I How to define addition and multiplication using Peano's axioms?

[Logical Dog]

How can one define addition using peanos axioms?

Number, successor, zero are terms which we presume to know the meaning of.

We then use five propositions:

1. 0 is a number.
2. Every number has a successor.
3. 0 is not the successor of any number.
4. Any proeprty common to zero and its successor and its successors successors is a property of all numbers. (basically any property common to zero, 1 and 2 is a property of all numbers)
5. No two numbers have the same successor.

I started with M + 0 = M
and M + 1 = successor of M.

and for multiplication M * N = M + M ...(n times)

 

[mfb]

Let f(M) be the successor of M, define 1=f(0), 2=f(1) and so on.
Define M+0=M, N+M=M+N and f(M)+N = M+f(N), everything else follows from induction.

Using N=0, we get f(M)=f(M)+0=M+f(0)=M+1
Setting N=1, we get f(f(M))=f(M)+1=M+f(1)=M+2
...

 

[Svein]

Let f(M) be the successor of M, define 1=f(0), 2=f(1) and so on.
Define M+0=M, N+M=M+N and f(M)+N = M+f(N), everything else follows from induction.

Using N=0, we get f(M)=f(M)+0=M+f(0)=M+1
Setting N=1, we get f(f(M))=f(M)+1=M+f(1)=M+2
...

First, you need to prove that N+M = M+N (commutation does not follow directly from the axioms). It usually starts with 0+0 = 0 (your definition) and 1+0 =1.

 

[mfb]

First, you need to prove that N+M = M+N (commutation does not follow directly from the axioms).

I don't have to prove it if I include it in the definition. If we can show it based on the other parts of the definition, even better.

 

[Logical Dog]

First, you need to prove that N+M = M+N (commutation does not follow directly from the axioms). It usually starts with 0+0 = 0 (your definition) and 1+0 =1.

I don't have to prove it if I include it in the definition. If we can show it based on the other parts of the definition, even better.

I am a slow learner in mathematics lol. Is it fair to say that addition is the operation and multiplication, division and subtraction are all defined in terms of this?

 

[Svein]

I am a slow learner in mathematics lol. Is it fair to say that addition is the operation and multiplication, division and subtraction are all defined in terms of this?

If you want a lucid and fun treatise on the subject, get hold of an exemplar of "Gödel, Escher, Bach - an Eternal Golden Braid".

 

[Logical Dog]

If you want a lucid and fun treatise on the subject, get hold of an exemplar of "Gödel, Escher, Bach - an Eternal Golden Braid".

Yes, I will try..But can you answer my question please? :) Isnt multiplication and division and subtraction all defined in terms of addition..

 

[Logical Dog]

Thanks for the recommendation I will get it...

"
Gödel, Escher, Bach: An Eternal Golden Braid, also known as GEB, is a 1979 book by Douglas Hofstadter. The tagline "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll" was used by the publisher to describe the book.[1]

By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher and composer Johann Sebastian Bach, the book expounds concepts fundamental to mathematics, symmetry, and intelligence. Through illustration and analysis, the book discusses how self-reference and formal rules allow systems to acquire meaning despite being made of "meaningless" elements. It also discusses what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself.

In response to confusion over the book's theme, Hofstadter has emphasized that Gödel, Escher, Bach is not about the relationships of mathematics, art, and music, but rather about how cognition emerges from hidden neurological mechanisms. At one point in the book, he presents an analogy about how the individual neurons of the brain coordinate to create a unified sense of a coherent mind by comparing it to the social organization displayed in a colony of ants.[2][3]

so I guess numbers are a hack concept

 

[Svein]

Yes, I will try..But can you answer my question please? :) Isnt multiplication and division and subtraction all defined in terms of addition..

Well - the basic operation is the succession. From that operation you can define the numbers (since the only number in the axioms is 0). Then you can define addition: N+0 = N; Succ(N + M) = N + Succ(M). Defining subtraction is hard since the result of a subtraction may not exist (remember that we only have defined the classic natural numbers). Multiplication may be defined: M⋅1 = M; M⋅Succ(N) = M.N + M. Division is even more tricky than subtraction, since the result usually is not a natural number.

 

[Logical Dog]

Well - the basic operation is the succession. From that operation you can define the numbers (since the only number in the axioms is 0). Then you can define addition: N+0 = N; Succ(N + M) = N + Succ(M). Defining subtraction is hard since the result of a subtraction may not exist (remember that we only have defined the classic natural numbers). Multiplication may be defined: M⋅1 = M; M⋅Succ(N) = M.N + M. Division is even more tricky than subtraction, since the result usually is not a natural number.

Yes, I realized that too that we had only come up with natural numbers..I don't know how other numbers were defined this way. I mean I know the set definitions for them but not this way.

 

[Svein]

Yes, I realized that too that we had only come up with natural numbers..I don't know how other numbers were defined this way.

The answer is rather long, but the short version is:

• We saw that subtraction did not always give meaningful answers, so we extended the natural number to the integers (containing both the natural numbers and the negation of them).
  
• We saw that division did not always give an integer answer, so we extended the integers to the rational numbers.
• We observed that square roots tended to give answers that were not rational numbers, so we extended the rationals to the algebraic numbers.
• The famous diagonal argument of George Cantor showed that we needed to extend the algebraic numbers to the real numbers.
• And since the square root of negative numbers could not be found in the real numbers, we extended the real numbers to the complex numbers.

 

[Logical Dog]

The answer is rather long, but the short version is:

• We saw that subtraction did not always give meaningful answers, so we extended the natural number to the integers (containing both the natural numbers and the negation of them).
  
• We saw that division did not always give an integer answer, so we extended the integers to the rational numbers.
• We observed that square roots tended to give answers that were not rational numbers, so we extended the rationals to the algebraic numbers.
• The famous diagonal argument of George Cantor showed that we needed to extend the algebraic numbers to the real numbers.
• And since the square root of negative numbers could not be found in the real numbers, we extended the real numbers to the complex numbers.

I know. I meant I thought they are also defined in the same way above with successor.

 

[Svein]

I know. I meant I thought they are also defined in the same way above with successor.

No. For the rationals and onward, there is no "successor" (given two real numbers a and b, a<b, there exists a rational number q such that a<q<b). When you extend the natural numbers to the integers, axiom 3 is violated.