undefined | Better HN

story

0 pointspcmonk10y ago0 comments

Where in math is the subset partial ordering used to describe one set as larger than another?

Diagonalization isn't showing that a number in set A isn't in set B - that's obviously true for reals and integers, but it's also true for rationals and integers. It's showing that there does not exist a mapping from B to A where there's an element in B for each element in A.

We're obviously not using the same definition of "size". I generally think in terms of cardinality, what are you thinking of?

0 comments

Retric10y ago

Cardinality is the number of elements in a set.

If every element in set A is in set B, and there are elements in set A left over it's larger because that's what larger means. {A,B} < {A,B,C}

There are countable and uncountable infinite set's. All countable set's have a bijection with N. However, there are more than two sizes of infinite sets. Real numbers < Imaginary numbers.

pcmonkOP10y ago

That's not what larger means, that's what (maps into a) strict subset means. In finite numbers, that's the same as larger (greater cardinality), but it's very much not the case for infinite numbers.

There are more than two sizes of infinite sets, but there are just as many real numbers as imaginary numbers for the same reason there's just as many integers as rational numbers.

Try reading this: https://en.wikipedia.org/wiki/Cardinality#Infinite_sets

Retric10y ago

We agree that {A,B,C} has a lower cardinality than {A,B}.

Now, feel free to try and map the set of Real numbers to the set of irrational numbers. ex: e + ei.

pcmonkOP10y ago

{A,B,C} is of greater cardinality than {A,B}. However, the argument that "a rule works for finite numbers, so it must work for infinite numbers" is clearly false.

For a your mapping, see: http://math.stackexchange.com/questions/512397/is-there-a-si...

1 more reply

yongjik10y ago

1. e + ei is not real.

2. f(x) = x + sqrt(2) if there exists an integer k>=0 such that x - k * sqrt(2) is rational; f(x) = x otherwise.

This function maps all real numbers to irrational numbers, 1-to-1.

1 more reply

stouset10y ago

The cardinality of the set of prime numbers and of rationals are both aleph-null[1]. Same with the list of all integers, the list of all positive integers, and the list of all even integers. They all have the same cardinality, and that cardinality is aleph-null.

But please, continue to argue with mathematical definitions that have been established for over a century.

[1]: https://en.wikipedia.org/wiki/Aleph_number#Aleph-naught

j / k navigate · click thread line to collapse

0 comments

Retric10y ago

Cardinality is the number of elements in a set.

If every element in set A is in set B, and there are elements in set A left over it's larger because that's what larger means. {A,B} < {A,B,C}

There are countable and uncountable infinite set's. All countable set's have a bijection with N. However, there are more than two sizes of infinite sets. Real numbers < Imaginary numbers.

pcmonkOP10y ago

That's not what larger means, that's what (maps into a) strict subset means. In finite numbers, that's the same as larger (greater cardinality), but it's very much not the case for infinite numbers.

There are more than two sizes of infinite sets, but there are just as many real numbers as imaginary numbers for the same reason there's just as many integers as rational numbers.

Try reading this: https://en.wikipedia.org/wiki/Cardinality#Infinite_sets

Retric10y ago

We agree that {A,B,C} has a lower cardinality than {A,B}.

Now, feel free to try and map the set of Real numbers to the set of irrational numbers. ex: e + ei.

pcmonkOP10y ago

{A,B,C} is of greater cardinality than {A,B}. However, the argument that "a rule works for finite numbers, so it must work for infinite numbers" is clearly false.

For a your mapping, see: http://math.stackexchange.com/questions/512397/is-there-a-si...

1 more reply

yongjik10y ago

1. e + ei is not real.

2. f(x) = x + sqrt(2) if there exists an integer k>=0 such that x - k * sqrt(2) is rational; f(x) = x otherwise.

This function maps all real numbers to irrational numbers, 1-to-1.

1 more reply

stouset10y ago

But please, continue to argue with mathematical definitions that have been established for over a century.

[1]: https://en.wikipedia.org/wiki/Aleph_number#Aleph-naught

j / k navigate · click thread line to collapse