I would appreciate some help on proving theorem #4 (last two images) using Peano's axioms and subsequent theorems. I have been working on it, off and on, for about two weeks.
The proof is divided into two parts, A and B. I understand neither half of the proof as given although I have put together my own form of proof for the first half which is perhaps just a more detailed version of the given proof.
For the second half of the proof I have nothing but questions but confine myself to the first half of the proof, perhaps that is all of the help I will need.
So the first question is this, is my version of the proof valid.
My version of the first half of the proof, Part A.
This theorem introduces and defines the addition operator designated by the infix symbol “+”, i.e “x + y”.
Using a prefix operator symbol one can write “x + y” = sum(x,y) which I prefer as an aid to my thinking process.
So the claim is that given that x, y are elements of N, where N is the set of Natural numbers, sum(x,y) is a function that produces a unique natural number, say “s” . sum(a,b) is required to have the properties:
Property 1: sum(x, 1) = x’
Property 2: sum(x, y’) = (sum(x, y))’
The first half of the proof says let x be a particular fixed natural number and show that sum(x,y) produces a unique “s” no matter what value of y is chosen.
The proof then refers to ay and by but does not say what either of those refer too. I assume to the experienced eye their meaning is obvious but I have considered three different possibilities and finally settles on this: (am I right? A critical question.)
Suppose sum(x,y) is not the only possible function that has the two required properties, suppose there is a
sum_A(x, y) = ay and a sum_B(x, y) = by ,
each of which subscribe to the two properties but when loaded with the same values for “x” and “y” produces different results. The proof then endeavors to show that each case produces the same result therefore they are only two different names for the same function and so
sum_A(x, y) = sum_B(x, y) = sum(x, y).
Now then with x a fixed natural number and assuming both sum_A(x,y) and sum_B(x,y) subscribe to the two required properties we say:
Let “M” be the set of natural numbers for which both functions, sum_A and sum_B produce the same results. We can write for both functions, since they each, by inductive hypthosis, subscribes to property 1, when y = 1 :
sum_A(x, 1) = x’ and
sum_B(x, 1) = x’
Therefore sum_A(x, 1) = sum_B(x, 1) when y =1 and so 1 is an element of M.
Now we come to the critical point for the first half of the proof. The author writes,
if “y” is an element of M then
1: sum_A(x, y) = sum_B(x, y) (by “inductive hypothesis” … is that correct?)
Since both sum_A(x, y) and sum_B(x, y) are expressions for the same natural number it follows by the uniqueness of a successor that:
2: [sum_A(x, y)]’ = [sum_B(x, y)]’
Finally, using property 2 from the right hand side back to the left hand side:
3: [sum_A(x, y)]’ = sum_A(x, y’), and
4: [sum_B(x, y)]’ = sum_B(x, y’),
So that 2: becomes
5: sum_A(x, y’) = sum_B(x, y’),
which shows that what is true for the inductive hypothesis with argument “y” : sum_A(x, y) = sum_B(x, y)
is also true for “y’ ” : sum_A(x, y’) = sum_B(x, y’),
therefore M contains all natural numbers y and so it is established that for a fixed “x” there is only one function:
sum(x, y), that subscribes to properties 1 and 2 and hence defines that defines addition for a fixed value of x.
If this proof is correct I will endeavor to understand the Professors proof for part A and part B, if not, help in where I went wrong would be appreciated.