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edited Feb 15, 2015 at 12:31

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You can see at least : The Early Development of Set Theory :

in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence with Dedekind, Cantor asked the question whether the infinite sets $\mathbb N$ of the natural numbers and $\mathbb R$ of real numbers can be placed in one-to-one correspondence. In reply, Dedekind offered a surprising proof that the set $A$ of all algebraic numbers is denumerable (i.e., there is a one-to-one correspondence with $\mathbb N$). A few days later, Cantor was able to prove that the assumption that $\mathbb R$ is denumerable leads to a contradiction. To this end, he employed the Bolzano-Weierstrass principle of completeness. Thus he had shown that there are more elements in $\mathbb R$ than in $\mathbb N$ or $\mathbb Q$ or $A$, in the precise sense that the cardinality of $\mathbb R$ is strictly greater than that of $\mathbb N$.

See :

Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (Engl.transl.by Philip Jourdain of "Beiträge zur Begründung der transfiniten Mengenlehre (I, 1895) and (II, 1897), 1915) , page 86,

for the definition of power or cardinal number of a set.

Georg Cantor discovered it.

You can see at least : The Early Development of Set Theory :

in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence with Dedekind, Cantor asked the question whether the infinite sets $\mathbb N$ of the natural numbers and $\mathbb R$ of real numbers can be placed in one-to-one correspondence. In reply, Dedekind offered a surprising proof that the set $A$ of all algebraic numbers is denumerable (i.e., there is a one-to-one correspondence with $\mathbb N$). A few days later, Cantor was able to prove that the assumption that $\mathbb R$ is denumerable leads to a contradiction. To this end, he employed the Bolzano-Weierstrass principle of completeness. Thus he had shown that there are more elements in $\mathbb R$ than in $\mathbb N$ or $\mathbb Q$ or $A$, in the precise sense that the cardinality of $\mathbb R$ is strictly greater than that of $\mathbb N$.

Georg Cantor discovered it.

You can see at least : The Early Development of Set Theory :

in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence with Dedekind, Cantor asked the question whether the infinite sets $\mathbb N$ of the natural numbers and $\mathbb R$ of real numbers can be placed in one-to-one correspondence. In reply, Dedekind offered a surprising proof that the set $A$ of all algebraic numbers is denumerable (i.e., there is a one-to-one correspondence with $\mathbb N$). A few days later, Cantor was able to prove that the assumption that $\mathbb R$ is denumerable leads to a contradiction. To this end, he employed the Bolzano-Weierstrass principle of completeness. Thus he had shown that there are more elements in $\mathbb R$ than in $\mathbb N$ or $\mathbb Q$ or $A$, in the precise sense that the cardinality of $\mathbb R$ is strictly greater than that of $\mathbb N$.

See :

Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (Engl.transl.by Philip Jourdain of "Beiträge zur Begründung der transfiniten Mengenlehre (I, 1895) and (II, 1897), 1915) , page 86,

for the definition of power or cardinal number of a set.

Source Link

created Feb 14, 2015 at 17:54

Mauro ALLEGRANZA

15.2k
1
40
53

Georg Cantor discovered it.

You can see at least : The Early Development of Set Theory :

in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence with Dedekind, Cantor asked the question whether the infinite sets $\mathbb N$ of the natural numbers and $\mathbb R$ of real numbers can be placed in one-to-one correspondence. In reply, Dedekind offered a surprising proof that the set $A$ of all algebraic numbers is denumerable (i.e., there is a one-to-one correspondence with $\mathbb N$). A few days later, Cantor was able to prove that the assumption that $\mathbb R$ is denumerable leads to a contradiction. To this end, he employed the Bolzano-Weierstrass principle of completeness. Thus he had shown that there are more elements in $\mathbb R$ than in $\mathbb N$ or $\mathbb Q$ or $A$, in the precise sense that the cardinality of $\mathbb R$ is strictly greater than that of $\mathbb N$.