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Re: E. T. Bell, Adolf Fraenkel and Ordered Pairs
Posted:
May 12, 2007 1:31 PM
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On p. 36, Bell wrote, "The human element is there whether we agree to
recognize it or not." I think it's easier to deal with the question if you
consider it possible to banish the human element: to establish
communication between machines. That is more familiar now than then. By
this means we can more readily concentrate on one aspect at a time of such
problems.
In programming, it's generally easier to specify ordered finite sets than
unordered. The difference is not the data stored, but the data structure
used to store them. A structure for an ordered finite set can make use of
the physical order of a machine's memory. A structure for unordered finite
sets is generally more complicated: actually, it's the equality relation, a
component of that data structure, that's more complicated.
We use data structures in everyday discourse as well as information
technology. For example, we might question how observers on opposite sides
of a printed transparency might distinguish the ordered pair (I , T) from
(T , I). Answer: the difference is in the data structure (.. , ..) : the
comma points to the first entry. That may be why we don't use (.. . ..).
How about human speech? It has a physical order, too: time. Fraenkel
noted that.
I'd say that Bell was confused because he was trained from infancy to ignore
the time structure (e.g., to adeptly reshuffle things when he changes his
mind). He thought that an unordered structure was inborn.
It might be held that our standard axiomatic set theory recognizes
"unordered set" as more fundamental; hence it needs to define "ordered
pair" in terms of "unordered pair", as Fraenkel noted. But that would be a
little off, because the fundamental notion in our standard axiomatic set
theory is "membership in an unordered set". I wonder, is there a formalized
theory of "membership in an ordered finite set"?
--------------------------------------
James T. Smith
Professor of Mathematics
San Francisco State University
smith@math.sfsu.edu
<http://math.sfsu.edu/smith> http://math.sfsu.edu/smith
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