Logic Quantifiers

Having a look at some higher level calculus that i never came to through physics.
I am stuck conceptually at a first hurdle. If anyone can help me frame this in my mind it would be much appreciated.
I'm looking at the ordering of the universal and existential quantifiers.

I have this statement

\left ( \forall x \right )\left ( \exists y \right )\left ( x< y \right )

.
So for all x there exists a y that is greater than x. This is true.

However i don't get the statement

\left ( \exists y \right )\left ( \forall x \right )\left ( x< y \right )

.
There exists a y for all x that is greater than x. This is apparently false but i just can't seem to make it make sense for me.

Any help? Cheers

(edited 11 years ago)

Reply 1

ghostwalker

Original post by stanante

I have this statement

\left ( \forall x \right )\left ( \exists y \right )\left ( x< y \right )

.
So for all x there exists a y that is greater than x. This is true.

"for all" can be read as "for each", which sometimes makes the meaning clearer.

However i don't get the statement

\left ( \forall x \right )\left ( \exists y \right )\left ( x< y \right )

.
There exists a y for all x that is greater than x. This is apparently false but i just can't seem to make it make sense for me.

Any help? Cheers

That's the same logical statement.

I presume you meant

\left ( \exists y \right )\left ( \forall x \right )\left ( x< y \right )

There exists a y such that for all x, x < y. Which means there is some number bigger than every other number. Which is clearly false - y is not bigger than y+1 for instance.

Reply 2

stanante

Original post by ghostwalker

That's the same logical statement.

I presume you meant

\left ( \exists y \right )\left ( \forall x \right )\left ( x< y \right )

Yes i did. Sorry about that. I'll make an edit.

It's still not clicking. So there's a y that for every x can be bigger than x (x<y). What if x and y tend to infinity? Can there not be a y that is always larger?

Reply 3

ghostwalker

Original post by stanante

y doesn't tend anywhere. The proposition is stating that a y exists. So, what is it? It's not 1, 10^10, or any number. We can't put infinity there, as it's not a number.

What ever number we choose, we can always find a larger number by adding one to it.

Paraphrasing, it's saying there is a number bigger than all numbers.

Another counter example is whatever number we choose, it's not bigger than itself.

(edited 11 years ago)

Reply 4

stanante

Original post by ghostwalker

y doesn't tend anywhere. The proposition is stating that a y exists. So, what is it? It's not 1, 10^10, or any number. We can't put infinity there, as it's not a number.

What ever number we choose, we can always find a larger number by adding one to it.

Paraphrasing, it's saying there is a number bigger than all numbers.

Another way of looking at it is whatever number we choose, it's not bigger than itself.

So basically, for all x, means 'all' numbers. So there can't be a number, y, larger than all numbers.

Reply 5

Noble.

Original post by stanante

Sometimes, it helps to think of

\exists, \forall

like this:

Using your statement

(\exists y) (\forall x) (x < y)

You can think of it like this, someone is giving you an

x

, it can be any number, and no matter what number they give you there is a

y

that is bigger than it, and this

y

works for any

x

they give you.

This is clearly not true, because no matter what

y

you set, if they gave you the number

y+1

it doesn't work.

Reply 6

stanante

Original post by Noble.

Someone is giving you an

x

, it can be any number, and no matter what number they give you there is a

y

that is bigger than it, and this

y

works for any

x

they give you.

This is clearly not true, because no matter what

y

you set, if they gave you the number

y+1

it doesn't work.

How can we consider

y+1

? Isn't the statement that, a 'certain'

y

is bigger than all

x

. If we consider

y+1

then is it not a different number being considered?
And if

Unparseable latex formula:

(x < y)

for a

y

then

y+1

is still bigger than an

x

.

I understand what ghostwalker is saying that there can't be a

y

larger than all numbers.

Although i get it once it's pointed out, it wasn't obvious to me.

I'm throwing it all out there. Sorry if it sounds ridiculous. Got to follow the routes coming up in my mind.