Question about Successor Function

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In summary, one of the Peano Axioms specifies that if Sa = Sb, then a = b, where S is the successor function. To establish that S is a function, the axioms state that there exists a successor function. Peano Arithmetic is a theory with equality, so the equality axiom states that all functional symbols of arity 1 will result in the same output if the inputs are equal.
  • #1
agapito
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One of the Peano Axioms specifies

Sa = Sb --> a = b

where S is the successor function. How does one establish from the axioms that S is, in fact, a function, that is the converse

a = b --> Sa = Sb?

Probably a very simple matter, but I would appreciate any help in clarifying. Many thanks in advance,

Agapito
 
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  • #2
That should be part of the definition! If we say, as part of, say, the Peano axioms, "there exist a successor function" then we are saying this is a function. The Wikipedia entry on the Peano axioms say ". The naturals are assumed to be closed under a single-valued "successor" function S." (my emphasis)
 
  • #3
Peano Arithmetic is by definition a theory with equality. One of the equality axioms is $x=y\to f(x)=f(y)$ for all functional symbols of arity 1, and similarly for other arities.
 

Related to Question about Successor Function

1. What is the successor function?

The successor function is a mathematical function that takes a number as its input and outputs the next number in the natural number sequence. It is denoted as "n+1" and is a fundamental concept in mathematics and computer science.

2. How does the successor function work?

The successor function works by taking a number as its input and adding 1 to it. For example, the successor of 5 would be 6. This process can be repeated indefinitely, resulting in an infinite sequence of natural numbers.

3. What is the importance of the successor function?

The successor function is important because it forms the basis for the concept of counting and arithmetic operations such as addition and multiplication. It also plays a crucial role in the foundations of mathematics and logic.

4. Can the successor function be applied to other types of numbers?

The successor function is typically defined for natural numbers, but it can also be extended to other types of numbers such as integers, rational numbers, and even complex numbers. However, the concept of "next number" may vary depending on the type of numbers being used.

5. How is the successor function used in practical applications?

The successor function is used in various practical applications, such as computer programming, where it is used to iterate through loops and data structures. It is also used in cryptography, where it is used to generate random numbers and in algorithms for prime number generation.

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