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Since the aleph numbers refer to cardinality of sets, aleph -1 would not make any sense. Not sure what a negative cardinality might mean?
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What is an aleph number?
An aleph number is any of a sequence of numbers used to represent the cardinality of infinite sets, denoted by the Hebrew letter aleph.
Is infinity a real number?
Infinity is not a specific number but a cardinality. The cardinality of a finite set is the number of elements in the set. For example, the cardinality of the set {1, 2, 3, 4, 5} is 5. Simple enough, but what about the cardinality of all natural numbers? There is no end to natural numbers so the cardinality cannot be a number in the normal sense. The cardinality is an infinity, called aleph-null. [As an aside, aleph is the first letter of the Hebrew language – which, along with the next letter, beth, gives us the word alphabet.]The cardinality of any set which can be put into one-to-one correspondence with the set of natural numbers (or conversely) is also aleph-null. You then have the curious result that, using the mappings x-> 2x-1 and x -> 2x the cardinality of positive odd number is also aleph-null as is the cardinality of positive even numbers. Comparing cardinalities, you get the aleph-null + aleph-null = aleph-null or 2* aleph-null = aleph-null.This result can be extended to all integers so that n * aleph-null = aleph-null for all integers n. This leads to the counter-intuitive result that aleph-null * aleph-null = aleph-null! It is possible to devise a diagonal scheme which gives a one-to-one correspondence between all rational numbers and all natural numbers. So there are aleph-null rational numbers. The classic exposition for this is Hilbert’s Grand Hotel. See https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_HotelThe above sets are said to have countably infinite elements (since they can be put into 1-1 correspondence with the counting numbers.You may have noted that, when introducing aleph-null, I used the phrase “an infinity”. This is because there is another, higher cardinality: the uncountably infinite. The cardinality of the set of all subsets of a set with countably infinite elements, or 2-to-the-power-aleph-null. This infinity is also known as the continuum. Cantor proved that the cardinality of irrational numbers (and therefore the real numbers) is the continuum and also that there are no orders of infinity between aleph-null and the continuum.
How many rational numbers are between -1 and 0?
There are an infinite amount of rational numbers between 0 and 1.
How many rational numbers are there between 1 and 1000?
There are infinitely many. More precisely, there are Aleph-null of them, or countably infinite.
Can infinity be an element of a set?
Yes.The set of {Aleph-null, Aleph-one, ...}, which is the set of the different infinities, has infinity as an element.Aleph-null is the countable infinity.
When was Aleph Zadik Aleph created?
Aleph Zadik Aleph was created in 1924-05.
Which Hebrew letter has the numerical value of 1?
Aleph (א)
When was Aleph Samach created?
Aleph Samach was created in 1893.
What is an aleph number?
An aleph number is any of a sequence of numbers used to represent the cardinality of infinite sets, denoted by the Hebrew letter aleph.
What is the cardinality of set b?
If set b is finite then the cardinality is the number of elements in it. If it is not finite then it depends on whether its elements can be put into 1-to-1 correspondence with the natural numbers (cardinality = Aleph Null) or with irrationals (Aleph-One).
How many psalms did aleph write?
Aleph is the name of a Hebrew letter, not a person. All 150 psalms in the Bible include the letter aleph (א).
What is infinity to the power infinity?
Also infinity. If you are concerned about the size of sets, it is a higher-level (larger) infinity. For example, 2 to the power aleph-zero, or aleph-zero to the power aleph-zero, is equal to aleph-one.
What are the classifications of a set?
Sets are classified according to their cardinality: a measure of the number of elements that it contains. The cardinality of a set may be finite, aleph-null or aleph-one. Aleph-null is the number of integers (or rational numbers). Aleph-one
When was The Aleph - short story - created?
The Aleph - short story - was created in 1945-09.
Is infinity a real number?
Infinity is not a specific number but a cardinality. The cardinality of a finite set is the number of elements in the set. For example, the cardinality of the set {1, 2, 3, 4, 5} is 5. Simple enough, but what about the cardinality of all natural numbers? There is no end to natural numbers so the cardinality cannot be a number in the normal sense. The cardinality is an infinity, called aleph-null. [As an aside, aleph is the first letter of the Hebrew language – which, along with the next letter, beth, gives us the word alphabet.]The cardinality of any set which can be put into one-to-one correspondence with the set of natural numbers (or conversely) is also aleph-null. You then have the curious result that, using the mappings x-> 2x-1 and x -> 2x the cardinality of positive odd number is also aleph-null as is the cardinality of positive even numbers. Comparing cardinalities, you get the aleph-null + aleph-null = aleph-null or 2* aleph-null = aleph-null.This result can be extended to all integers so that n * aleph-null = aleph-null for all integers n. This leads to the counter-intuitive result that aleph-null * aleph-null = aleph-null! It is possible to devise a diagonal scheme which gives a one-to-one correspondence between all rational numbers and all natural numbers. So there are aleph-null rational numbers. The classic exposition for this is Hilbert’s Grand Hotel. See https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_HotelThe above sets are said to have countably infinite elements (since they can be put into 1-1 correspondence with the counting numbers.You may have noted that, when introducing aleph-null, I used the phrase “an infinity”. This is because there is another, higher cardinality: the uncountably infinite. The cardinality of the set of all subsets of a set with countably infinite elements, or 2-to-the-power-aleph-null. This infinity is also known as the continuum. Cantor proved that the cardinality of irrational numbers (and therefore the real numbers) is the continuum and also that there are no orders of infinity between aleph-null and the continuum.
When was Aleph at Hallucinatory Mountain created?
Aleph at Hallucinatory Mountain was created on 2009-05-18.
What are the release dates for Aleph - 1966?
Aleph - 1966 was released on: USA: 3 March 2009 (DVD premiere)
