Quote:
Originally Posted by winders
Bill,
The set of natural numbers has some ambiguity depending on what discipline is defining it. Because of that, extra clarity is often called for to avoid confusion.
I have never said the "actual math" ever changes. Just how we write it and how we interpret what is written. Ambiguity exists otherwise there would not be pages and pages written about it all over the Internet and we wouldn't have these arguments all over the Internet about equations and their results. Just because there shouldn't be any ambiguity doesn't mean there isn't any.
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the set of natural #s has no ambiguity it is a well defined set
it is commonly represented in set notation as {1,2,3,...}, the ellipsis(...) means to continue in the same, manner is also well defined and standard
whole #s is also a well defined set, {0,1,2,3,...}
integers is the set, {..., -2, -1, 0, 1, 2, ... }
Rational #s is another well defined set, this one has to be defined by rule rather than iteration, {any # that can be represented as an integer divided by a natural #} alternately {any # that can be written as the ratio of an integer and a natural #}
irrational #s is the next set that needs to be defined, again by rule, {any # that cannot be written as an integer divided by a natural #} or {any # that cannot be expressed as the ratio of an integer and a natural #}
next is the real #s, { set of rational #s + the set of irrational#} this last is what makes up a Real # line, such as an axis used in graphing
there are further # sets such as {Complex #s}, {Diophantine #s}, {Transcendental #s} etc. that can also be defined
Anyone that claims other wise is making their own idiosyncratic rule or definition.