How can the objects in all object theories be defined?
Do meta-theories refer to things which aren't objects (so defined)?
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The uses of 'theory' in the theory of object theories, are in the sense of the 'Theory of a Complex Variable'. They refer to a characterization of a set of theorems that make up a compelling and cohesive whole, not to any hypothetical model. Good examples of object theories are the kinds of structures described in finite algebra.
Take Group Theory. Each concrete group, for instance, clock arithmetic, is described in a declarative manner, as the points on the clock, or as a formal ratio of the integers modulo a given integer, or as the possible positions of a single cycle. Each of these is a different object theory.
Group Theory itself is the metatheory which identifies why all of these separate concrete representations for the same cyclic group, are in effect equivalent, and how similar such structures fit together.
The real (or informal) theory behind group theory is that there are intuitively useful aspects to the notion of a single arithmetic operator in isolation, across models, and many things act 'enough like arithmetic' to be thought of as alternative perspectives on how number theory itself applies to different situations.
According to Tarski I think it can be said that no truth value can be assigned to the propositions of an object language without a metalanguage.
The "propositions" of a language with no metalanguage then have no truth value, and so this object language can be defined without propositions - what doesn't have a truth value (or relation to truth) is not either true or false.
Definition: A proposition or statement is a sentence which is either true or false
Wittgenstein thought that mathematics expresses no propositions (and later, no thought).