How to represent each natural number?

Question

Assume we get the set of natural numbers $\mathbb{N}$ from any model of the Peano axioms.

We're given the symbols: $0,1,2,3,4,5,6,7,8,9$, or rather, we're given $0$ and we choose to use the symbols $1,2,3,4,5,6, 7, 8,9$.

Of course $0$ is the same from the model.

Then we'll have, by definition, $1=S(0)$, $2=S(1)$, $3=S(2)$, $4=S(3)$, $5=S(4)$, $6=S(5)$, $7=S(6)$, $8=S(7)$ and $9=S(8)$.

But how do we represent the rest of the natural numbers the way we expect them to be represented?

I understand that $1, 2, 3, 4, 5, 6, 7, 8, 9$ are just shorthand representations for the entities written above.

I guess my question can be particularized by: how do you know that the short hand notation for $S(999)$ is $1000$?

I'm assuming the way to get a representation for each natural number starts the way I write it. If that's not the case, please do it from the top.

Answer 1

Given a PA number, to translate it back into a string of digits is by the following recursive function:

• $f(x) = d(x)$ where $x < 10$
• $f(x \hat + SSSSSSSSSS0 \hat \times y) = f(y)"d(x)"$ where $x < 10$ and $y > 0$.

Justifying this recursion principle 9that a function defined in this way is well defined and total) is by Euclid's Division algorithm.

Answer 2

There is an obvious function from strings of digits to natural numbers:

• $f("") = 0$

• $f("3534") = 3 + 10\cdot f("534") = 3 + 10 \cdot (5 + 10 \cdot f("34")) = \ldots = 3 + 10(5 + 10(3 + 10(4 + 0))))$

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Recall that we define + and * for peano arithmetic:

• $0 \hat + y = y$

• $Sx \hat + y = S(x \hat + y)$

• $0 \hat \times y = x$

• $Sx \hat \times y = x \hat + x \cdot y$

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therefore, define

• $g("") = 0$

• $g("dssss") = p(d) \hat + SSSSSSSSSZ\hat \times g("ssss")$

and the gives the peano arithmetic number or a digits expression.

(p is the function that gives $p(2) = SSZ$ for the first 10 digits for example, that you already mentioned)

Answer 3

first you must define the "remain" and "quotient" of one natural number to another then you can represent any natural number by its "quotient" and "remain" to the powers of ten

Actually in set theory you don't represent numbers like 10 or 11 or something like that, number 3 in set theory is S(S(S(0))), but because you are already familiar with this representation of natural numbers, you use 3, actually using "3" in set theory is wrong

Answer 4

You need to notice that the standard decimal representation of the natural numbers is not a representation from inside the Peano model. The language only contains a constant symbol for $0$. Then one introduces $1=S(0)$ as shorthand notation not as a new symbol. Then comes $2=S(1)=s(S(0))$ again as shorthand notation, not a new symobol, and so on.

So, whenever you write something in PA and you use $0$ then you mean the constant symbol in the language. When you write $1$, $2$, and so on mean that this is shorthand notation that you use just because you are lazy to write out an expression of the form $S(S(0))$.

I hope this helps.

Answer 5

Suppose you have $9$ stones in a pile and you add another stone to make a pile of $S(9)$ stones. Rather than invent a new symbol, the convention is to give a pile of this size its own name. Henceforth, any pile of exactly $S(9)$ stones will be called a "ten". So, the successor of $9$ is a ten and no other stones. This is denoted by placing a $1$ in the ten column and a $0$ in the single stone column, which gives $S(9) = 10$. The process is similar for larger piles of stones: "hundred", "thousand", etc.