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Is there any difference between

$(\forall e) ((\exists N=N(e)) (P(N,e)))$.

and

$(\forall e) ((\exists N)(P(N,e)))$

Should we read this two statements differently? What does this $N=N(e)$ stand for? To me it is clear that $N$ depends somehow on $e$ from both statements.

Evgenii.Balai
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    The second one is entirely rigorous. The first one tries to stress the fact that N may depend on e, and while doing this sacrifices rigor, which is useless since the fact is indicated nonambiguously by the order of the quantificators in the second formulation. Better to use the second form and to add afterwards something like "Let N(e) denote some N such that P(N,e) holds". (+1 to the question.) – Did Aug 25 '13 at 11:55
  • Actually,if one does introduce the notion of asuch a function, one should rather say: There exists a function $N\colon e\mapsto N(e)$ such that $(\forall e)(P(N(e),e))$. – Hagen von Eitzen Aug 25 '13 at 13:01

2 Answers2

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Trying to convey in more or less conventional words your formulas, we can say the first one says "For all $\,e\,$ there exists a function $\,N\,$ of $\,e\,$ s.t. $\,P(N,e)\,$ , whereas the second one omits the words "a function" above.

We're assuming, of course, that everybody knows what kind of beings $\,N,e ,P\,$ etc. are...

DonAntonio
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The second formulation is entirely rigorous. The first formulation aims at stressing the fact that N may depend on e, and while doing this sacrifices rigor. This is useless since the fact that N may depend on e is indicated nonambiguously by the order of the quantificators.

One can advise to stick to the second form and to add afterwards a formulation similar to: Let N(e) denote some N such that P(N,e) holds.

Did
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