Applying the negation of the axiom of choice
Quantum mechanics
household names
About the very numerous
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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
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
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