Remarkable cardinal
Remarkable cardinals were introduced by Schinder in
(Schindler, 2000) to provide precise
consistency strength of the statement that cannot be
modified by proper forcing.
Definitions
A cardinal is remarkable if for each regular
, there exists a countable transitive and an
elementary embedding with and also a countable transitive and an elementary
embedding such that:
- the critical point of is ,
- is a regular cardinal in ,
- ,
- .
Remarkable cardinals could be called virtually
supercompact,
because the following alternative definition is an exact analogue of the
definition of supercompact cardinals by Magidor [Mag71]:
A cardinal is remarkable iff for every , there is such that in a set-forcing extension there is an elementary embedding
with .(Gitman & Shindler, n.d.)
Equivalently (theorem
2.4(Bagaria et al., 2017))
- For every and every , there is such
that in there is an elementary embedding with and .
- For every in and every , there is also in such that in there is
an elementary embedding with and
.
- There is a proper class of such that for every in the
class, there is such that in there is
an elementary embedding with
Note: the existence of any such elementary embedding in
is equivalent to the existence of such elementary
embedding in any forcing extension (see
Elementary_embedding#Absoluteness).(Bagaria et al., 2017).
Results
Remarkable cardinals and the constructible universe:
Relations with other large cardinals:
-
Strong
cardinals are remarkable.
(Schindler, 2000)
- A
-iterable
cardinal implies the consistency of a remarkable cardinal: Every
-iterable cardinal is a limit of remarkable cardinals.
(Gitman & Welch, 2011)
- Remarkable cardinals imply the consistency of
-iterable cardinals:
If there is a remarkable cardinal, then there is a countable
transitive model of ZFC with a proper class of -iterable
cardinals. (Gitman & Welch, 2011)
- Remarkable cardinals are
totally indescribable.
(Schindler, 2000)
- Remarkable cardinals are totally
ineffable.
(Schindler, 2000)
-
Virtually
extendible
cardinals are remarkable limits of remarkable
cardinals.(Gitman & Shindler, n.d.)
- If is virtually
measurable,
then either is remarkable in or .(Nielsen & Welch, 2018)
- Remarkable cardinals are strategic
-Ramsey
limits of -Ramsey
cardinals.(Nielsen & Welch, 2018)
- Remarkable cardinals are
-reflecting.(Wilson, 2018)
Equiconsistency with the weak Proper Forcing
Axiom:(Bagaria et al., 2017)
- If there is a remarkable cardinal, then holds in a
forcing extension by a proper poset.
- If holds, then is remarkable in .
(this section from
(Wilson, 2018))
A cardinal is weakly remarkable iff for every , there is
such that in a set-forcing extension there is an elementary
embedding with . (the
condition is dropped)
A cardinal is remarkable iff it is weakly remarkable and
-reflecting.
The existence of non-remarkable weakly remarkable cardinals is
equiconsistent to the existence of an
-Erdős
cardinal (equivalent assuming ; Baumgartner definition of -Erdős
cardinals):
- Every -Erdős cardinal is a limit of non-remarkable weakly
remarkable cardinals.
- If is a non-remarkable weakly remarkable cardinal, then some
ordinal greater than is an -Erdős cardinal in .
-remarkability is equivalent to remarkability. A cardinal is
virtually -extendible iff it is -remarkable (virtually
extendible cardinals are virtually -extendible). A cardinal is
called completely remarkable iff it is -remarkable for all . Other definitions and properties in Extendible#Virtually
extendible
cardinals.(Bagaria et al., 2017)
References
- Schindler, R.-D. (2000). Proper forcing and remarkable cardinals. Bull. Symbolic Logic, 6(2), 176–184. https://doi.org/10.2307/421205
- Gitman, V., & Shindler, R. Virtual large cardinals. https://ivv5hpp.uni-muenster.de/u/rds/virtualLargeCardinalsEdited5.pdf
- Bagaria, J., Gitman, V., & Schindler, R. (2017). Generic Vopěnkaś Principle, remarkable cardinals, and the weak Proper Forcing Axiom. Arch. Math. Logic, 56(1-2), 1–20. https://doi.org/10.1007/s00153-016-0511-x
- Gitman, V., & Welch, P. (2011). Ramsey-like cardinals II. J. Symbolic Logic, 76(2), 541–560. http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinalsii.pdf
- Nielsen, D. S., & Welch, P. (2018). Games and Ramsey-like cardinals.
- Wilson, T. M. (2018). Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle.
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