[FOM] RE: Real Numbers
Victor Makarov
viktormakarov at hotmail.com
Sat May 10 09:24:02 EDT 2003
I just have found an amazing web page titled "What are the "real numbers,"
really?" at:
http://www.math.vanderbilt.edu/~schectex/courses/thereals/
Below are a few excerpts from the web page:
It is true that the real numbers are 'points on a line,' but that's not the
whole truth. This web page explains that the real number system is a
Dedekind-complete ordered field.
.............
Finally, the real definition of the reals
(No pun intended.)
Definition. The real number system is that unique algebraic structure
represented by all Dedekind-complete ordered fields.
You might wonder why mathematicians want to use such a complicated
definition. Wouldn't it be easier to simply define the real numbers to be
the Dedekind cuts, or define the real numbers to be the decimal expansions,
or something like that? That is the approach taken in some elementary
textbooks, but ultimately it is less productive. When we actually use the
real number system in proofs, the properties that we need are not
specifically the properties of (for instance) Dedekind cuts or of decimal
expansions. Rather, the properties we need are the axioms of a Dedekind
complete ordered field. It is much simpler to think in terms of those
axioms. To think of "numbers" as being cuts or expansions would just
encumber us with extra baggage. The cuts or expansions are models -- they
are useful for the job proving Theorem 1, but they are useful for little
else. Once they've done that job, we can discard and forget them.
------------------------------------------------------------------------------------------(end
of the excerpts)
The only thing I want to add as a formalist that "unique algebraic structure
represented by all Dedekind-complete ordered fields" can be naturally
represented by the theory T of Dedekind-complete ordered fields. This
theory T in my formalism (see http://home.nyu.edu/~yvm204/vm/DL-alg.htm )
can be defined as follows:
T := def[C,0,+,1,x,<; A]; (A is a conjunction of the axioms of the theory
T)
Because for formalists mathematics is just "playing with symbols" and every
mathematical concept is represented by some symbol, a formalist's answer on
the question "What the real numbers are?" can be as follows:
The concept "the real numbers" is represented by the name C
in the theory T of Dedekind-complete ordered fields.
The Theorems I and 2 from the Eric Schechter's web page can be stated as
follows:
Theorem 1. The theory T of Dedekind-complete ordered fields is consistent
(i.e. it has a model).
Theorem 2. The theory T is cathegorical (i.e. all models of T are
isomorphic).
Victor Makarov
http://home.nyu.edu/~yvm204/vm/vm.htm
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