Your axioms still don't work as a definition of the natural numbers, if that is your intent. (Earlier today, you assured us that was no longer your intent. Is that why you changed your handle, Wolfgang?)
With Peano's axioms alone you can easily prove that 1+1=/=1. Unfortunately , your "axioms" at not up to job -- not without much vigorous hand-waving and insults. You cannot even prove that there two distinct numbers in your |N, never mind all of the natural numbers! Yes, I know, it should be "obvious", but that's not how we do things with formal proofs. In formal proofs, your axioms are your entire world. You cannot use any other "facts" no matter how "obvious" they might be.
I'm afraid it's back to drawing board, Wolfgang. Or should we call you Heinrich now?
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Recently we saw a discussion about equality. It was claimed that 1 + 1 can be same as 1.
This is true of course, as long it remains undefined what equivalence relation is expressed by "being equal".
Consider the expressions 0 + 0 and 0.
With respect to the script they are different. Even the two zeros in 0 + 0 are different, one of them being that one on the left-hand side and the other one being just the "other". We can distinguish the zeros. We could not, if they were identical in all respects.
If we know that both expressions are meant to represent numbers, we know that they are equal with respect to property "being numbers" (and not being cars or stars).
With respect to numerical value we cannot know the result unless we know what "+" and "=" are meaning. As soon as we know the foundations of arithmetic, we see that 0 + 0 = 0. (This situation is comparable to having apples cut to pieces in closed boxes. Before opening the boxes, we cannot know in how many pieces the contained apple has been cut.)
With respect to angular diameter, sun is as large as moon. With respect to physical diameter sun is much larger than moon. With respect to volume sun is much, much larger than moon.
Conclusion: Before knowing what kind of comparison is meant, we cannot obtain a result.
With respect to the Peano axioms in their truncated version, we see for instance that S(x) = S(y) implies x = y. Here equality is not defined, so the expression is meaningless. If the script is meant, the sequence could be 0, 0 + 0, 0 + 0 + 0, etc. or 1, 1^1, 1^1^1, etc. Of course we "guess" somehow that arithmetical equality is meant as soon as numbers get involved. That means, the reader is not only expected to be able to read and to understand written text, but also to decide when two "successors" are equal or different. A reader who is able to recognize the numerical equality or inequality of numbers would know +1 and obtain the sequence |N from the three axioms:
1 in M n in M ==> n + 1 in M |N is a subset of every such M.
If unable to read this text, the prospective reader should learn to read. If unable to understand the used logic, the prospective reader should learn its basics. If unable to understand the meaning of +1 and =, the prospective reader should learn the basics of arithmetic.
Then the reader would be far better off than with the five Peano axioms in their truncated version which do not define the natural numbers unless their definition is taken from elsewhere.