# It's impossible to choose a number at random Source: https://www.youtube.com/watch?v=5U-V0y-fkSw Recap page: https://rapidrecap.app/video/5U-V0y-fkSw Generated: 2026-10-01T18:03:54.333+00:00 --- ## The Gist Math cannot truly pick random numbers because formulas always yield deterministic results, forcing mathematicians to rely on specific rules like choosing the smallest option. However, applying this rule to the infinite set of real numbers creates an impossible paradox where no starting point or subsequent number can be identified. ## Quick Overview Computers and mathematical systems lack true random number generators and instead use deterministic algorithms tied to local time to simulate randomness. When attempting to select numbers through strict rules, positive integers and prime numbers are straightforward because they have a clear smallest value. Real numbers break this logic completely by stretching endlessly toward negative infinity without a definable first or next value, sparking a historical mathematical crisis in 1870. **Key Points:** - Formulas always generate the same result in mathematics, which prevents computers from possessing true random number generators. - Computers typically run algorithms on local time to generate numbers that only appear random. - Selecting values without true randomness requires following a specific rule, such as always choosing the smallest option available. - Positive integers and prime numbers easily yield to this rule because their smallest values are one and two respectively. - Real numbers encompass positive, negative, whole, fractional, and irrational values like pi and the square root of two. - Applying the rule to real numbers fails because they stretch indefinitely toward negative infinity with no smallest possible value. - Attempts to specify a starting point immediately after one fail because numbers like 1.01 and 1.0001 leave an infinite gap of smaller values. - The mission to resolve the ordering of real numbers and find a definitive way to select them began with one man in 1870. ![Screenshot at 1:32: The historical effort to put real numbers in a definitive order began with one man in 1870.](https://ss.rapidrecap.app/screens/5U-V0y-fkSw/00-01-32.jpg) **Context:** Mathematicians and computer scientists have long grappled with the fundamental limits of randomness and deterministic computation. The core challenge of defining order within infinite sets like the real numbers underpins modern mathematical set theory. ## Detailed Analysis Mathematical operations rely on deterministic formulas that inherently prevent true randomness, meaning computers must simulate random selection using algorithms based on local time. To select a value in the absence of true randomness, a rule must be established, such as selecting the smallest possible number. While this method succeeds with positive integers and prime numbers due to clear starting points like one and two, it completely collapses when applied to real numbers. Real numbers include every conceivable fraction, integer, negative number, and irrational value like pi, stretching infinitely in both directions without a definitive smallest value. Even highly specific rules, such as trying to select the number immediately following one, fail because infinite subdivisions like 1.01, 1.0001, and beyond leave no absolute next step. This failure to sequence real numbers exposes a profound paradox where infinite options exist, yet selecting a single one through strict logic remains impossible. The monumental task of solving this ordering problem began in 1870 under the weight of severe mathematical stakes. ### The Illusion of Randomness in Mathematics - Formulas in mathematics always produce identical outputs, which makes true random selection impossible. - Computers bypass this limitation by running algorithms on local system time to produce pseudorandom numbers. ![Screenshot at 0:05: Computers do not have true random number generators because formulas always give the same result.](https://ss.rapidrecap.app/screens/5U-V0y-fkSw/00-00-05.jpg) ### Selecting Numbers Through Strict Rules - Choosing values mathematically requires enforcing a specific rule, such as selecting the smallest available option. - Positive whole integers easily resolve to a minimum value of one. - Prime numbers similarly resolve to a minimum value of two. ![Screenshot at 0:30: Positive integers and prime numbers yield easily to the rule of choosing the smallest option.](https://ss.rapidrecap.app/screens/5U-V0y-fkSw/00-00-30.jpg) ### The Collapse of Order in Real Numbers - Real numbers encompass all positive, negative, whole, fraction, and irrational numbers like pi and the square root of two. - Attempting to find the smallest real number fails because the set stretches infinitely toward negative infinity. - Refining the rule to find the number immediately after one fails because infinite values like 1.01 and 1.0001 prevent a definitive next step. ![Screenshot at 1:21: Infinite options exist in the real number system, but mathematicians cannot figure out how to pick just one.](https://ss.rapidrecap.app/screens/5U-V0y-fkSw/00-01-21.jpg)