On 4/8/2013 2:46 AM, William Elliot wrote: > On Sun, 7 Apr 2013, fom wrote: > >>>> A proof >>> >>> Yes. Assume not GCH. Thus there's myriads of superfluously conceptual >>> infinite cardinals As this violates Occam's Razor, GCH, QED. >>> >>> Similarly Occam's Razor shows there's no inaccessible cardinals. >> >> I will take your statement as confirmation that you >> believe GCH to be true. It is, however, not a proof >> in the sense which had been intended. > > By Occam's Razor, GCH + no inaccessibles. > Thus AxC and some forcing arguments are vacuous. > >> :-) > > Remember the enginers' KISS and the beauty of simplicity. > What more simple than invoking Occam for V = L and no inaccessibles? > Face it, that's all the set theory needed for all of math.
Do you believe that?
What about Grothendieck universes arising from category theory?
MacLane, as a matter of parsimony, assumes one universe satisfying Tarski's axiom. Grothendieck assumes one such universe for every set.
> > BTW, Quine's NF denies AxC. >
I need to look at Quine's work more carefully at this point. I doubt I would like it because I do not agree with his views on the nature of identity.
But, it is interesting for other reasons.
The axiom of choice, however, may be construed as a necessary axiom for model theory. Models interpret the sign of equality as the diagonal of a Cartesian product. The statement of axiom of choice in terms of Cartesian products is a guarantee that models of the sign of equality will exist.