Continuum (set theory)

In the mathematical field of

real numbers, or the corresponding (infinite) cardinal number

{\mathfrak {c}}

.[1]^[2] Georg Cantor proved that the cardinality

{\mathfrak {c}}

is larger than the smallest infinity, namely,

\aleph _{0}

. He also proved that

{\mathfrak {c}}

is equal to

2^{\aleph _{0}}\!

, the cardinality of the

natural numbers

.

The

natural numbers

, $\aleph _{0}$ , or alternatively, that ${\mathfrak {c}}=\aleph _{1}$ .^[1]

Linear continuum

Main article: Linear continuum

According to
Raymond Wilder
(1965), there are four axioms that make a set C and the relation < into a linear continuum:

C is
simply ordered
with respect to <.

If [A,B] is a cut of C, then either A has a last element or B has a first element. (compare Dedekind cut)

There exists a non-empty,
countable subset S of C such that, if x,y ∈ C such that x < y, then there exists z ∈ S such that x < z < y. (separability axiom
)

C has no first element and no last element. (Unboundedness axiom)

These axioms characterize the
real number line
.

See also

Aleph null

Suslin's problem

Transfinite number

References

^ ^a ^b Weisstein, Eric W. "Continuum". mathworld.wolfram.com. Retrieved 2020-08-12.

^ "Transfinite number | mathematics". Encyclopedia Britannica. Retrieved 2020-08-12.

Bibliography

Raymond L. Wilder (1965) The Foundations of Mathematics, 2nd ed., page 150,
John Wiley & Sons
.

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Retrieved from "https://en.wikipedia.org/w/index.php?title=Continuum_(set_theory)&oldid=1213235022"

[:0-1] Weisstein, Eric W. "Continuum". mathworld.wolfram.com. Retrieved 2020-08-12.

[2] "Transfinite number | mathematics". Encyclopedia Britannica. Retrieved 2020-08-12.

[2]

[1]