Re: Rudin and Dedekind cuts

Mathman1271 wrote:

But surely there is a fundamental difference. To define
rationals we can start with the Peano axioms for integers
and then define rationals as ordered pairs of integers.
The idea of a 'number' is very clear, there is no ambiguity.

With Dedekind cuts, there IS ambiguity.

The Peano axioms are for the natural numbers -- integers
greater than or equal to zero. From the natural numbers
the integers can be defined as certain equivalence classes
of ordered pairs of natural numbers (the idea is that -6, say,
can be thought of as 0 - 6 <---> (0,6), or 1 - 7 <---> (1,7),
or 2 - 8 <---> (2,8), etc.). With this construction, each
integer (including those that are the non-negative integers,
which wind up being a "copy" of the natural numbers in the
integers; for example, the natural number 8 winds up being
the integer {(8,0), (9,1), (10,2), (11,3), (12,4), ...})
is an infinite set of certain objects, namely ordered pairs
of natural numbers.

A similar situation occurs in the standard way of constructing
the rational numbers from the integers. Each rational number
winds up being an infinite set of ordered pairs of integers.
Thus, each rational number is an infinite set of elements,
each element of which is an infinite set of ordered pairs of
natural numbers.

With the real numbers, a similar situation occurs, although
this time ordered pairs of the previous objects are not used.
Another distinction that occurs with the real number constructions
is that each equivalence class of objects that defines a real
number is an uncountable infinite set rather than a countable
infinite set.

Dave L. Renfro
.

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