Order of quantifiers in this implication

Question

Is the following implication true? $$\exists x \forall y ( x < y) \implies \forall y \exists x (x<y)$$

I'm not sure how these two formulas differ?

Answer 1

$\exists x \forall y ( x < y)$ says that there is an object 'smaller than' all objects.

$\forall y \exists x ( x < y)$ says that for all objects you can find a 'smaller' object.

Assuming we're talking about the natural numbers, both claims would be false.

Assuming we're talking about the integers, the first claim would be false, but the second one would be true. Note that this shows that the claims are not saying the same thing .. and that the second claim does not imply the first.

Now, assuming we're talking about any kind of numbers, the first claim is never true, since no number is smaller than itself.

But the question is: if the first claim is true (so, we must be talking about some different domain, or some different interpretation of '$<$'), would that make the second claim true? So, if there is some object 'smaller than' all objects (including itself), will it then be true that for all objects there is some objects that is 'smaller' than it?