Vicious vs Benign Infinity

Infinity means endlessness, boundlessness, or something larger than any number, representing unlimited space, time, quantity, or extent, symbolized by ∞. It signifies a concept of "without end," applying to vast physical scales like the cosmos or abstract ideas like eternal cycles, and is used in mathematics for values beyond any finite number.

The Vicious vs Benign Infinity is useful to point out to critics who try to attack the first premise of the Kalam Cosmological Argument (KCA) - "Whatever begins to exist has a cause" or Prime Mover Argument, by trying to invoke an infinite regress. 

They will invariably try to point out a mathematical infinity/regress along the lines of (2) below and try to infer that it applies to (1) below. But this is the equivocation fallacy - is a logical error where a single word or phrase is used with different meanings in the same argument, making it seem valid when it's not. This is done by shifting meaning from one premise to the conclusion. Like using "right" to mean "entitled" in one part and "moral" in another. 

 

1. Vicious Infinity (The "borrowing money" analogy)

A vicious infinity, or vicious infinite regress, describes a situation where a task or object cannot exist because it depends on an infinite number of prior steps being completed first. Since an infinite number of steps can never be completed, the final result should not exist. If it does exist, the explanation leads to a logical contradiction.

• The money analogy: "It’s like a chain of people borrowing money from each other... if Darren pays Evan $10, and Darren got the $10 from Charlie... so on into infinity, no one actually took the money out of their pocket to start with, Evan never get paid."

• Why it is "Vicious": Because the payout depends on a source that never arrives. The chain has no foundation, so the event (Evan getting paid) is impossible. In the context of the universe, this argues that if the past were infinite, we would never have arrived at "today."
  

2. Benign Infinity (The "Fibonacci" analogy)

A benign infinity is an infinite series that exists conceptually or mathematically but does not prevent the current reality from existing. It often applies to abstract concepts (like math) or situations where the current step doesn't require the completion of the infinite series to exist.

• The Fibonacci sequence as an example: "There are still logically consistent continuations of the Fibonacci sequence that extend it all the way to negative infinity."

• Why it is "Benign": Because the number 10 doesn't "wait" for the negative infinity numbers to be counted before it can exist. The numbers are abstract and exist simultaneously. The regress is just a property of the number line, not a barrier to the existence of the number.

Summary Comparison

Feature	Vicious Infinity	Benign Infinity
Core Problem	Lack of foundation or starting point.	Infinite quantity without functional issues.
Causality	Dependent: Step B cannot happen until Step A finishes.	Independent: Steps typically exist abstractly or simultaneously.
Result	The event never happens (Logical Absurdity).	The concept is valid (Logical Consistency).
Example	An infinite chain of borrowers; no one ever gets paid.	The negative integers (... -3, -2, -1); they just exist.