About the very numerous Mr Andreas Blass was kind to clarify the following about alephs: The alephs are a proper class. The alephs are indexed by ordinals such as omega, omega+1,... 2 omega,... Alephs are cardinalities of well orderable sets. Let us consider the ordinal corresponding to the cardinality of the set of real numbers, with the notation c. Let us consider the aleph indexed by c, the alephs smaller than that one would be as numerous as the real numbers. One should start to feel some dizziness from the very numerous. Let us consider the ordinal corresponding to the cardinality of the set of subsets of the set of real numbers, with the notation s. Let us consider the aleph indexed by s, the alephs smaller than that one would be more numerous than the real numbers. Aleph indexed by s is itself big beyond imagination. It is difficult to get an intuition of it. Such an aleph appears almost never in mathematical texts. Physical space could not have the dimension of such an aleph (I explained in "Physical space is infinite", in ASL Winter Meeting 2006 2007, why physical space is infinite). So, for what could be used a very big aleph ? May be the number of Big Bangs, each associated with a previous Big Crunch, is infinite and a very big aleph, as it probably never ends (although it takes an infinite time for a Big Crunch to happen). Indeed, it is in the fields of cosmology and cosmogony that we can hope to find use for the very big alephs. I think there was no beginning for the cosmos, it was always there and that is why very big cardinals should be used to study the cosmos. Adib Ben Jebara
About urelements and astronomical black holes In "About space and time in quantum mechanics" of ASL Winter Meeting 2007 2008, it is not clear at what time will happen the Big Crunch. It is, since the Big Bang, at time p, p being the Dedekind cardinal cardinality of U set of urelements. ZFU, set theory with urelements can be applied to space and time which are U1xU2. Let us assume that there are no urelements where matter is, in the astronomical black holes. It means that space and time are not well defined there. It enables us to avoid an infinite density and paradoxes for time. The assumption is of some importance as black holes may be the missing mass to explain gravitational forces in a galaxy. Mr Andreas Blass commented that if the missing mass is known to be rather far from the centers of the galaxies, may be there are black holes there, more difficult to detect. Adib Ben Jebara.
About research in mathematics and philosophy of mathematicsMathematics are giving shape to the physical world. To reduce mathematics to logic is like reducing physics to mathematics. Mathematics are a way to contemplate Good which is a way to contemplate God. The evolution of the philosophy of mathematics is slow, for instance platonism is always here. Either set theory will be increasingly used in physics and other sciences or it will remain just a part of mathematics. What will happen is probably something in the middle with isolated researchers finding out how to apply it. There will be more set theory taught and less logic as set theory will show even more its usefulness in the remaining parts of mathematics. Platonism will be more spread because less dogmatic about what exists. Nowadays, research in mathematics and in philosophy of mathematics make complicated things even more so. This occurs as a lack of creativity. Some are fatalist about it, thinking that the great theories are already made. The future will contradict them as new theories will appear. The new theories are not always accepted at once. Adib Ben Jebara.
