[FOM] alternative foundations?

[MartDowd at aol.com]

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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