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Goldbach's conjecture in English reads:

“Every even integer greater than 2 is the sum of two primes.”

Which can be written in terms of quantifiers as:

$$\forall n \in \text{Even}. \exists p \in \text{Primes} \exists q \in \text{Primes}. n= p +q. $$

Where Even = {set of all even numbers greater than or equal 2} and Primes = {the set of all primes numbers}.

I was told that changing the order of the quantifiers a little bit changes the meaning of the statement, but I don't understand why it does that:

$$ \exists p \in \text{Primes} \exists q \in \text{Primes}. \forall n \in \text{Even}. n=p +q. $$

How is the second one different? I was told that it means in English that:

Every even number $\geq 2$ is the sum of the same two primes.

However, I cannot convince myself that, that is correct. Also, how come that the second interpretation is false?

雨が好きな人
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Charlie Parker
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2 Answers2

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It is standard practice in mathematics to instantiate quantifiers from left to right. Hence the standard way to interpret the second statement is: there is a prime $p$, and a prime $q$, such that for all even $n$, the equation $n=p+q$ holds.

That is, $p,q$ are chosen before $n$ is instantiated, so the statement must hold without changing either $p$ or $q$, for every $n$. This is absurd, and false.

In English we often rely on context to disambiguate sentences, so we are used to interpreting them "however makes sense"; however in mathematics we want extra precision, including the ability to make statements that are absurd if we so desire.

vadim123
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  • Sorry if this is obvious to you, but what do you mean that, "it must hold for the same p and q"? The same p and q, means $p = q$? Your explanation that if we fix p and q, then p+q is some number say k, and that number cannot possibly be every even number once p and q are fixed. That does make sense. However, I just don't understand what the word "same" means here though. – Charlie Parker Oct 20 '14 at 16:46
  • Sorry, will clarify. Your understanding is correct. – vadim123 Oct 20 '14 at 16:53
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You can try with a simpler example :

$\forall n \in \mathbb N \exists m \in \mathbb N (n < m)$.

It is clearly true, because for every natural number $n$ we can always find a greater number $m$ : it is enough to choose as $m$ the number $n+1$.

Now, swapping the quantifiers we get :

$\exists m \in \mathbb N \forall n \in \mathbb N (n < m)$

that must be read as : there exists a natural number $m$ that is greater then every number $n$, which is clearly false.

Mauro ALLEGRANZA
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