limit cardinals, inaccessible cardinals and other monstruous things ...
David Sanchez Molina
i. A cardinal aleph-a whose index is a limit
ordinal is a limit cardinal (e.g. aleph-0,
aleph-omega, ...).
ii. Cofinality of A: cf (A) = the least limit
ordinal B such that there is an increasing
B-sequence whose limit is precisely A.
iii. A infinite !a cardinal is regular if cf
(aleph-a) = aleph-a, and it is singular if
cf(aleph-a) < aleph-a.
iv. An uncountable cardinal is weakly inaccessible
if it is a limit cardinal and is regular.
v. An uncountable cardinal is inaccessible
(strongly) if it is regular an is a strong limit.
... and I hope someone can answer some of
following questions:
1) Can you give another characterisation of limit
cardinals?
2) Can someone explain in deep the meaning of
cofinality?
3) Can someone enumerate some singular cardinal,
and explain his construction?
4) Is aleph-omega an weakly inaccessible cardinal?
5) Why if Generalised Continuum Hypothesis holds
every weakly inaccessible cardinal is also
inaccesible?
... and finaly:
6) Why existence of inaccesible cardinals is not
provable in Zermelo-Fraenkel + AC system, and
moreover why it cannot be shown that existence of
inaccessible cardinals is consistent with other
Zermelo-Fraenkel axioms ?
Click on the attachment to see the same message on
other format !
Richard Carr
:Date: Wed, 05 Apr 2000 12:22:59 +0200
:From: David Sanchez Molina <david.sanc...@upc.es>
:Newsgroups: es.ciencia.matematicas, sci.math
:Subject: limit cardinals,
: inaccessible cardinals and other monstruous things ...
:
:I'll give some definitions ...
:
:i. A cardinal aleph-a whose index is a limit
:ordinal is a limit cardinal (e.g. aleph-0,
:aleph-omega, ...).
:ii. Cofinality of A: cf (A) = the least limit
:ordinal B such that there is an increasing
:B-sequence whose limit is precisely A.
You can also define cofinality for successor ordinals to be 1 and for 0 to
be 0. (It can also be done for other orders.)
:iii. A infinite !a cardinal is regular if cf
:(aleph-a) = aleph-a, and it is singular if
:cf(aleph-a) < aleph-a.
:iv. An uncountable cardinal is weakly inaccessible
:if it is a limit cardinal and is regular.
:v. An uncountable cardinal is inaccessible
:(strongly) if it is regular an is a strong limit.
You didn't define strong limit but I'll assume you know what it means.
:
:... and I hope someone can answer some of
:following questions:
:
:1) Can you give another characterisation of limit
:cardinals?
The one you gave is pretty much standard. If you assume GCH then they are
the same as the strong limit cardinals.
:2) Can someone explain in deep the meaning of
:cofinality?
Another way to look at it is this: if cf(A)>=aleph_0 then A cannot be
written as a union of fewer than cf(A) sets all having cardinality less
than card(A). Thus cf(aleph_1)=aleph_1 as a countable union of countable
sets is countable (with ZFC). cf(A) is always a regular cardinal-
i.e. cf(cf(A))=cf(A).
:3) Can someone enumerate some singular cardinal,
:and explain his construction?
aleph_omega is the union of aleph_n for n<omega. Each set has size less
than aleph_omega and there are less than aleph_omega sets, so aleph_omega
is singular. The sequence you can use is f:omega->aleph_omega
f(n)=aleph_n. The supremum of f is aleph_omega but omega<aleph_omega so
aleph_omega is singular.
:4) Is aleph-omega an weakly inaccessible cardinal?
:
No, it isn't regular- see 3.
:5) Why if Generalised Continuum Hypothesis holds
:every weakly inaccessible cardinal is also
:inaccesible?
:
In this situation the limit cardinals and the strong limit cardinals are
the same thing (if A<B then 2^A=A^+<B too as B is a limit cardinal).
:... and finaly:
:
:6) Why existence of inaccesible cardinals is not
:provable in Zermelo-Fraenkel + AC system, and
:moreover why it cannot be shown that existence of
:inaccessible cardinals is consistent with other
:Zermelo-Fraenkel axioms ?
:
If ZFC|- exists an inaccessible cardinal then let kappa be the least
inaccessible cardinal. V_kappa is then a model of ZFC+ there is no
inaccessible cardinal, a contradiction.
I haven't time to answer the other bit as I have to be somewhere in 8
minutes- basically you could get ZFC|-Con(ZFC) in contradiction to GIT as
V_kappa in the appropriate model would be a set model of ZFC.
:Click on the attachment to see the same message on
:other format !
:
Miguel A. Lerma
:
: I'll give some definitions ...
:
: i. A cardinal aleph-a whose index is a limit
: ordinal is a limit cardinal (e.g. aleph-0,
: aleph-omega, ...).
: ii. Cofinality of A: cf (A) = the least limit
: ordinal B such that there is an increasing
: B-sequence whose limit is precisely A.
: (aleph-a) = aleph-a, and it is singular if
: cf(aleph-a) < aleph-a.
: iv. An uncountable cardinal is weakly inaccessible
: if it is a limit cardinal and is regular.
: v. An uncountable cardinal is inaccessible
: (strongly) if it is regular an is a strong limit.
: ... and I hope someone can answer some of
: following questions:
:
: 1) Can you give another characterisation of limit
: cardinals?
The one you gave is good enough.
: 2) Can someone explain in deep the meaning of
: cofinality
Cofinality of a cardinal is just the minimum "number"
of steps that you have to climb through the ordinal
ladder to reach the given cardinal. Paradoxically
some larger cardinals such as Aleph-omega have lower
cofinality than some smaller cardinals such as
Aleph-1, because you can reach them by stepping on
a small set of intermediate cardinals.
: 3) Can someone enumerate some singular cardinal,
: and explain his construction?
Aleph-omega is the limit of the countable sequence
Aleph-0, Aleph-1, Aleph-2,..., so its cofinality
is Aleph-0, but Aleph-omega is clearly uncountable.
: 4) Is aleph-omega an weakly inaccessible cardinal?
No, because it is singular, not regular.
: 5) Why if Generalised Continuum Hypothesis holds
: every weakly inaccessible cardinal is also
: inaccesible?
Because if the Generalised Continuum Hypothesis holds
then the power set operation cannot take you beyond
the "next" cardinal, so it does not provide you with
any additional advantage respect to the "successor
cardinal" operation.
: 6) Why existence of inaccesible cardinals is not
: provable in Zermelo-Fraenkel + AC system, and
: moreover why it cannot be shown that existence of
: inaccessible cardinals is consistent with other
: Zermelo-Fraenkel axioms ?
This is a rather technical matter, but there is an
intuitive explanation. An inaccessible cardinal is
closed respect to the set-building operations (union,
power set...), so it provides a model for ZF. By the
completeness theorem, a first-order theory is consistent
iff has a model, so if we were able to prove in ZF that
there is an inaccessible cardinal, then we would have
proven that ZF is consistent. But as a consequence of
Goedel's incompleteness theorem, if ZF is consistent then
we cannot prove in ZF that ZF is consistent. So we cannot
prove in ZF that there is an inaccessible cardinal because
that would amount to a proof for the consistency of ZF.
Miguel A. Lerma
denis-feldmann
Miguel A. Lerma <le...@math.nwu.edu> a écrit dans le message :
8cfj2t$t2b$2...@news.acns.nwu.edu...
inaccessible cardinal) is strictly stronger than Consis ZF, for essentially
the same reason.
>
> Miguel A. Lerma
>