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There is a mathematical way of doing it, through set theory. Discuss a little how that goes, how you define addition, and so on. | There is a mathematical way of doing it, through set theory. Discuss a little how that goes, how you define addition, and so on. | ||
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| + | '''A Brief History on Natural Numbers''' | ||
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| + | As most of us know, natural numbers are the numbers that we use to count (1, 2, 3, …). But, where did these numbers come from? | ||
| + | Whether or not to include 0 has been debated throughout history. Zero was included to make the set more “complete.” This was because 0 represents an empty set. I said before that natural numbers are counting numbers. If you do not count any, you have 0 (an empty set). Whether or not 0 is included in the set of natural numbers depends on the field of mathematics, but most define the set starting at 1. | ||
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| + | '''Defining Addition of Natural Numbers''' | ||
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| + | There are many ways to define the addition of natural numbers: recursion, induction, set theory. However, for recursion and induction, we must look at the basic axiom of mathematics that a number, a, added to zero, is still a, or with the symbols: | ||
| + | a + 0 = a | ||
| + | Set theory on the other hand defines “0” so that defining “1” becomes easier. Peano axioms, which states that a+0 =a, can be derived from set theory of natural numbers. The definition of “0” starts from the definition of zero, defined by John von Neumann, as an empty set (denoted as ∅), which can be written as 0 = ∅. John von Neumann also defines a function s(a), where s is the successor function. The successor function is defined as s(a) = a+1. So if we use this idea of succession, we can define our successor function such that s(a) = a ∪ {a}. | ||
[[2014 Spring MA 375 Walther|Back to MA375 Spring 2014]] | [[2014 Spring MA 375 Walther|Back to MA375 Spring 2014]] | ||
[[Category:MA375Spring2014Walther]] [[Category:Math]] [[Category:Project]] | [[Category:MA375Spring2014Walther]] [[Category:Math]] [[Category:Project]] | ||
Revision as of 10:20, 26 April 2014
We all "understand" what the number 1 is. Right? How do you really explain what that is to a smart person that has never heard of it?
There is a mathematical way of doing it, through set theory. Discuss a little how that goes, how you define addition, and so on.
A Brief History on Natural Numbers
As most of us know, natural numbers are the numbers that we use to count (1, 2, 3, …). But, where did these numbers come from? Whether or not to include 0 has been debated throughout history. Zero was included to make the set more “complete.” This was because 0 represents an empty set. I said before that natural numbers are counting numbers. If you do not count any, you have 0 (an empty set). Whether or not 0 is included in the set of natural numbers depends on the field of mathematics, but most define the set starting at 1.
Defining Addition of Natural Numbers
There are many ways to define the addition of natural numbers: recursion, induction, set theory. However, for recursion and induction, we must look at the basic axiom of mathematics that a number, a, added to zero, is still a, or with the symbols: a + 0 = a Set theory on the other hand defines “0” so that defining “1” becomes easier. Peano axioms, which states that a+0 =a, can be derived from set theory of natural numbers. The definition of “0” starts from the definition of zero, defined by John von Neumann, as an empty set (denoted as ∅), which can be written as 0 = ∅. John von Neumann also defines a function s(a), where s is the successor function. The successor function is defined as s(a) = a+1. So if we use this idea of succession, we can define our successor function such that s(a) = a ∪ {a}.
