Re: Definition of finite.

On May 8, 10:03 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On May 8, 6:12 pm, zuhair <zaljo...@xxxxxxxxx> wrote:





On May 8, 7:25 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
Theorem:
x is Dedekind infinite
<->
Eyz(y subset of x & ~ z=0 & z/\y=0 & ~ y strictly dominated by zuy)

Proof:
Suppose x is Dedekind infinite.
So let y proper subset of x and y equinumerous with x.
So x\y subset of x and ~ x\y = 0 and (x\y)/\y = 0.
But y equinumerous with (x\y)uy. So ~ y strictly dominated by (x\y)uy.

So far so good. this proves that: x is Dedekind infinite -> x is z-
infinite.
that's correct.

Let y subset of x and ~ z=0 and z/\y = 0 and ~ y strictly dominated by
zuy.
So either ~ y dominated by zuy or y equinumerous with zuy.
But y dominated by zuy. So y equinumerous with zuy.
Since ~ z=0 and z/\y = 0, we have y proper subset of zuy.
So zuy is Dedekind infinite.

Yes.

So, since y equinumerous with zuy, we have y is Dedekind infinite.

Strange???

What is strange???

that's what we want to prove isn't it?

It's not what we want to prove finally but you can see by the end of
the proof that it's a step toward what we want to prove, can't you?

Why y is Dedekind infinite??

What don't you understand about a set being Dedekind infinite if it is
equinumerous with a Dedekind infinite set??

I don't understand?

You don't?

Your argument is vague, you should show that there exist a proper
subset of y that is equinumerous with y in order to prove that y is
Dedekind infinite, did you show that???

What don't you understand about a set being Dedekind infinite if it is
equinumerous with a Dedekind infinite set???

Perhpas, I don't know.

Obviously, you don't.

So, since x subset of y, we have x is Dedekind infinite.

So why don't you just read the proof and ask whatever legitimate
questions you have about it instead of asking unnecessary questions as
a childishly petulant way of acting out your umbrage regarding an
exchange we had in another thread??? [Should that have been three
question marks or just two???]

MoeBlee- Hide quoted text -

- Show quoted text -


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