Re: abundance of irrationals!)


mueck...@xxxxxxxxxxxxxxxxx wrote:
> Randy Poe wrote:
> > W. Mueckenheim wrote:
>
> > > There is no other reliable way to prove
> > > anything about properties and existence of n e N than by
induction.
> >
> > That may or may not be true, but at any rate induction
> > doesn't run 1, 2, 3, ... to N. It runs 1, 2, 3, ... n
> > for any finite n you are interested in.
>
> But how do you find Card(N) = aleph_0 > n e N from the Peano-axioms?

I could be wrong, but I think aleph_0 is defined
to be card(N).

What you can easily prove from the Peano axioms is
that card(N) is larger than any finite value. That
follows directly from the existence of a successor
to any finite value.

> > > All differences between two finite natural numbers exist by
> > > definition, because the sum of two n does exist and is a natural
> > > number.
> >
> > Why does that imply there is a largest one? Why does
> > "exist" imply "there's a maximum"?
>
> No.

Good. If later you claim there is a maximum, I'll come
back to this post.

> But EVERY difference that can be defined by two natural numbers
> does exist. Therefore, there are only finite differences (=
distances)
> in N.

Yes. All differences between values in N are finite, because
all values in N are finite.

But I'm a little suspicious of the wording of that last
sentence. Are you trying to do implication-by-omission?
By leaving out the word "sized" in that sentence:

"There are only finite[-sized] differences in N."

are you going to later pretend that you can conclude

"There are only finite[-ly many] differences in N"?

After all, they can both be shortened to

"There are only finite differences in N."

- Randy

.

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