Why is 2nd First Order Logic Statement of "Infinitely Many Primes" Wrong? | Help

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Homework Statement



The following are two first order logic statements of the statement "There are infinitely many prime numbers"

1. [PLAIN]http://uploadpie.com/3PZlO
2. [PLAIN]http://uploadpie.com/PN5i8

Can anyone explain why the second one is wrong? Thanks for help!

Homework Equations


The Attempt at a Solution

 
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They are equivalent, so both are wrong.

They are equivalent because the scope of a quantifier, when not otherwise specified, is taken to be the entire statement to its right; so the outer pair of parentheses in the second statement is superfluous.

They are wrong because they assert, in part, that every integer p is prime.
 
[PLAIN]http://uploadpie.com/mHyHp

Is this correct?
 
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Yes, this is a correct statement that given any prime there is a greater prime. (Which implies there are infinitely many primes, but to state directly that there are infinitely many primes you need to formalize enough set theory to have a concept of "finite set".)