Sets {infinite set} can have unlimited numbers of elements, with no greatest element.
The smallest infinite set {aleph naught} is enumerable. Enumerable sets include cardinal numbers, ordinal numbers, integers, and rational numbers {algebraic-equation solutions}, which all have same size.
The next-smallest infinite set is all points on a line {power of the continuum} {aleph one}, which is equivalent to all n-dimensional space points and to all real numbers. The second-smallest infinite set is all curves {aleph two}. Even-higher infinite sets exist, up to infinity {aleph infinity} {aleph naught naught}.
Rational-number sets have same infinity level as counting-number sets, as proved by Cantor's diagonal process {diagonalization} {Cantor diagonal process} {diagonal proof}.
rational numbers
Make table with infinite rows and columns. Cells have positive rational numbers. First row is a positive-integer series. Second row is a positive-integer series, each divided by 2. nth row is a positive-integer series, each divided by n.
To count fractions, start at top left 0. Go down one row to 0/2. Go diagonally up and right to first row at 1. Go right one column to 2. Go diagonally down and left to first column at 0/3. Repeat to cover all cells and count all fractions. Fractions count only once, establishing one-to-one correspondence between counting numbers and rational numbers.
real numbers
The real-number set has higher infinity than counting-number set. List real numbers in sequence as table. Rows are real numbers. Columns are digits. Along diagonals through table are real numbers, with one digit from each row and column. Change all digits of main-diagonal real number. Resulting real number is not any real number already in table, because all row and column digits have changed. Therefore, real numbers number more than counting numbers, and no one-to-one correspondence exists between counting numbers and real numbers.
No set has size between integer-set size and real-number-set size {continuum hypothesis} {continuum problem}. No aleph is between aleph naught and aleph one. Raising any positive integer to aleph-naught power results in aleph one.
indeterminible
Continuum hypothesis is indeterminable under set theory.
generalized continuum hypothesis
Positive integers raised to aleph n power equal aleph n+1 {generalized continuum hypothesis}. Generalized continuum hypothesis is independent of set theory.
In continuous intervals, continually removing inner third of remaining continuous interval still leaves infinitely many points, and total empty distance is interval length {Cantor set}. Cantor sets are the same at all scales. 1/f noise is like Cantor sets.
Power sets are larger than their basis sets {Cantor's paradox} {Cantor paradox}. Therefore, there can be no largest set and no set of all sets.
Are infinite sets countable {paradox of Galileo} {Galileo paradox}?
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Date Modified: 2022.0225