Integration described by first-order logic?

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Describing a Riemann integral using only first-order logic (FOL) poses significant challenges, as it typically requires quantifiers that extend beyond FOL's capabilities. The discussion highlights that any attempt to define integration necessitates qualifiers like "for all subsets," which FOL cannot express. The original poster struggles to formulate a definition without using existential quantifiers or infinite disjunctions, which are not permissible in standard FOL. There is an openness to being proven wrong about the limitations of FOL in this context. Overall, the consensus leans towards the belief that first-order logic is insufficient for a complete description of Riemann integration.
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Is it possible, to describe a Riemann integral with just first-order logic? And if so could someone point me to somewhere that has such a description of it.
 
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As far as I know any description of integration requires some qualifier of the form "##\forall A \subset X##" which is not expressible in first order logic. But I'd be happy to be proven wrong!
 
I don't know much formal logic beyond my intro model theory class, but...

In my attempt to define it, I couldn't do it without being able to say ``\exists n\in\mathbb N:\exists x_1,...\exists x_n:\enspace...\text{''} or ``\bigvee_{n=1}^\infty \exists x_1,...\exists x_n:\enspace...\text{''} which isn't kosher in any version of FOL I know.
 

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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