On 2010-11-06, herbzet <email@example.com> wrote: > I guess the problem people are having with my thesis is that they > are willing to accept (a) the mathematical existence of S, similar > to that of the object 6, and they are willing to accept (b) the > mathematical existence of a bijection f from w_1 to P(N), similar > to that of the object 6, but they are not willing to accept both > (a) and (b), because the object 6 is special -- it really and truly > exists in some sense
I'm not sure I believe that N "really exists" in the same sense as the number 6, let alone objects defined in terms of functions involving the power set of N.
Even if one accepted that N "really exists", does it really have a power set? If you accept that all objects definable in any conceivable consistent mathematical system exist in some sense, then P(N) does exist in that sense. However almost all of its elements don't and can't. Is it reasonable for a set to "really exist" in some sense if it has members that don't exist in that sense?
At some point, most of math seems to me a game of rules and inferences. One with a very surprising structure and practical usefulness, which perhaps explains why it seems to have a stronger sense of existence than arbitrary babble.