Why is mathematical induction included in the Peano postulates but not in the definition of a ring such as the ring of integers?

[PseuTonym]

It seems that there should be an obvious answer to the question, but for some reason I cannot at the moment think of what it is.

Of course, the integers include negative numbers, but we can compensate for that with a minor variation of the usual formulation of mathematical induction. The change is small enough to hardly merit mention. For example, suppose that mathematical induction is formulated as follows: “If S is a set of numbers and zero is an element of S and for every n, (if n is an element of S then n+1 is an element of S) then every number is an element of S.”

To handle arbitrary integers, we can revise that to the following:
“If S is a set of numbers and zero is an element of S and for every n, (if n is an element of S then n+1 is an element of S and n-1 is an element of S) then every number is an element of S.”

Links for the curious who aren’t familiar with the terminology:

mathworld.wolfram.com/PeanosAxioms.html

mathworld.wolfram.com/Ring.html

If somebody has a better link for algebraic rings, then please post it in this thread. The one above does not seem very conceptually oriented and merely includes the bare minimum of the formal details.

 

[Tomdstone]

It seems that there should be an obvious answer to the question, but for some reason I cannot at the moment think of what it is.

Of course, the integers include negative numbers, but we can compensate for that with a minor variation of the usual formulation of mathematical induction. The change is small enough to hardly merit mention. For example, suppose that mathematical induction is formulated as follows: “If S is a set of numbers and zero is an element of S and for every n, (if n is an element of S then n+1 is an element of S) then every number is an element of S.”

To handle arbitrary integers, we can revise that to the following:
“If S is a set of numbers and zero is an element of S and for every n, (if n is an element of S then n+1 is an element of S and n-1 is an element of S) then every number is an element of S.”

Links for the curious who aren’t familiar with the terminology:

mathworld.wolfram.com/PeanosAxioms.html

mathworld.wolfram.com/Ring.html

If somebody has a better link for algebraic rings, then please post it in this thread. The one above does not seem very conceptually oriented and merely includes the bare minimum of the formal details.

. The set of all continuous real-valued functions defined on the real line forms a commutative ring. The operations are addition and multiplication of functions.For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations. I don’t see mathematical induction as being appropriate in these cases, although perhaps it can be stated to yield true results, the results would not be particularly useful as they are in the case of integers.

 

[Weejee]

Make a ring, and make another ring…

 

[tabycat]

 

[PseuTonym]

The set of all continuous real-valued functions defined on the real line forms a commutative ring. The operations are addition and multiplication of functions.

That’s a good example to show that ordinary induction isn’t going to work. Perhaps, for that example of a ring, we could formulate some kind of transfinite induction, but even if we succeed we will be relying upon the specifics of the example, and there may be no way to generalize beyond the example to an arbitrary algebraic ring.

I don’t see mathematical induction as being appropriate in these cases, although perhaps it can be stated …

The Peano postulates give the impression of inviting people to amend the list to include additional self-evident properties of the natural numbers that might be discovered in future. Godel’s incompleteness theorem suggests that this might happen. In contrast, it doesn’t seem that there is any process of discovery that would allow for the list of properties in the definition of an algebraic ring to be amended to include new properties. Of course, if the Peano postulates were to be amended, then some new name would have to be chosen to distinguish the amended list from the old list of Peano postulates. Nevertheless, we have a contrast here between the Peano postulates and the properties in the definition of an algebraic ring.

So, what is the moral? It seems that although there is much to be gained by generalizing, there can also be significant costs. For example, the integers under the ordinary operations of addition and multiplication are not merely an algebraic ring. There would always be more to the integers under the ordinary operations of addition and multiplication, no matter what revisions were made to the definition of an algebraic ring. Furthermore, in contrast with a concept that is obviously associated with the Peano postulates, there is no concept of an algebraic ring that allows for a future process of unlimited discovery and amendment to the definition.