Questions regarding peculiar statement involving empty sets

Question

I have some questions about the following statement involving empty sets:

"1) $ \forall a\in\{x|x\in \Re, x^2+1=0\} $ , we have $ a^{17}-72a^{12}+39=0 $

"1) says that all elements of the empty set have a certain property; this is true because there are no elements in the empty set. Any similar "$\forall$" statement involving the empty set is true"

My questions are as follows:

1. What does statement 1) mean?

2.Is statement 1) true?

3.Why is the sentence that I've put in bold true?

Answer 1

The set $ \{x|x\in \Re, x^2+1=0\} $ is, in fact, the empty set. Thus, 1) reduces to:

$$ \forall a\in\emptyset...\text{something}. $$

To see why a statement of this form is always true, we look at its negation:

$$ \exists a\in\emptyset...\text{(NOT) something}. $$

But, there is no $a\in\emptyset$ in the first place, so a statement of this form is clearly false. Thus, its negation:

$$ \forall a\in\emptyset...\text{something}. $$

is true.

Answer 2

A similar statement is:

$\forall a \in S=\{x|x=\lim (1-1+1-1\dots) \}, \quad a=1-a$

This is true, but we cannot conclude that $\lim (1-1+1-1\dots)=\frac{1}{2}$. The correct conclusion is $S=\emptyset$.

Answer 3

1) The meaning of the statements is comparable with the meaning of a statement like: "all fishes that are human like hamburgers". So there is no meaning at all, I would say.

2) Can we find any element $a$ of $\{x\in\mathbb R\mid x^2+1=0\}$ for wich $a^{17}-72a^{12}+39=0$ is not true??… No! Even stronger: we cannot find any element in $\{x\in\mathbb R\mid x^2+1=0\}$ at all, because this set has no elements. This justifies the conclusion that the statement it is true for every element $a$ of $\{x\in\mathbb R\mid x^2+1=0\}$.

3) It works whenever the set $A$ in formula $\forall a\in A[\cdots]$ can be recognized as the empty set.