I came up with this question- how would you show 4 not equal to 6 (or m not equal to m+n ( n not 0))

Jaiden Bowman

Jaiden Bowman

Answered question

2022-05-09

I came up with this question- how would you show 4 not equal to 6 (or m not equal to m+n ( n not 0)), using only Peano's Postulates?

I can see a number of things go wrong- for instance the Principle of Mathematical Induction seems to fail. Also possibly 0 seems to be in the image of the successor function in that case.

Answer & Explanation

gudstrufy47j

gudstrufy47j

Beginner2022-05-10Added 16 answers

Assume 4 = 6, where 4 = S ( S ( S ( S ( 0 ) ) ) ) and 6 = S ( S ( S ( S ( S ( S ( 0 ) ) ) ) ) ). One of the Peano axioms states that the successor function S is injective. So if S ( 3 ) = 4 = 6 = S ( 5 ), then 3 = 5. Similarly, S ( 2 ) = 3 = 5 = S ( 4 ) 2 = 4. Doing this two more times, you arrive at 0 = 2 = S ( S ( 0 ) ). But another of the Peano axioms states that there is not integer n such that S ( n ) = 0, so this is a contradiction.

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