/sci/ - Science & Math

>>14888514
The statement [math] \forall y \exists x : A(x,y) [/math] means that for every [math] y [/math] we can find a [math] x [/math] such that [math] A(x,y) [/math] is true. The important point here is that for EVERY [math] y [/math] the [math] x [/math] that exists might be different, let me make this example more concrete by letting [math] A(x,y) \Longleftrightarrow y < x [/math] so the statement becomes [eqn] \forall y \exists x : y < x [/eqn]This statement is basically saying for every number there exists a number which is bigger than it.
So let's try to follow the logic you presented with this particular [math] A(x,y) [/math] [eqn] 1. \quad \exists x : y < x \quad \text{Universal instantiation of y} \\ 2. \quad y < x(y) \quad \text{Existential instantiation of x to a constant DEPENDENT on y}[/eqn]Your second step is actually invalid, if [math] x [/math] was not dependent on [math] y [/math] then you're basically saying there's a constant that is bigger than any arbitrary number which is clearly false (if we are working with a number system that does not include infinite numbers)

Bill Shillito has an excellent lecture on predict logic which you're studying, i highly recommend it:
https://www.youtube.com/watch?v=YbNmPievBak