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Topic: The Limit Concept in MatheRealism
Replies: 22   Last Post: Jun 21, 2014 7:48 AM

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 JÃÂ¼rgen R. Posts: 1,083 Registered: 12/13/04
The Limit Concept in MatheRealism
Posted: Jun 18, 2014 12:34 PM

Recently Mueckenheim made the following pronouncement:

"Contrary to the correct understanding of lim in mathematics,
Cantor and set theory understand the limit set as the collection
of "all" its elements, not as a limit that is only approached.
(The latter would never prove that all elements are collected,
or all rationals would be enumerated and the like.)
This is usually confused by set theorists, often deliberately."

As usual there are numerous murkily defined terms in this
statement, making it incomprehensible, most prominently
"the correct understanding of lim in mathematics".

I think the time has come for Mueckenheim to let everybody
know what this hitherto secret "correct understanding of lim" is.

Meanwhile, just in case WM has forgotten, here is the only
definition that is used in mathematics:

Let {x_n} be a sequence of elements of a topological space S.
Then {x_n} converges to the limit x in S if for every open set
G in S, with x in G, there is an index m s.t. n > m implies
x_n in G.

*Every* limit process in current mathematics is reducible
to this definition.

A limit therefore is never determined by
the sequence alone, but rather by the sequence,
the space S and the particular topology in S.