[permalink] [id link]
The uncountability of the real numbers was already established by Cantor's first uncountability proof, but it also follows from the above result.
from
Wikipedia
Some Related Sentences
uncountability and real
The diagonal argument was not Cantor's first proof of the uncountability of the real numbers ; it was actually published much later than his first proof, which appeared in 1874.
* Cantor's diagonal argument ( diagonal metho ), a mathematical proof of the uncountability of real numbers
The original statement of the paradox, due to Richard ( 1905 ), has a relation to Cantor's diagonal argument on the uncountability of the set of real numbers.
This formulation of compactness is used in some proofs of Tychonoff's theorem and the uncountability of the real numbers ( see next section )
uncountability and numbers
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.
* Proof of the uncountability of the set of all subsets of the set of natural numbers ( Cantor's theorem 1891 )
Infinite divisibility alone implies infiniteness but not uncountability, as the rational numbers exemplify.
uncountability and was
This was proven by Georg Cantor in his 1874 uncountability proof, part of his groundbreaking study of different infinities, and later more simply in his diagonal argument.
uncountability and Cantor's
For a proof, see Cantor's first uncountability proof.
uncountability and .
There are many equivalent characterizations of uncountability.
So it is not obvious which one is the appropriate generalization of " uncountability " when the axiom fails.
real and numbers
For example, suppose that X is the set of all non-empty subsets of the real numbers.
But some subsets of the real numbers do not have least elements.
One might say, " Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering.
For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.
Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proven to exist using the axiom of choice, it is consistent that no such set is definable.
** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.
For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬ AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.
* There exists a model of ZF ¬ C in which there is a function f from the real numbers to the real numbers such that f is not continuous at a, but f is sequentially continuous at a, i. e., for any sequence
The object of study is the real numbers.
The real numbers are uniquely picked out ( up to isomorphism ) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound.
The graph of a function | graph of the absolute value function for real numbers
Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings.
As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.
real and was
`` They knew I was a good sharecrop farmer back in Carolina, but out West was a chance to build a real farm of our own.
There was no real sign of the river now, just a roiling, oily ribbon of liquid movement through muddy waters that reached everywhere.
The girl took a couple of steps toward the man in shorts when Benson, in that barefoot courtliness Ramey could never decide was real, said, `` You don't want to go around there, Ma'am ''.
Anyone who tried to remedy some of the most glaring defects in our form of democracy was denounced as a traitorous red whose real purpose was the destruction of our government.
After that, he declared, `` to return to freedom was to fall to one's knees before the real world and adore it ''.
The lyricist's father was a lawyer who had branched out into real estate.
The real Franco-German frontier was beyond the town's limits.
Both Baker and Fosdick knew that a substitute was necessary, that a verboten approach was not the real answer.
J. T. Shotwell was appalled by such spurious history as that which attributed the fall of the Carolingian empire to the woolen trade, and he urged Adams to `` transform his essay into a real history, embodying not merely those facts which fit into his theory, but also the modifications and exceptions ''.
When he was in the war, he was in Law or Supplies or something like that, and an old buddy of his told me he would come down on Sundays to the Pentagon and read the citations for medals -- just like the one we sent in for Trig -- and go away with a real glow.
The bank which held the mortgage on the old church declared that the interest was considerably in arrears, and the real estate people said flatly that the land across the river was being held for an eventual development for white working people who were coming in, and that none would be sold to colored folk.
There was, of course, no real need to rearrange everything.
She was getting real dramatic.
He appeared in the hopples about November 14, was treated for worms on the 18th, the latter date being the first time he struck a real pace.
Ruger reports that on his recent African safari the little Magnum cartridge was a real work horse.
`` That House & Home Round Table was the real starting point for today's revolution in materials handling '', says Clarence Thompson, long chairman of the Lumber Dealers' Research Council.
However, there was no real question of the justice of creating a strong Poland, both industrially and agriculturally, and one unplagued by large minorities of Germans or Russians.
It is the similarity between Estella's hands and Molly's ( `` The action of her fingers was like the action of knitting '' ) that provides Pip with a vital clue to the real identity of both and establishes a symbolic connection between the underworld of crime and the genteel cruelty of Satis House.
After his pains got worse, Tom decided to see a real doctor, from whom he learned he was suffering from cancer of the lung.
The real question was how one passed from anti-Semitism of this sort to murder, and the answer to this question is not to be found in anti-Semitism itself.
0.302 seconds.