Block matrix pseudoinverse

In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. Template:Harvtxt (extending a result of Robert M. Solovay) showed via forcing that

$\kappa <\operatorname {cf} (2^{\kappa })\,$

and, for $\kappa <\lambda \,$, that

$2^{\kappa }\leq 2^{\lambda }\,$

are the only constraints on permissible values for 2^κ when κ is a regular cardinal.

Statement of the theorem

Easton's theorem states that if G is a class function whose domain consists of ordinals and whose range consists of ordinals such that

1. G is non-decreasing,
2. the cofinality of $\aleph _{G(\alpha )}$ is greater than $\aleph _{\alpha }$ for each α in the domain of G, and
3. $\aleph _{\alpha }$ is regular for each α in the domain of G,

then there is a model of ZFC such that

$2^{\aleph _{\alpha }}=\aleph _{G(\alpha )}\,$

for each $\alpha$ in the domain of G.

The proof of Easton's theorem uses forcing with a proper class of forcing conditions.

All conditions in the theorem are necessary. Condition 1 is a well known property of cardinality, while condition 2 follows from König's theorem.

No extension to singular cardinals

Template:Harvtxt proved that a singular cardinal of uncountable cofinality cannot be the smallest cardinal for which the generalized continuum hypothesis fails. This shows that Easton's theorem cannot be extended to the class of all cardinals. The program of PCF theory gives results on the possible values of $2^{\lambda }$ for singular cardinals $\lambda$. PCF theory shows that the values of the continuum function on singular cardinals are strongly influenced by the values on smaller cardinals, whereas Easton's theorem shows that the values of the continuum function on regular cardinals are only weakly influenced by the values on smaller cardinals.

See also

• Singular cardinal hypothesis
• Aleph number
• Beth number

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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