[FOM] alternative foundations?
MartDowd at aol.com
MartDowd at aol.com
Mon Mar 24 14:18:01 EDT 2014
The quote from Godel included below prompted me to consider the distinction
between adding higher types to (an extension of) ZFC, and collecting the
universe, so that higher types above the original universe are sets of rank
a small amount greater than that of the original universe. One resolution
is to note that "the universe" in this context has not been fully
collected. Although one might believe in a fully collected universe, discussions of
an axiom system for it must always allow for extending it. The
prototypical example is to add the existence of inaccessible cardinals rather than
higher types. Of course, one can then add higher types, but this is subsumed
(at least in terms of giving properties that assure that the universe has
been collected to a given extent) by assuming the existence of
1-inaccessible cardinals.
All this can be seen as a methodology for considering the endlessness of
the cumulative hierarchy in connection with the non-recursive enumerability
of truth.
- Martin Dowd
In a message dated 3/22/2014 5:05:29 P.M. Pacific Daylight Time,
martin at eipye.com writes:
Now if the system under consideration (call it S) is based on the theory
of types, it turns out that ... this proposition becomes a provable theorem
if you add to S the next higher type and the axioms concerning it. … the
construction of higher and higher types … is necessary for proving theorems
even of a relatively simple structure."
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