A computer science major here (not really that much into math) have to solve this problem but cannot find any solutions. Really appreciate it if someone can clear things out for me (sorry but the more I do research the more it becomes confusing).
Question:
Let Γ0 be Peano Arithmetic. Let c be a new constant. Consider Γ1 = Γ0 ∪ {c> 1, c> 1 + 1, c> 1 + 1 + 1,. . . }. PA proves several standard facts about IN. What does this allow you to say about the interpretation of the constant c in M (M is a structure for Γ0)? Do the elements c-1 and upper-bound of (c)^(1/2) exist? If so, are these elements standard or non- standard elements? Is it correct to think of M as being IN ∪ {∞} or is it something more complicated?
I realized from my googling that when you write a constant c in that form it makes the language nonstandard(I don't know why though, I couldn't even grasp what the hell these Non-standard models are, are they numbers such as 1,2,3? or relations such as "c>1" altogether is a nonstandard model? I really have no idea, an example to show what these things are would really be helpful!!!)! By reading from wiki I realized these numbers cannot be computed (based on Tennenbaum's theorem), so the answer to that question is no, a form such as c-1 does not exist? not really sure.
Your constant $c$ is a value greater than any natural number. The problem boils down to how does one extend the successor function $S(n)$ in a way that is consistent with the natural numbers. One can extend the successor function by setting the value of $S'(n)$ to $n+1$ as usual for any natural number $n$, leaving only the value $S'(c)$ to be determined. By definition, any natural $n$ is less than $c$, and $c \le S'(c)$, so $S'(c)$ must be greater than $n$. There is only one value, $c$, which meets this requirement, so $S'(c)$ is forcibly equal to $c$.
The reason why arithmetic with $c$ becomes unintuitive is because there is no number after $c$ and by definition of $S'$ no number immediately before $c$. It is an absolute upper limit which can arise only as the limit of a divergent sequence. Thus, $c + 1 = c + S'(0) = S'(c + 0) = S'(c) = c$ from the definition of addition, and it can be seen by induction that $c + n = c$ as well. $c + c$ must also be $c$ otherwise there would be a contradiction on the order. Since $c + c = c$, it follows that $c \times n = 0$ when $n = 0$ or $c$ when $1 \le n < c$. $c \times c$ again, because of order restrictions, must be equal to $c$.
Under this definition we have an element $x$ for which $x \times x = c$, thus $c$ is its own square root. $c$ also has a unique predecessor, itself, so, by abuse of notation, one could say that $c - 1 = c$. $c$ turns out to not be all that interesting arithmetically. Essentially the only reason one would consider such a constant is to ensure that any set of natural numbers (and consequently any set of extended natural numbers) has a least upper bound.
What if $S'$ had been defined differently? If for instance there was an $n$ for which $S'(n) = c$ then we would get a contradiction of the Peano axioms as $S'(c) = c = S'(n)$ which would imply $c = n$ which cannot be true. This shows that under any alternate definition the value $c$ must still be treated separately from the natural numbers as above.