# Difference between ∃\exists-rudimentary and generalised ∃\exists-rudimentary formulas

On Boolos’ Computability and Logic, chapter 16:

“A rudimentary formula [in the language of arithmetics] is a formula constructed from atomic formulas using only negation, disjunction, conjunction and bounded quantification. A ∃\exists-rudimentary formula is a a formula of the form ∃xϕ\exists x \phi, where ϕ\phi is rudimentary.”

Then,

“A generalised ∃\exists-rudimentary formula is any formula that can be obtained from a rudimentary formula by conjunction, disjunction, bounded quantification or unbounded existential quantification”

[Not exact words as my book is not in English]

By this definition, it seems that ∃\exists-rudimentary formulas are the same as generalised ∃\exists-rudimentary formulas. So I am believing that an extra “∃\exists” is missing on the second definition – i.e. “a generalised ∃\exists-rudimentary formula is any formula that can be obtained from a ∃\exists-rudimentary formula…”, making it so that the ∃\exists-rudimentary formulas are those with exactly one unbounded existential quantifier, and the generalised ones are those with any number of those (but no unbounded universal quantifiers of course), as that makes more sense, explains the omission of “negation” in the second definition and is consonant with Enderton’s definition of ∃n\exists_{n}-formulas.

Is this correct or am I misreading something?

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Correct; see George Boolos & John Burgess & Richard Jeffrey, Computability and Logic (5th ed – 2007), page 204 : rudimentary, ∃\exists-rudimentary and ∀\forall-rudimentary.
– Mauro ALLEGRANZA
2 days ago

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