The Most Controversial Idea In Math

In mathematics, you cannot truly pick things at random because formulas always give the same result. The central challenge is how to select anything mathematically when randomness is not permitted.

• Real numbers have no smallest value due to extending to negative infinity • Even trying to find the smallest number after one is impossible (1.01, 1.0001, 1.00000000001, etc.) • Without a clear ordering rule, mathematicians cannot determine which number comes first

How can mathematicians resolve the paradox of having infinite options but being unable to select even one without following a rule?

In 1870, Georg Cantor began a mission to place the real numbers in a definitive order, a challenge that nearly consumed his life.

Galileo's 1638 work showed that natural numbers and square numbers can be perfectly matched one-to-one, leading centuries of mathematicians to conclude that all infinities are the same size.

In 1874, Cantor wondered if two infinite sets that don't map perfectly to each other would represent different infinities, questioning the prevailing view.

• Cantor assumed he could list all real numbers between 0 and 1 in correspondence with natural numbers • He constructed a new number by taking the first digit of the first number and adding one, second digit of second number and adding one, continuing down the diagonal • This new number differs from every number on the list by at least one digit • Therefore, there are more real numbers between 0 and 1 than natural numbers extending to infinity

Cantor revealed that infinity comes in different sizes: countable infinities (like integers and rationals) and uncountable infinities (like real and complex numbers).

• A well-ordered set requires a clear starting point • Every subset must also have a clear starting point • Natural numbers are well-ordered with 1 as the starting point • Integers can be well-ordered starting with 0, then 1, -1, 2, -2, etc., ranking by absolute value

Cantor published a theorem claiming that every set, including uncountably infinite ones like the real numbers, could be well-ordered.

Cantor could not actually prove his theorem because he could not construct a well-ordering of the real numbers, yet he remained confident in it based on religious faith rather than mathematical proof.

Leopold Kronecker, Cantor's former teacher and head of mathematics at University of Berlin, viciously attacked Cantor's work, labeling him a charlatan and corrupter of youth, leading to Cantor's nervous breakdown and rejection from academic positions.

In 1904, at an international mathematics congress, Julius König announced a proof that Cantor's well-ordering theorem was false. However, Ernst Zermelo quickly identified a fatal flaw in König's proof and proved the theorem within a month.

Zermelo realized that Cantor had been unknowingly assuming throughout his work that one could make an infinite number of choices at once from any set, including uncountably infinite sets. This assumption had never been formalized as a mathematical axiom.

The axiom of choice states: if you have infinitely many non-empty sets, there is a way to choose one element from each set simultaneously. For finite sets this is obvious, but for infinitely many uncountable sets, you cannot specify how you are choosing, only that it is possible.

• Zermelo uses the axiom to choose a number from all real numbers, placing it as X1 • The axiom allows choosing another number from the remaining reals as X2 • This continues indexing with natural numbers until running out, then extends beyond infinity with omega numbers (omega, omega+1, omega+2) • Every real number is eventually assigned a position, creating a well-ordering

• Giuseppe Vitali in 1905 assigned every real number between 0 and 1 to groups based on rational differences • He used the axiom of choice to select one representative from each group • He created infinite copies of this set, shifted by different rational numbers • The result is a non-measurable set with no consistent definition of size or length

When infinite copies are combined, they cover all numbers between 0 and 1, yet no positive or zero size can be added infinitely many times to produce a value between 1 and 3, creating a mathematical impossibility.

• In 1924, Stefan Banach and Alfred Tarski proved a single solid ball can be split into five non-measurable pieces • Through rotation and movement, these pieces can be reassembled into two identical balls of the same volume • The process can be repeated infinitely, creating infinite balls from one • This violates intuitive understanding of conservation of volume

The mathematical community was in crisis for over 30 years. When Tarski submitted equivalent results to a journal, one editor called them obviously false while another called them obviously true, reflecting the confusion about the axiom's validity.

In 1938, Kurt Godel proved that the axiom of choice is consistent with all other accepted axioms of set theory. Adding choice to these axioms does not create contradictions.

In 1963, Paul Cohen proved that there exists a valid mathematical world where all axioms of set theory hold true except for the axiom of choice, proving the axiom cannot be proven or disproven from other axioms.

• Like the parallel postulate in geometry, the axiom of choice selects which mathematical universe to work in • Choosing no parallel lines creates spherical geometry • Choosing one parallel line creates flat Euclidean geometry • Choosing multiple parallel lines creates hyperbolic geometry • All are mathematically valid depending on the system you wish to develop

The axiom of choice is now almost universally accepted in mathematics because it makes proofs concise and enables essential theorems. However, some mathematicians still study universes without choice to understand alternative systems, and using choice without it becomes harder but provides additional insight.

The axiom of choice allows mathematicians to replace lengthy explicit proofs with concise arguments. Proofs that could span 20 pages can be reduced to half a page by extending finite cases to infinite ones.

Many theorems in modern mathematics cannot be proven without using the axiom of choice somewhere. The general case often requires choice even when specific cases don't.

Without the axiom of choice, mathematicians work with both hands tied behind their backs, making progress on modern mathematics extremely difficult. Including choice comes with counterintuitive consequences like non-measurable sets.

The real question was never whether the axiom of choice is right in absolute terms, but rather whether it is right for what you want to do. Mathematicians can choose to work with or without it based on their needs and comfort with its consequences.