Hello, Carl Futia wrote:
Is it known just how much mathematics can be done using Lawvere's axiom scheme? My confusion on this point arises partly from a remark by Gray in his paper "The categorical comprehension scheme" in which he asserted that Lawvere's axioms couldn't do all that was claimed for them. Of course all of this predates the recognition that an elementary topos (with nn object) can be treated as a "universe of sets".
As I see it, the problem is with the replacement scheme. Even if you start with ZF and classical logic, you have problems to express replacement in terms of first-order properties of the category of sets. An infinite cardinal a is called a strong limit cardinal, if b<a implies 2^a<b. For every strong limit cardinal a the category of all sets of cardinality < a is en elementary topos, regardless whether a is regular or not; regular strong lomit cardinals are inaccessible cardinals. Replacement should yield regularity of a, but how to express it? The easiest way to find a singular strong limit cardinal a is to start with an arbitrary infinite cardinal b(0), define b(n+1):=2^b(n), and a the sum of al b(n) for all natural number n. Then a is obviously singular because it is uncountable but the sum of countably many strictly smaller cardinals; a is a strong limit cardial because every cardinal <a is <b(n) for some n. Greetings Reinhard 12-Mar-2005 19:27:06 -0400,7017;000000000001-00000000