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

 
Last edited by a moderator:
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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?
 
Last edited by a moderator:
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".)
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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