Umbriel (moon)

Name: Jodi Junker
My age: 32
Country: Netherlands
Home town: Oudkarspel
Post code: 1724 Xg
Street: Waterlelie 22

my page - www.hostgator1centcoupon.info

In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

Characterizations

There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions holds:

• There is no injective function from X to the set of natural numbers.
• X is nonempty and every ω-sequence of elements of X fails to include at least one element of X. That is, X is nonempty and there is no surjective function from the natural numbers to X.
• The cardinality of X is neither finite nor equal to $\aleph _{0}$ (aleph-null, the cardinality of the natural numbers).
• The set X has cardinality strictly greater than $\aleph _{0}$.

The first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.

Properties

• If an uncountable set X is a subset of set Y, then Y is uncountable.

Examples

The best known example of an uncountable set is the set R of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers and the set of all subsets of the set of natural numbers. The cardinality of R is often called the cardinality of the continuum and denoted by c, or $2^{\aleph _{0}}$, or $\beth _{1}$ (beth-one).

The Cantor set is an uncountable subset of R. The Cantor set is a fractal and has Hausdorff dimension greater than zero but less than one (R has dimension one). This is an example of the following fact: any subset of R of Hausdorff dimension strictly greater than zero must be uncountable.

Another example of an uncountable set is the set of all functions from R to R. This set is even "more uncountable" than R in the sense that the cardinality of this set is $\beth _{2}$ (beth-two), which is larger than $\beth _{1}$.

A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω or ω₁. The cardinality of Ω is denoted $\aleph _{1}$ (aleph-one). It can be shown, using the axiom of choice, that $\aleph _{1}$ is the smallest uncountable cardinal number. Thus either $\beth _{1}$, the cardinality of the reals, is equal to $\aleph _{1}$ or it is strictly larger. Georg Cantor was the first to propose the question of whether $\beth _{1}$ is equal to $\aleph _{1}$. In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that $\aleph _{1}=\beth _{1}$ is now called the continuum hypothesis and is known to be independent of the Zermelo–Fraenkel axioms for set theory (including the axiom of choice).

Without the axiom of choice

Without the axiom of choice, there might exist cardinalities incomparable to $\aleph _{0}$ (namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above but not the fourth characterization. Because these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.

If the axiom of choice holds, the following conditions on a cardinal $\kappa \!$ are equivalent:

• $\kappa \nleq \aleph _{0};$
• $\kappa >\aleph _{0};$ and
• $\kappa \geq \aleph _{1}$, where $\aleph _{1}=|\omega _{1}|$ and $\omega _{1}\,$ is least initial ordinal greater than $\omega .\!$

However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.

See also

• Aleph number
• Beth number
• Injective function
• Natural number

References

• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition).
• 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
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

• Proof that R is uncountable