On Friday, July 21, 2017 at 10:13:35 PM UTC-4,

*Post by Jim Burns*On Friday, July 21, 2017 at 9:31:47 PM UTC-4,

On Friday, July 21, 2017 at 6:04:12 PM UTC-7,

*Post by Dan Christensen*Given the real numbers, what axioms of ZFC would you

use to extend them to include an infinite number?

Axiom of Union

The union of what set(s)?

Using one-sided Dedekind cuts,

the union of the set of real numbers is just

the set of rational numbers.

Yes, the set of rational numbers is infinite,

That's a different kind of infinity, cardinality.

but it would be a stretch to call it the extension

of the reals to "infinity."

Recall that you gave the real numbers.

For each real number x (of the unextended reals),

there is exactly one set of real numbers X = {y| y < x}

the set of reals y less than x.

Define an injection L: R -> P(R) so that

L(x) = X = { y | y < x }

Notice that

x =< y <-> L(x) sub L(y)

We can extend the image of L in R (the Dedekind cuts

in R) with the sets R and {}.

(Remember, L(R) doesn't have reals,

it has sets of reals.)

Let's call L(R) U { R, {} } = L(R)+

How do you go from this set to the notion of

some kind of "infinite number?"

And that it might somehow be equal to 1/0?

I'm going to assume that you've read all the way to

the end of my post. An important of the answer to

your question is "Those things I said next".

I'm assuming that we already have the usual set of real

numbers with their usual operations, and that we have

ZFC too.

What I'm doing is extending those usual real numbers with

a positive infinite number _larger than all_ usual, finite

real numbers and a negative infinite number _less than all_

usual, finite real numbers. It's certainly not the only way

to extend the reals to infinite numbers, but it's one way.

I'm not sure if you are concerned that the infinite

numbers I named +inf and -inf exist. Supposing you

are concerned, consider that the sets R and {}

that correspond to them do exist (by assumption, given

the reals and ZFC) and there is a bijection between R+ and

L(R)+ . Anything non-self-contradictory that we can say

about the elements of L(R)+ we should be able to translate

(using L(x)) into a statement about the elements of R+ .

As far as +inf and -inf are concerned, they exist because

(i) I said so, and (ii) they do not create contradictions

(no more than the reals or ZFC do). These are the same

reasons we say 0, 1, 2, ... exist.

It's not enough just to say that +inf and -inf exist.

Do they play well with others? How do the extended versions

of <, +, -, *, / work?

One thing I assumed is that we want the _usual_ part of

the extended operations to look like the _unextended_

operations. That bypasses deep discussions of whether

the finite, extended reals are "really" the reals. This

would be my great preference, to avoid that discussion.

I already showed how to extend =< by placing +inf at

the upper end of the extended reals and -inf at the

lower end of the extended reals.

If that was all I needed to do, there would have been

little point to introducing Dedekind cuts. But we can

define the extended +, -, *, / by looking at the

operation on finite unextended reals in those cuts.

This is what's done in the usual construction of the

real numbers from Dedekind cuts of the rationals.

By the way, _I won't know_ until I look whether this will

give us 1/0. I'm taking "larger (smaller) than all the

finite numbers" to be the defining characteristic of

+inf (-inf) and then seeing what follows from that.

(This is not the only way to do things.)

Notation:

[x] = { y e R | y < x }

Define an "addition" operation [+] on L(R)+

[x] [+] [y] = [z] <->

[z] = { w e R | (Eu e [x])(Ev e [y]( u + v = w ) }

where the '+' inside the definition of [z] is the usual

addition on the reals.

From this definition, we can see that, for all

_finite_ x and y

[x] [+] [y] = [x + y]

which is precisely what we want.

We also have, from the same definition,

[x] [+] [+inf]

= [x] [+] R

= R

= [+inf]

Therefore, it seems reasonable to define, for R+, x finite,

x + +inf = +inf

in order to preserve, for infinite numbers,

[x] [+] [y] = [x + y]

By similar arguments,

x + -inf = -inf

+inf + -inf = -inf

(This last one surprised me.)

We can try extending the definitions for

neg([x]) = [-x]

inv([x]) = [1/x]

[x][*][y] = [x*y]

as we extended

[x][+][y] = [x + y]

It might or might not give us something for 1/0.

*Post by Jim Burns*Notice that

all x in L(R)+, x sub R

and

all x in L(R)+, {} sub x

If we extend R to R+ with two constants +inf, -inf

We could extend our infection L(x) to L'(x) so that

L': R+ -> L(R)+

with

L(+inf) = R

and

L(-inf) = {}

Then we would still have

x < y <-> L'(x) sub L'(y)

only now with two more elements such that

all x in R+, x =< +inf

and

all x in R+, -inf =< x

If I'm wrong, I hope someone corrects me, but that

looks like the reals extended with positive and negative

infinity.