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In meta-logic, it's routine to consider sets of sentences. However, are there rules of syntax that permit a sentence to be literally an element of a set?

Is there always an explicit or implicit encoding of a sentence, so that we replace each sentence with an object that can be an element of a set, and strictly speaking study sets of those non-sentence objects, rather than study sets of sentences?

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  • $\begingroup$ What's the problem? A sentence is an element of a set of sentences. A book is a set of sentences. $\endgroup$ Jan 8, 2021 at 12:48
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    $\begingroup$ See e.g. Kenneth Kunen, The Foundations of Mathematics (College, 2009), Ch.2, page 86-on: "A lexicon is a pair $( \mathcal W, \alpha)$ where $\mathcal W$ is a set of symbols and $\alpha : \mathcal W \to \omega$." $\endgroup$ Jan 8, 2021 at 13:45

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The most common set theory, ZFC, has no prohibition on anything being a member of a set. It does not require them, but you can have a model with things that are not sets being members of sets. Those things might as well be sentences. This will not allow you to consider the structure of the sentences themselves. Each one is an atom. You could add some axioms to define the structure of sentences and how to relate them to each other.

There is also the technique used in the original Gödel proof of independent propositions in arithmetic. It maps each string of characters or set of strings to a natural number. He gave a function that would take a natural number and report whether it was a syntactically correct sentence. Any set of properly chosen natural numbers can then be interpreted as a set of sentences.

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    $\begingroup$ There are variants of ZFC which allow atoms, but ZFC itself is incompatible with atoms: The axiom of extensionality implies that any object without elements is equal to the empty set. $\endgroup$ Jan 8, 2021 at 15:47
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    $\begingroup$ Instead, the usual way to handle things like sentences formally in ZFC is to encode them as certain sets. This is no different from the way that numbers, or any other mathematical objects, are handled formally in ZFC. $\endgroup$ Jan 8, 2021 at 15:49

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