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

• zohapmkoftid
In summary, the two first order logic statements of the statement "There are infinitely many prime numbers" are equivalent, but both are wrong. This is because they incorrectly assert that every integer is prime. A correct statement would be that given any prime, there is a greater prime, implying that there are infinitely many primes.
zohapmkoftid

Homework Statement

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

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

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.

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".)

1. What is the 2nd First Order Logic Statement of "Infinitely Many Primes"?

The 2nd First Order Logic Statement of "Infinitely Many Primes" is a mathematical statement that claims there are infinitely many prime numbers. In other words, there is no largest prime number and the list of prime numbers goes on forever.

2. Why is the 2nd First Order Logic Statement of "Infinitely Many Primes" considered wrong?

The 2nd First Order Logic Statement of "Infinitely Many Primes" is considered wrong because it was proven to be a fallacy by the mathematician Euclid. He showed that there are infinitely many prime numbers by contradiction, meaning that assuming there is a largest prime number leads to a contradiction.

3. What is the difference between the 1st and 2nd First Order Logic Statement of "Infinitely Many Primes"?

The 1st First Order Logic Statement of "Infinitely Many Primes" is a correct statement that states there are infinitely many prime numbers. The 2nd First Order Logic Statement, on the other hand, is a false statement that claims there is a largest prime number. They differ in their truth value and validity.

4. How does the 2nd First Order Logic Statement of "Infinitely Many Primes" impact the study of prime numbers?

The 2nd First Order Logic Statement of "Infinitely Many Primes" has no impact on the study of prime numbers because it has been proven to be false. Mathematicians continue to use the 1st First Order Logic Statement, which has been proven to be true, in their research and calculations related to prime numbers.

5. How can I understand the proof that the 2nd First Order Logic Statement of "Infinitely Many Primes" is wrong?

The proof that the 2nd First Order Logic Statement of "Infinitely Many Primes" is wrong can be understood by studying Euclid's proof of the infinitude of prime numbers by contradiction. This proof uses basic principles of number theory and logic, and can be easily understood with some background knowledge in these areas. There are also many resources available online and in books that explain the proof step by step for a deeper understanding.

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