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I am interested in this question. We all know that for example the natural number seven consists of seven 1 (7=1+1+1+1+1+1+1) and it is located between 6 and 8. But is there any law or axiom that would confirm this, or all people just agreed among themselves about it? Thanks for your answers.

Honza Prochazka
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    I don't know what answer you expect... To use and talk about a particular symbol or word, we must first agree upon the meaning of the symbol. The symbol $7$ and word seven are indeed defined to be equal to $S(S(S(S(S(S(1))))))$ where $S$ is the successor function, giving the natural number one larger than the input. – JMoravitz Jun 15 '20 at 01:17
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    Yes; the definition of seven as the successor of six, and that of eight as the successor of seven. – Mauro ALLEGRANZA Jun 15 '20 at 09:25

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It might depend on the formal framework you are using, the arithmetic theory, like some of the most common: PRA, HA or PA. In all those mentioned axiomatization of arithmetic, you find some kind of construction process formalized, usually a "successor function" satisfying some stuff like "for every well-formed expression n, S(n) is a well-formed expression" and "there is no well-formed expression n, s.t. for the Symbol '0' 0 = S(n) holds.". If you identify now 0 with 0, S(0) with 1, S(S(0)) with 2 and so on, the set $\mathbb{N}$ becomes a model for the theory. So the location of a number between two other numbers is in the end a result of the construction process (the successor function) and "binding" the natural numbers to it.

farud
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I think you might be looking for something like the Peano axioms. @JMoravitz hinted at it in their comment when they referred to the successor function S(n).

As far as the choice of assigning the symbol 7 to the value of $S(S(S(S(S(S(1))))))$, that is completely cultural.

ReGuess
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