0
$\begingroup$

On a re-reading of D.J.Vellemann's book - "How to prove it" (2nd edition, pg. 69), it reads

It should be clear that if $A =\varnothing$, then $\exists x \in A P(x)$ will be false no matter what the statement $P(x)$ is. There can be nothing in $A$ that, when plugged in for $x$, makes $P(x)$ come out true as well.

This second line actually means that suppose $P(x)$ is true, then we can't find anything that proves it to be true. In other words, one cannot prove that $P(x)$ is true rather than proving $P(x)$ is false. Am I right in this reasoning?

$\endgroup$
5
$\begingroup$

It is not a question about whether $$P(x)$$ is true or false, it is whether $$\exists\ x\in\varnothing\quad P(x)$$ is true or false, and this is a different statement. This statement says that there is an element $x$ in $\varnothing$ such that...

...well, I don't need to go any further - there is no element at all in $\varnothing$, so whatever I was about to say, it will be a false statement because it asserts, among other things, that there is an element in $\varnothing$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks, it should have been obvious to me. $\endgroup$ – Ricky Jan 21 '16 at 6:26
2
$\begingroup$

$\exists x \in S : P(x)$ is a shorthand for $\exists x : x \in S \land P(x)$. So (as David explains well) when $S = \emptyset$ it is always false.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.