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In Cohen, Set Theory and the Continuum Hypothesis, page 44 the ability to form Partial Truth Formulae is described :

"We leave as an exercise for the reader the proof of the following fact: For each r, there is a formula A(n) in Z$_1$ such that if we enumerate all statements Tn in Z$_1$ (ie arithmetic), which have fewer than r quantifiers, in a natural way, then Tn <=> A(n) is true for all n"

I am surprised by this, as I have only read that there is no definable formulae that can express the Truth of all expressions. I just assumed (incorrectly) that this also meant that no partial truth formulae would be possible for countable subsets of expressions in a language.

With the above it looks like for a given structure M, its possible to determine the Truth of increasingly large expressions, represented by number of quantifiers r , so that if A(r,n) is a formula defining the truth of all logical expressions with <= r quantifiers, then X:= {A(1,n),A(2,n),....} would be sufficient to define truth in the structure M for any expression E in the language, by determining the number of quantifiers 'e' in E and finding the appropriate A(e,n). I am starting to suspect that this technique is used in Set Theory Forcing to create the infinite set of expressions that describe a new element not in M (as the language can't describe by a formula the element in M as a contradiction would exist).

So my question is : are there any references that describe (and prove) how this is partial truth definition is possible and consider further this ability in more mathematical detail, as it seems quite a general finding (and also looks related to classes in Set Theory) ?

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We can't whip up truth predicates for arbitrary "small" sets of sentences, but when we bound the complexity - in terms of the Levy hierarchy in set theory - we can. The reason we can't "glue" these definitions together to get a full truth predicate is that the definitions themselves get increasingly complex; since there's no bound on their complexity, we can't find a single formula which does the job.

This phenomenon also happens - and may be easier to understand - in the context of arithmetic; here we use the arithmetical hierarchy instead of the Levy hierarchy, but the abstract idea is the same.

In the context of arithmetic - which again will be easier to understand, and uses the same ideas - the book Metamathematics of first-order arithmetic by Hajek and Pudlak has a good explanation. In the context of set theory, I believe Kunen's book is a good source (but I don't have it on hand to check); Jech's book also probably covers it.

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  • $\begingroup$ Many thanks for the enlightenment! I'm chasing up your references. $\endgroup$ Jun 28 '19 at 8:37

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