having trouble understanding various exact statements of conservative extensions. Although I have a vague idea of numerous explanations of what's going on, they all get too technical for me, so I thought that maybe its best to instead ask about an example, like the one above.

Some places make it sound like an extension is conservative if it expands the language and/or axioms/rules of inference of the old theory, but all theorems of the new system which are written only in the symbols of the old language are already theorems of the old system. Technically since adding the axiom $\forall x(\text{Fish}(x)\to\text{Flies}(x))$ doesn't allow you to prove any new statements in the language of ZFC, is it a conservative extension? It certainly doesn't feel that way, but is it?

Idk what to tag this question but I've been trying to learn model theory on my own just in order to answer it, and have nearly given up.

As an update, after lot of reading about posts like this, I want to say that the answer seems (in heavy italics) to be yes, but I really have no intuition as for why this should be the case. The only thing I can find explicitly to the contrary is the wiki article on extensions by definitions, which kind of implicitly says that new statements need or have some translation or another back into the old language. But I still don't really know what the right answer to this is, about whether or not the extension in the opening paragraph, for example, would be considered as conservative.

Pineapple Fish
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    Pick any model of $\mathsf{ZFC}$ and declare $\operatorname{Fish}(x)$ and $\operatorname{Flies}(x)$ false for all sets, and voila, you have turned it into a model of your extension without adding any new sets. – Hagen von Eitzen Jan 03 '22 at 21:18
  • @HagenvonEitzen woah. Mind blown - I wasn't even thinking about applying models like that. So its an extension, but is it conservative? – Pineapple Fish Jan 03 '22 at 21:21

1 Answers1

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It certainly doesn't feel that way, but is it?

This is absolutely a conservative extension. In fact, it's a conservative extension in a very strong sense: every model $M$ of $\mathsf{ZFC}$ has an expansion to a model of your theory.

I suspect that your concern comes from the sense that the behavior of the symbols you introduce is unconstrained by $\mathsf{ZFC}$ alone; model-theoretically, every model $M$ of $\mathsf{ZFC}$ has many distinct expansions to models of your theory. This doesn't contradict conservativity, however; it just means that your theory is not an expansion by definitions of $\mathsf{ZFC}$.

Noah Schweber
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  • Does this also mean that technically every theory is a conservative extension of the empty theory (ie the theory with no non-logical symbols)? – Pineapple Fish Jan 04 '22 at 01:34
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    @PineappleFish No. There are still (slightly) nontrivial things we can say in the empty language, such as "$\exists x,y(x\not=y)$." – Noah Schweber Jan 04 '22 at 01:38
  • Are $\forall x\forall y(x=y)$ and its negation the only such statements? In any case, I get what you're saying and this really clears things up. – Pineapple Fish Jan 04 '22 at 01:51
  • @PineappleFish No: for each finite set of natural numbers $F$, we can write a single sentence saying "The number of objects in the universe is an element of $F$." We can also write the negations of such sentences. This, however, is all we can say. – Noah Schweber Jan 04 '22 at 03:23
  • I guess this is really more like a corollary to my original question, but in every conservative extension who's newly introduced symbols are constrained by ZFC (we can write a non-logical axiom relating the new and the old symbols), does it follow that extension/expansion must be an expansion by definitions (-of new relations, constants, or functions) in order for it to remain conservative? – Pineapple Fish Jan 04 '22 at 16:37
  • -actually its more like an inverse; 1) you seem to suggest that if M models T but there is not unique M' which models the conservative extension T', then T' is not an extension by definition - is this true in general? and 2) inversely, if the extension does have a unique model then will it always be an extension by definitions? Perhaps that is a clearer way of stating what I said above. – Pineapple Fish Jan 04 '22 at 17:21
  • one more thing, and I'm not going to mind if you don't answer it, but in the light of this question, what do you think about this; https://cs.nyu.edu/pipermail/fom/1998-October/002306.html ? I don't really see in the context of proving things why to focus on conservative extensions rather than just definitions, given that conservative extensions in general can behave so weirdly. I don't understand his last paragraph on constructors. I don't know what they are and I can't even parse what it's really supposed to mean. – Pineapple Fish Jan 04 '22 at 20:00
  • (continued) I also don't see why people say that something is "provable in ZFC" rather than "provable in a conservative extension of ZFC" or something analogous. From the point of view of proving things, even working with an extension by definitions changes the feel of the system. So if you were being pedantic, is it technically correct to say that the most common foundation of math is 'ZFC' or 'various extensions by definitions of ZFC'? – Pineapple Fish Jan 04 '22 at 20:07