Constructing the integers from set theory without equivalance classes

[Tim Sweeney]

There exist many simple ways to construct the natural numbers from set
theory, such as:

0 = {}
n+1 = {n,{n}}

Then one typically constructs the integers from equivalance classes of
pairs of natural numbers, such that:

{a0,{a1}}={b0,{b1}} iff a0+b1=a1+b0

Is there a simple way to construct the integers directly from set
theory, without making use of equivalance classes? In other words,
I'm looking for a simple representation of the integers such that each
integer is represented by a unique set, rather than an equivalance
class of sets.

As a bonus it would be nice if the representation admitted a
straightforward definition of addition, subtraction and
multiplication.

Thanks,

Tim Sweeney

[Torkel Franzen]

t...@epicgames.com (Tim Sweeney) writes:

> There exist many simple ways to construct the natural numbers from set
> theory, such as:
>
> 0 = {}
> n+1 = {n,{n}}
>
> Then one typically constructs the integers from equivalance classes of
> pairs of natural numbers, such that:
>
> {a0,{a1}}={b0,{b1}} iff a0+b1=a1+b0
>
> Is there a simple way to construct the integers directly from set
> theory, without making use of equivalance classes?

Yes. You can, for example, define an integer as the *first* pair
<a,b> in an equivalence class, where "first" refers to a canonical
ordering of the hereditarily finite sets.

[Denis Feldmann]

"Torkel Franzen" <tor...@sm.luth.se> a écrit dans le message news:
vcbk7pk...@beta13.sm.luth.se...

In a completely different direction, the Conway construction defines all
numbers , great and small, as pairs of sets of earlier constructed numbers:
0={|}, 1={0|}, -1={|0}, n+1={n|} (and 1/2={0 | 1}; w={0,1,2,...,n,...|},
etc...) See the firs pages of ONAG, or "surreal numbers", by Knuth.

>

[Franz Fritsche]

>
> Is there a simple way to construct the integers directly from set

> theory, without making use of equivalence classes?
>

Yes. ( Note: IN_0 := IN u {0} )

First lets introduce the notation:

-n := (0,n) : n e IN (and 0 e IN_0)

Now (for example) you could construct the following set:

IN- := { -n | n e IN }

And then define

Z := IN u {0} u IN-

(Comment: We can also allow to write +n for n e IN and instead of 0
also +0 or -0.)

With properly defined addition and multiplication, etc. we have in
fact constructed the "integers".

Here just the definition for addition in Z:

For all n,m e IN we define [n+m for n,m e IN and subtraction in IN is
already defined] :

n + (-m) := (-m) + n :=

n - m, if n > m
0, if n = m
-(m - m), if n < m

(-n) + (-m) := -(m + n)

q + 0 := 0 + q := q

0 + 0 := 0

(-q) + 0 := 0 + (-q) := -q

F.

[David C. Ullrich]

On 31 May 2002 02:33:41 -0700, t...@epicgames.com (Tim Sweeney) wrote:

>There exist many simple ways to construct the natural numbers from set
>theory, such as:
>
> 0 = {}
> n+1 = {n,{n}}
>
>Then one typically constructs the integers from equivalance classes of
>pairs of natural numbers, such that:
>
> {a0,{a1}}={b0,{b1}} iff a0+b1=a1+b0
>
>Is there a simple way to construct the integers directly from set
>theory, without making use of equivalance classes? In other words,
>I'm looking for a simple representation of the integers such that each
>integer is represented by a unique set, rather than an equivalance
>class of sets.

Well, of course an equivalence class of sets _is_ a unique set...

But never mind that. Certainly you can do what you ask. For example
you could define an integer to be an ordered pair <n, s>, where
n is a natural number and s = 0 or 1, with the restriction that
if n = 0 then s is required to be zero (to give uniqueness).
The idea being that <n,0> is the integer equal to n, while
<n,1> is -n.

>As a bonus it would be nice if the representation admitted a
>straightforward definition of addition, subtraction and
>multiplication.

That's easy. Although it requires breaking things into
cases - the definition of integers as equilvalence classes
allows much more elegant definitions of the arithmetic
operations.

>Thanks,
>
>Tim Sweeney

David C. Ullrich

[Franz Fritsche]

Ullrich's mentioned a notable variant:

>
> For example you could define an integer to be an ordered pair <n, s>,
> where n is a natural number and s = 0 or 1, with the restriction that
> if n = 0 then s is required to be zero (to give uniqueness).
> The idea being that <n,0> is the integer equal to n, while
> <n,1> is -n.
>

So lets try the following:

IN+ := { (0,n) | n e IN\{0} }

IN- := { (1,n) | n e IN\{0} }

and

0_ := (0,0)

Now we may define:

Z := IN+ u {0_} u IN-

So the "integers" are defined. Now let us agree on writing

n or +n instead of (0,n), with n e IN,
-n instead of (1,n), and, with n e IN,
0, +0 or -0 instead of 0_,

since no misunderstanding can arise if we always make clear that we
are working in Z, i.e. by writing n e Z, -n e Z, 0 e Z (with n e IN)
etc. (Thus its just a notational question.)

Now we can give proper definitions for addition and multiplication
again...

F.

In addition: since IN\{0} and IN+ can be identified (because they are
isomorphic), we can state that IN is a proper subset of Z.