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Hi all,

I am learning the axiomatic construction of Naturals, from the Peano axiom. Hence proving every property of naturals based solely on these axioms. I have come to some point where i am not very sure that what i am doing is okay, since there is not an explicit proof of this, it is left as an exercise to the reader.

I am trying to prove that for every a,b from naturals, a+b=b+a.

Here it is how i am going about it

[tex] S=\{b\in N | a+b=b+a\}[/tex] where a is any fixed natural numeber.

My attempt is to show that S is an inductive set.

We first need to prove that

[tex] 1\in S(?)[/tex]

Here it is how i go about it, this is the part that i am not sure,

Using another theorem, where it says that a+1=a', where a' is the successor of a...this in fact is the theorem where they define the operation '+'. But to prove this, the author defines this: 1+b=b' for every b in N.------(1)

Now on my proof above, i am not using the definition of '+' but rather this other definition 1+b=b' for every b in N. ... so my question is am i allowed to do this, or there is another way of going about it.

Here is how i go about it

a+1=a' from the definition 0f +

=1+a from definition (1)

this actually means that 1 is an element of S. the rest i can easily prove if this is correct.

So?

thnx

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# Prove that a+b=b+a?

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