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

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Homework Help Overview

The discussion revolves around two first order logic statements that aim to express the concept of "there are infinitely many prime numbers." Participants are examining the correctness of these statements and the implications of their formulations.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the equivalence of the two statements and the implications of their logical structure. There is a focus on the scope of quantifiers and the assertion that every integer is prime.

Discussion Status

Some participants have provided guidance on the logical structure of the statements and the necessary conditions for asserting the infinitude of primes. Multiple interpretations of the statements are being explored, particularly regarding their correctness and implications.

Contextual Notes

There is a mention of the need for formalizing set theory to adequately express the concept of a "finite set," which may influence the understanding of the statements in question.

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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".)
 

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