On 4/6/2013 3:51 PM, Virgil wrote: > In article > <email@example.com>, > WM <firstname.lastname@example.org> wrote: > >> On 5 Apr., 23:54, Virgil <vir...@ligriv.com> wrote: >> >>>> If not, how do you prove its finiteness? >>> >>> By finding its largest member. >> >> Find the largest line of the list >> >> 1 >> 1, 2 >> 1, 2, 3 >> ... >> >> that cannot be removed without changing the union of the remaining >> lines. >> >> Regards, WM > > What makes you think that there is such a line? > > As far as I can see removing any one line alone from the union of all > lines has no effect on the union. > > So which lines does WM claim satisfy his criterion? >
I did this analysis elsewhere.
The sense of his question is that all the lines satisfy the criterion. When all the lines are removed, he perceives a contradiction because the union over the empty set is not the initial union over all of the monotonic inclusive constructive sets of marks.
The problem with this reasoning is that a contradiction is defined in terms of truth and falsity of statements. The "property" one would use for this induction, however, does not change its truth value because the empty set cannot be diminished by removing lines. The inductive property becomes vacuously satisfied at the "completion" of the induction to the extent that one may extend mathematics to WMytheology to make such a statement.