First of all many thanks for the answer!
> If by "contain x" you mean "has x as a subset" then what you
> say is false (i give an example below).
> Here's an example of the definition:
> If 0 denotes the emptyset, consider
> S = {0, {0}, {0,{0}}}
> with the ordering relation x<y being "x is a proper subset
> of y".
> Then if x is an element of S, either
> 1) x=0
> 2) x={0},
> or
> 3) x={0,{0}}.
> In either case, you can easily check that the set {z in S:
> z<x} is exactly x. So in fact S is an ordinal (usuall known
> as 2 :)
So, I still don't get it :(.
1) take x=0, then everything is fine:
{z in S: z<x} is exactly x
2) take x={0}, why
{z in S: z<x} dosn't contain 0, and {0}???, since 0 IS a subset of
{0}
3) the same for x={0,{0}}.
Sorry, but perhaps I just don't know something :(((.
Another confusing thing is the following:
> with the ordering relation x<y being "x is a proper subset
> of y".
if you look at the entry "ordering relation", then you will see that there two notions are defined:
1) the relation "<=" which enables a<=a
2) and the relation "<" which doesn't enable a<a
since in this entry one opeates with notation "<", then it is indirectly assumed that a<a can't be, and thus your example
> "x is a proper subset
> of y".
is not an example of the ordering "<", but of "<=". |
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