Return to Answer

I provided a direct axiomatization of integers for instance under the MO question Axiomatic definition... by @Victor Makarov, in the post Part 2: Cyclands and integers. This axiomatization makes no direct reference to the natural numbers, or to any linear order.

I provided a direct axiomatization of integers for instance under the MO question Axiomatic definition... by @Victor Makarov, in the post Part 2: Cyclands and integers. This axiomatization makes no direct reference to the natural numbers, or to any linear order.

I provided a direct axiomatization of integers for instance under the MO question Axiomatic definition... by @Victor Makarov, in the post Part 2: Cyclands and integers. This axiomatization makes no direct reference to the natural numbers, or to any linear order.

On the other hand, a clean direct constructions of integers was mentioned by Gerald Edgar in a comment above--let's copy it for the sake of visual clarity:

each integer is uniquely defined as a sequence   $(a_0\ a_1\ \ldots)$ of integers   (called digits): $-1\ \,0\ \,1$,   where all digits but a finite number are equal to   $0$.

(The rest is obvious). By using only the first   $n$   digits we get the continuous range of   $3^n$   integers: $$-\frac{3^n-1}2\ \ldots\ 0\ \ldots\ \frac{3^n-1}2$$

Furthermore, the ring   $\mathbb Z[\frac 13]$   can be defined as all sequences   $f:\mathbb Z\rightarrow \{-1\ \,0\ \,1\}$   such that all values   $f(n)$   but a finite number are   $0$.

I provided a direct axiomatization of integers for instance under the MO question Axiomatic definition... by @Victor Makarov, in the post Part 2: Cyclands and integers. This axiomatization makes no direct reference to the natural numbers, or to any linear order.

I provided a direct axiomatization of integers for instance under the MO question Axiomatic definition..., in the post Part 2: Cyclands and integers. This axiomatization makes no direct reference to the natural numbers, or to any linear order.

On the other hand, a clean direct constructions of integers was mentioned by Gerald Edgar in a comment above--let's copy it for the sake of visual clarity:

each integer is uniquely defined as a sequence   $(a_0\ a_1\ \ldots)$ of integers   (called digits): $-1\ \,0\ \,1$,   where all digits but a finite number are equal to   $0$.

(The rest is obvious). By using only the first   $n$   digits we get the continuous range of   $3^n$   integers: $$-\frac{3^n-1}2\ \ldots\ 0\ \ldots\ \frac{3^n-1}2$$

Furthermore, the ring   $\mathbb Z[\frac 13]$   can be defined as all sequences   $f:\mathbb Z\rightarrow \{-1\ \,0\ \,1\}$   such that all values   $f(n)$   but a finite number are   $0$.

I provided a direct axiomatization of integers for instance under the MO question Axiomatic definition... by @Victor Makarov, in the post Part 2: Cyclands and integers. This axiomatization makes no direct reference to the natural numbers, or to any linear order.

On the other hand, a clean direct constructions of integers was mentioned by Gerald Edgar in a comment above--let's copy it for the sake of visual clarity:

each integer is uniquely defined as a sequence   $(a_0\ a_1\ \ldots)$ of integers   (called digits): $-1\ \,0\ \,1$,   where all digits but a finite number are equal to   $0$.

(The rest is obvious). By using only the first   $n$   digits we get the continuous range of   $3^n$   integers: $$-\frac{3^n-1}2\ \ldots\ 0\ \ldots\ \frac{3^n-1}2$$

Furthermore, the ring   $\mathbb Z[\frac 13]$   can be defined as all sequences   $f:\mathbb Z\rightarrow \{-1\ \,0\ \,1\}$   such that all values   $f(n)$   but a finite number are   $0$.