Huge cardinal
Huge cardinals (and their variants) were introduced by Kenneth Kunen
in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to
prove that the consistency of the existence of a huge cardinal implies
the consistency of +”there is a -saturated
-ideal
on ”. It is now known that only a
Woodin
cardinal is needed for this result. However, the consistency of the
existence of an -complete -saturated
-ideal on , as far as the set theory world is
concerned, still requires an almost huge cardinal.
(Kanamori, 2009)
Definitions
Their formulation is similar to that of the formulation of
superstrong
cardinals. A huge cardinal is to a
supercompact
cardinal as a superstrong cardinal is to a
strong
cardinal, more precisely. The definition is part of a generalized
phenomenon known as the “double helix”, in which for some large cardinal
properties n- and n-, n- has less consistency strength
than n-, which has less consistency strength than (n+1)-,
and so on. This phenomenon is seen only around the n-fold
variants
as of modern set theoretic concerns.
(Kentaro, 2007)
Although they are very large, there is a first-order definition which is
equivalent to n-hugeness, so the -th n-huge cardinal is
first-order definable whenever is first-order definable. This
definition can be seen as a (very strong) strengthening of the
first-order definition of
measurability.
Elementary embedding definitions
The elementary embedding definitions are somewhat standard. Let
be a nontrivial elementary
embedding
of into a
transitive
class with critical point . Then:
- is almost n-huge with target iff
and is closed under all of its sequences
of length less than (that is, ).
- is n-huge with target iff
and is closed under all of its sequences
of length ().
- is almost n-huge iff it is almost n-huge with target
for some .
- is n-huge iff it is n-huge with target for
some .
- is super almost n-huge iff for every , there
is some for which is almost n-huge
with target (that is, the target can be made arbitrarily
large).
- is super n-huge iff for every , there is some
for which is n-huge with target
.
- is almost huge, huge, super almost huge, and
superhuge iff it is almost 1-huge, 1-huge, etc.
respectively.
Ultrahuge cardinals
A cardinal is -ultrahuge for
if there exists a nontrivial elementary embedding
for some transitive class such that
, and
. A cardinal is ultrahuge if it is
-ultrahuge for all .
[1]
Notice how similar this definition is to the alternative
characterization of
extendible
cardinals. Furthermore, this definition can be extended in the obvious
way to define -ultra n-hugeness and ultra n-hugeness, as well
as the “almost” variants.
Hyperhuge cardinals
A cardinal is -hyperhuge for
if there exists a nontrivial elementary embedding
for some inner model such that , and . A
cardinal is hyperhuge if it is -hyperhuge for all
. (Usuba, 2017; Boney, 2017)
Huge* cardinals
A cardinal is -huge* if for some , is
the critical point of an elementary embedding such
that .(Gitman & Shindler, n.d.)
Hugeness* variant is formulated in a way allowing for a virtual variant
consistent with : A cardinal is virtually -huge* if for
some , in a set-forcing extension, is the critical
point of an elementary embedding such that .(Gitman & Shindler, n.d.)
Ultrafilter definition
The first-order definition of n-huge is somewhat similar to
measurability.
Specifically, is measurable iff there is a nonprincipal
-complete
ultrafilter,
, over . A cardinal is n-huge with target
iff there is a normal -complete ultrafilter, ,
over , and cardinals
such that:
Where is the
order-type
of the poset . (Kanamori, 2009)
is then super n-huge if for all ordinals there is a
such that is n-huge with target
, i.e. can be made arbitrarily large. If
is such that (i.e.
witnesses n-hugeness) then there is a ultrafilter as above such
that, for all , , i.e. it is not
only that is an iterate of by ; all
members of the sequence are.
As an example, is 1-huge with target iff there is a
normal -complete ultrafilter, , over
such that
. The
reason why this would be so surprising is that every set
with every set of order-type would be in
the ultrafilter; that is, every set containing
as a subset is
considered a “large set.”
As for hyperhugeness, the following are
equivalent:(Boney, 2017)
- is -hyperhuge;
- and a normal, fine, κ-complete ultrafilter exists on
;
- is --compact for type
omission.
Coherent sequence characterization of almost hugeness
--huge cardinals
(this section from (Bagaria, 2012))
is
--huge
iff it is -huge and (-huge if it is huge
and ).
Equivalent definition in terms of normal measures: κ is
--huge iff it is uncountable and there is a -complete
normal
ultrafilter
over some and cardinals , with and such that for each , .
It follows that “ is --huge” is expressible.
Every huge cardinal is -huge.
The first --huge cardinal is not --huge, for
all and greater or equal than . For suppose is the least
--huge cardinal and witnesses that is
--huge. Then since “x is --huge” is
expressible, we have “ is --huge”.
Hence, since , “ “δ is huge””. By elementarity, there is a
--huge cardinal less than in – contradiction.
Assuming ,
if is a limit cardinal (instead of a successor of a limit cardinal –
Kunen’s Theorem excludes other cases), it is equal to where is the elementary embedding. Then and are
--huge (inter alia) in , for all and .
If is -, then it is --huge, for
all , and there is a normal ultrafilter over such
that
is --huge for every .
Consistency strength and size
Hugeness exhibits a phenomenon associated with similarly defined large
cardinals (the n-fold
variants)
known as the double helix. This phenomenon is when for one n-fold
variant, letting a cardinal be called n- iff it has the property,
and another variant, n-, n- is weaker than n-, which
is weaker than (n+1)-. (Kentaro, 2007) In
the consistency strength hierarchy, here is where these lay (top being
weakest):
-
measurable =
0-superstrong =
0-huge
- n-superstrong
- n-fold supercompact
- (n+1)-fold strong, n-fold extendible
- (n+1)-fold Woodin, n-fold Vopěnka
- (n+1)-fold Shelah
- almost n-huge
- super almost n-huge
- n-huge
- super n-huge
- ultra n-huge
- (n+1)-superstrong
All huge variants lay at the top of the double helix restricted to some
natural
number n,
although each are bested by
I3
cardinals (the critical
points
of the I3 elementary embeddings). In fact, every I3 is preceeded by a
stationary set of n-huge cardinals, for all n.
(Kanamori, 2009)
Similarly, every huge cardinal is almost huge, and there is a
normal measure over which contains every almost huge cardinal
. Every superhuge cardinal is
extendible
and there is a normal measure over which contains every
extendible cardinal . Every (n+1)-huge cardinal
has a normal measure which contains every cardinal
such that ” is super n-huge”
(Kanamori, 2009), in fact it contains every
cardinal such that ” is ultra
n-huge”.
Every n-huge cardinal is m-huge for every m<n. Similarly with almost
n-hugeness, super n-hugeness, and super almost n-hugeness. Every almost
huge cardinal is
Vopěnka
(therefore the consistency of the existence of an almost-huge cardinal
implies the consistency of Vopěnka’s principle).
(Kanamori, 2009) Every ultra n-huge is super
n-huge and a stationary limit of super n-huge cardinals. Every super
almost (n+1)-huge is ultra n-huge and a stationary limit of ultra n-huge
cardinals.
In terms of size, however, the least n-huge cardinal is smaller than the
least
supercompact
cardinal (assuming both exist).
(Kanamori, 2009) This is because n-huge
cardinals have upward reflection properties, while supercompacts have
downward reflection properties. Thus for any which is
supercompact and has an n-huge cardinal above it, “reflects
downward” that n-huge cardinal: there are -many n-huge
cardinals below . On the other hand, the least super n-huge
cardinals have both upward and downward reflection properties, and are
all much larger than the least supercompact cardinal. It is notable
that, while almost 2-huge cardinals have higher consistency strength
than superhuge cardinals, the least almost 2-huge is much smaller than
the least super almost huge.
While not every -huge cardinal is
strong,
if is almost -huge with targets
, then is
-strong as witnessed by the generated . This
is because is
measurable
and therefore and so
and because
, for each
and so
.
Every almost -huge cardinal with targets
is also
-supercompact
for each , and every -huge cardinal with
targets is also
-supercompact.
For -huge , is a model of +“there are proper
class many hyper-huge
cardinals”.(Usuba, 2017) Hyper-huge
cardinals are extendible limits of extendible
cardinals.(Usuba, 2019)
An -huge* cardinal is an -huge limit of -huge cardinals. Every
-huge cardinal is
-huge*.(Gitman & Shindler, n.d.)
As for virtually
-huge*:(Gitman & Shindler, n.d.)
- If is virtually huge*, then is a model of proper class
many virtually
extendible
cardinals.
- A virtually -huge* cardinal is a limit of virtually -huge*
cardinals.
- A virtually -huge* cardinal is an
-iterable
limit of -iterable cardinals. If is -iterable, then
is a model of proper class many virtually -huge*
cardinals.
- Every
virtually rank-into-rank
cardinal is a virtually -huge* limit of virtually -huge*
cardinals for every .
The -huge cardinals
A cardinal is almost -huge iff there is some
transitive model and an elementary embedding with
critical point such that where
is the smallest cardinal above such that
. Similarly, is -huge iff
the model can be required to have .
Sadly, -huge cardinals are inconsistent with ZFC by a version
of Kunen’s inconsistency theorem. Now, -hugeness is used to
describe critical points of
I1 embeddings.
Relative consistency results
Hugeness of
In
[2]
it is shown that if “there is a huge cardinal” is
consistent then so is “ is a huge cardinal”
(with the ultrafilter characterization of hugeness).
Generalizations of Chang’s conjecture
Cardinal arithmetic in
If there is an almost huge cardinal then there is a model of
in which every successor cardinal is a
Ramsey
cardinal. It follows that (1) for all inner models of
and every singular cardinal , one has and that (2) for all ordinal there is no injection
. This in turn imply the
failure of the
square principle
at every infinite cardinal (and consequently
, see
determinacy).
[3]
In set theoretic geology
If is hyperhuge, then has many
grounds
(so the
mantle
is a ground itself).(Usuba, 2017) This
result has been strenghtened to
extendible
cardinals(Usuba, 2019). On
the other hand, it s consistent that there is a
supercompact
cardinal and class many grounds of (because of the indestructibility
properties of
supercompactness).(Usuba, 2017)
References
- Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3
- Kentaro, S. (2007). Double helix in large large cardinals and iteration of elementary embeddings. Annals of Pure and Applied Logic, 146(2-3), 199–236. https://doi.org/10.1016/j.apal.2007.02.003
- Usuba, T. (2017). The downward directed grounds hypothesis and very large cardinals. Journal of Mathematical Logic, 17(02), 1750009. https://doi.org/10.1142/S021906131750009X
- Boney, W. (2017). Model Theoretic Characterizations of Large Cardinals.
- Gitman, V., & Shindler, R. Virtual large cardinals. https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf
- Bagaria, J. (2012). -cardinals. Archive for Mathematical Logic, 51(3–4), 213–240. https://doi.org/10.1007/s00153-011-0261-8
- Usuba, T. (2019). Extendible cardinals and the mantle. Archive for Mathematical Logic, 58(1-2), 71–75. https://doi.org/10.1007/s00153-018-0625-4
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