Properties of Natural Numbers and Mathematical Induction When working with natural numbers how to check that the property we consider is "permissible" to speak about? And not like the property "The smallest positive integer not definable in under eleven words".
Is there a strict answer or it's just the question of experience:  some properties behave well, like "$1 + ... + n = \frac{n(n+1)}{2}$" or "2|$n$", but some one day fail to be good properties.
I know that me may translate our vision of arithmetic to some formal language and deduce our informally well understood operations and relations from several axioms. But how we work with "properties" in this approach? Do we allow consideration of only those which may be formulated in this language?
But don't we need well-established theory of natural numbers to define this formal language and to prove statements about its formulas?
It seems to be a vicious circle:  we need theory of natural numbers to define formal languages which we need to establish theory of natural numbers.
Please, help me to understand what I'm missing here.
Thank you very much in advance.
 A: In principle you are right.
In order to define a formal language you must be able at least "to count", in order to build expression, that are sequnces of symbols, and derivations, taht are sequences of formula, and so on.
In addition, a lot of elementary (meta-)theorems about formal languages and theories nedd induction.
So, at least arithmetic is unavoidable.
At the same time, when you work with formal arithmtics, the concept of "admissible properties" is well defined: if you work with first-order language, the admissibleproperties are those expressible in your language, like : $x = 0$, $S(x) \ne 0$, $x = 0 \lor S(x) \ne 0$, ...
But this "circularity" does not means that you cannot work with mathematics.
First of all, mathematics is not restricted to the "formal one".
And then, the search of an "absolute" beginning is a dream, also in mathematics.
Think to natural language : a dictionary is a very useful tool and you sometimes need it; but this does not mean that you must use a dictionary in order to learn and practice in speaking, nor that the obvious "circularity" of a dictionary frustrates its usage ...
