Cantor's first set theory article: Difference between revisions

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Suppose a set '''R''' is

*#[[total order|linearly ordered]], and

*#contains at least two members, and
#is densely ordered, i.e., between any two members there is another, and

*has#is no[[complete gapslattice|complete]], i.e., if it is [[partition of a set|partitioned]] into two nonempty sets ''A'' and ''B'' in such a way that every member of ''A'' is less than every member of ''B'', then there is a boundary point ''c'' (in '''R'''), so that every point less than ''c'' is in ''A'' and every point greater than ''c'' is in ''B''.

*has no "endpoints", i.e., smallest or largest members, and

*has no gaps, i.e., if it is partitioned into two nonempty sets ''A'' and ''B'' in such a way that every member of ''A'' is less than every member of ''B'', then there is a boundary point ''c'' (in '''R'''), so that every point less than ''c'' is in ''A'' and every point greater than ''c'' is in ''B''.

Then '''R''' is not [[countable]].

The set of [[real numbers]] with its usual ordering is a typical example of such an ordered set; in this setting the 'gaplessness' property is known as [[complete space|completeness]]. The set of [[rational numbers]] (which ''is'' countable) has properties 1-3 but does not have this property 4.

===The proof===

The proof begins by assuming some [[sequence]] ''x''₁, ''x''₂, ''x''₃, ... has all of '''R''' as its range. Define two other sequences (''a''_''n'') and (''b''_''n'') as follows:

:Pick ''a''₁ =< ''xb''₁ in '''R''' (possible because of property 2).

:''b''₁ =Let ''xa''_''in''+1, wherebe the first element in the sequence ''ix'' which is the smallest index such thatbetween ''xa''_''in'' is not equal toand ''ab''_1''n'' (possible because of property 3).

:Let ''ab''_''n''+1 = ''x''_''i'', where ''i'' is the smallest index ''greater thanbe the onefirst consideredelement in the previous step'' such thatsequence ''x''_''i'' which is between ''a''_''n''+1 and ''b''_''n''.

The two [[monotonic function|monotone sequences]] ''a'' and ''b'' move toward each other. By the "gaplessness" of '''R''', some point ''c'' must lie between them. (Define ''A'' to be the set of all elements in '''R''' that are smaller than some member of the sequence ''a'', and let ''B'' be the [[complement]] of ''A''; then every member of ''A'' is smaller than every member of ''B'', and so property 4 yields the point ''c''.) The claim is that ''c'' cannot be in the range of the sequence ''x'', and that is the contradiction. If ''c'' were in the range, then we would have ''c'' = ''x''_''i'' for some index ''i''. But then, when that index was reached in the process of defining ''a'' and ''b'', then ''c'' would have been added as the next member of one or the other of those two sequences, contrary to the assumption that it lies between their ranges.

:''b''_''n''+1 = ''x''_''i'', where ''i'' is the smallest index ''greater than the one considered in the previous step'' such that ''x''_''i'' is between ''a''_''n''+1 and ''b''_''n''.

The two monotone sequences ''a'' and ''b'' move toward each other. By the "gaplessness" of '''R''', some point ''c'' must lie between them. The claim is that ''c'' cannot be in the range of the sequence ''x'', and that is the contradiction. If ''c'' were in the range, then we would have ''c'' = ''x''_''i'' for some index ''i''. But then, when that index was reached in the process of defining ''a'' and ''b'', then ''c'' would have been added as the next member of one or the other of those two sequences, contrary to the assumption that it lies between their ranges.

==Real algebraic numbers and real transcendental numbers==