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Topic: Matheology § 259
Replies: 9   Last Post: May 2, 2013 4:46 PM

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Posts: 18,076
Registered: 1/29/05
Matheology § 259
Posted: May 1, 2013 4:59 AM
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Matheology § 259

A discussion in sci.logic yielded the following remarkable results
with respect to a list having infinitely many lines
{1} U {2}
{1} U {2} U {3}
Each line contains as many unioned sets as its line number indicates
but does not contain a line |N, since each line has a finite last
number n.

On the other hand, there are infinitely many lines and, as each line
adds one number, there are infinitely many numbers in the list. Since,
by construction, every finite initial segment s_n = {1, 2, 3, ..., n}
is in one single line, all finite initial segments are in one single
line. But |N is not more than all its finite initial segments.

Otherwise there must exist at least two finite initial segments such
exist j, k, m, n : m e s_j & ~(m e s_k) & ~(n e s_j) & n e s_k.

Further, if all lines of the list are written within one single line,
say the first one, then |N is in the first line.

Further if the list is prepared such that (for n > 1) after adding
line s_n the preceding line s_n-1 is removed, then the list, again
consisting of a single line only, is empty.

Regards, WM

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