[FOM] RE: FOM Friedman's Simplified Foundations

Matt Insall montez at fidnet.com
Mon May 12 23:43:02 EDT 2003


(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.

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.

More information about the FOM mailing list