∀x(I(x) → ∃y(I(y) ∧ (x < y))), I(x): x is an integer

Question

∀x(I(x) → ∃y(I(y) ∧ (x < y))), I(x): x is an integer

Is the following a correct translation? For all x, if x is an integer, then there exists an y such that y is an integer and x < y.

Is this a true or false statement? For numbers in R

Answer 1

Your translation is correct. Let $F_2(x,y)=x<y$, $$\forall x(I_1(x)\to\exists y(I_1(y)\wedge F_2(x,y)))=\forall x(\overline{I_1(x)}\vee\exists y(I_1(y)\wedge F_2(x,y)))=\forall x(\exists y(\overline{I_1(x)}\vee I_1(y))\wedge \exists y(\overline{I_1(x)} \vee F_2(x,y)))=\forall x\exists y(\overline{I_1(x)}\vee I_1(y))\wedge \forall x\exists y(\overline{I_1(x)} \vee F_2(x,y))=[\textrm{paste } I_1(x) \textrm{ and } F_2(x,y);x,y\in R]=(T\vee T) \wedge(T\vee T)=T$$

Answer 2

Your translation is a direct symbol-by-symbol translation ... but no one actually speaks like that in English ... which is also why you have trouble understanding the sentence.

Here is a more fluent translation: "For every integer there is a greater integer"

Do you now see whether that is a true or false statement?