Q: What is an aleph number?

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The highest place value in the decimal system is the aleph null'th place. It should be noted that aleph null is an infinite number (in fact, it is the smallest infinity possible) and represents the cardinality of the natural number set. In other words, there is no finite number which is the highest place value in the decimal system. (The Aleph character breaks Answers.com's WYSIWYG editor. The Aleph null character looks like: 0א except that the 0 is on the other side of the aleph character )

It is still infinity.If you add an infinite number plus a finite number, you get the same infinite number.If you multiply an infinite number times a finite number, you get the same infinite number.If you raise an infinite number to a finite power, you get the same infinite number.For example, aleph-zero squared is still aleph-zero.If you raise two (or a higher finite number) to an infinite power, you get a larger infinite number. To say it another way, the infinite number must be in the exponent, to get a larger infinity. For example, two to the power aleph-zero is equal to aleph-one; two to the power aleph-one is equal to aleph-two, etc.Read the Wikipedia article on "Aleph number" for more details. http://en.wikipedia.org/wiki/Aleph_numberAlso note that "infinity" has different meanings in different contexts, even if you concentrate only on mathematical meanings (as opposed to philosophical meanings, for example). I therefore suggest you also read articles about "Infinity", for example, at http://en.wikipedia.org/wiki/Infinity

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.

Since the aleph numbers refer to cardinality of sets, aleph -1 would not make any sense. Not sure what a negative cardinality might mean?

The set of odd whole numbers is countably infinite. It's cardinality is aleph null.

Related questions

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

The number of rationals is Aleph-null.

Aleph Zadik Aleph was created in 1924-05.

An aleph-one is the second of the transfinite cardinal numbers, which, according to the continuum hypothesis, corresponds to the number of real numbers.

The cardinality of composite numbers is Aleph-null.

The highest place value in the decimal system is the aleph null'th place. It should be noted that aleph null is an infinite number (in fact, it is the smallest infinity possible) and represents the cardinality of the natural number set. In other words, there is no finite number which is the highest place value in the decimal system. (The Aleph character breaks Answers.com's WYSIWYG editor. The Aleph null character looks like: 0א except that the 0 is on the other side of the aleph character )

The symbol for the number of rational numbers is Aleph-Null (Aleph being the first letter of the Hebrew alphabet - in fact that is where the word "alphabet" comes from!) The symbol for the number of real numbers is C (standing for continuum).

It is still infinity.If you add an infinite number plus a finite number, you get the same infinite number.If you multiply an infinite number times a finite number, you get the same infinite number.If you raise an infinite number to a finite power, you get the same infinite number.For example, aleph-zero squared is still aleph-zero.If you raise two (or a higher finite number) to an infinite power, you get a larger infinite number. To say it another way, the infinite number must be in the exponent, to get a larger infinity. For example, two to the power aleph-zero is equal to aleph-one; two to the power aleph-one is equal to aleph-two, etc.Read the Wikipedia article on "Aleph number" for more details. http://en.wikipedia.org/wiki/Aleph_numberAlso note that "infinity" has different meanings in different contexts, even if you concentrate only on mathematical meanings (as opposed to philosophical meanings, for example). I therefore suggest you also read articles about "Infinity", for example, at http://en.wikipedia.org/wiki/Infinity

It's the maximum.Probably C, the continuum.The cardinality (count) of the infinite set of integers is Aleph-null. Then C = 2^(Aleph-null).

There are countably infinite rational numbers. That is, it is possible to map each rational number to an integer so that the set has the same number of elements as all integers. This is the lowest order or infinity, Aleph-null. The number of irrationals is a higher order of infinity: 2^(Aleph-null). This is denoted by C, for continuum. There are no orders of infinity between Aleph-null and C.

Aleph Samach was created in 1893.

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.