[FOM] RE: FOM Friedman's Simplified Foundations
Matt Insall
montez at fidnet.com
Mon May 12 23:43:02 EDT 2003
Taylor:
(e) I don't understand the following at all - would someone please explain:
> One proves that every sentence is provably equivalent to a sentence
> that mentions only epsilon.
Insall:
This is what one has when one develops mathematical foundations from the
empty set. You see, if everything is a set and every concept is defined in
terms of sets, then every sentence can be replaced, in a very pedestrian
manner, by a sentence that uses only set membership to do everything. This
is somewhat like treating everything as an abbreviation for something more
complicated in a very simple language. For example, + is just an
abbreviation for a certain set whose members are triples of real numbers,
the first two of which sum to the third:
+ == {<x,y,z> | z is the sum of x and y}.
This enables us to express all of mathematics in terms of sets. But simple
statements like ``2+2=4'' become monstrously long and cumbersome when we do,
so we point out it can be done, and then do not usually do it.
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