Validate Peano Arithmetic Approach

In summary, the conversation discusses the process of proving commutativity and distributivity of multiplication for natural numbers. The main focus is on whether using distributivity in the proof of commutativity is valid and whether both left and right distributivity are necessary for the proof. The conversation also touches on the importance of the multiplicative identity in these proofs. Overall, it is suggested to first prove left distributivity, then use the multiplicative identity to prove right distributivity, and finally use both properties to prove commutativity.
  • #1
snipez90
1,101
5

Homework Statement


Show that the natural numbers satisfy commutativity of multiplication and distributivity of multiplication over addition.


Homework Equations





The Attempt at a Solution


I'm wondering if there is any potential circularity in this reasoning. I proved distributivity over addition first by say, fixing n, m and showing that M = {p in naturals | (n+m)p = np + mp} is equal to the set of natural numbers.

Then I used distributivity in the final step of my commutativity proof where the inductive hypothesis was mk = km and the inductive step m(k+1) = mk + m = km + m = k(m+1) (literally the last equality follows from distributivity).

This seems valid, but I know that distributivity is usually stated with both (n+m)p = np + mp and p(n+m) = pn + pm. Typically this follows from commutativity, but would this still follow even if I used the distributive law in my commutativity proof?
 
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  • #2
I suppose it would be fine to have a sequence like:
- Show left distributivity: (n+m)p = np + mp
- Show commutativity, only using left distributivity
- Right distributivity follows

However, it looks like you are using right distributivity in the proof of commutativity. If you had in the final step: mk + m = (k + 1)m it would be fine, as you have just proven that to be true (assuming you know that 1k = k, and fun stuff like that).

By the way, note that
m(k + 1) = k(m + 1)
is not true (m = 1, k = 2 is a counter example) -- you meant: km + m = (k + 1)m ?[edit]On closer inspection, I think you need both right and left distributivity, if you set it up this way. However, my guess is that right distributivity is an almost-copy of the proof of left distributivity [/edit]
 
  • #3
Sorry that was a huge typo. I mean m(k+1) = mk + m = km + m = (k+1)m, which should be permissible since I proved that 1 is the multiplicative identity for the base case of the same commutativity proof. I'll try to set up right distributivity and then prove commutativity I guess.

*EDIT* I guess the smartest route to take would be left distributivity -> multiplicative identity -> right distributivity -> commutativity (use multiplicative identity for base case).
 

1. What is Peano Arithmetic Approach?

Peano Arithmetic Approach is a mathematical framework used to formalize the properties of natural numbers and the operations performed on them. It was developed by the Italian mathematician Giuseppe Peano in the late 19th century.

2. Why is it important to validate Peano Arithmetic Approach?

Validating Peano Arithmetic Approach is important because it ensures that the axioms and rules used to define the natural numbers and their operations are consistent and free of contradictions. This allows us to use Peano Arithmetic as a foundation for other mathematical theories and proofs.

3. How is Peano Arithmetic Approach validated?

Peano Arithmetic Approach is validated through the use of formal proofs and logical reasoning. This involves starting from the basic axioms and using mathematical rules and principles to prove the validity of more complex statements and equations.

4. What are the challenges in validating Peano Arithmetic Approach?

One of the main challenges in validating Peano Arithmetic Approach is ensuring that the axioms and rules used are complete and consistent. This can be a difficult task, as it requires a deep understanding of mathematical logic and the ability to identify potential contradictions or inconsistencies.

5. What are the applications of Peano Arithmetic Approach?

Peano Arithmetic Approach has many applications in mathematics, computer science, and other fields. It provides a foundation for the study of number theory, abstract algebra, and mathematical logic. It is also used in the construction of computer programs and algorithms that deal with natural numbers and their properties.

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