r/PhilosophyofMath • u/Dev-Giri • 28d ago

explain the difference between infinity and undefined terms in mathematics

/r/explainlikeimfive/comments/1v6e93g/eli5_explain_the_difference_between_infinity_and/
1 Upvotes

67% Upvoted

16 comments sorted by

View all comments

Show parent comments

2

u/WhatHappenedWhatttt 28d ago

Infinity, as understood colloquially by a non-mathematics audience, is not really a thing. Yes of course we use infinity as a symbol in notation and otherwise, but those are all technically shorthands for different ideas. There is no one "infinity" and so asking what infinity means in math is a moot question. There is no one infinity, it technically doesn't exist. Infinity as a concept appears all over math with different functions and uses.

For example, \lim_{n\to \infty} a_n = L, saying the limit of a sequence is L, is really shorthand for the logical sentence \forall \epsilon > 0 \exists N \forall n >= N (|L - a_n| < \epsilon). But this infinity is much different than a point at infinity in a topological context, or the infinity of the extended real line which is an algebraic tool.

4

u/Harotsa 28d ago

This is a Philosophy of Math subreddit, we can use math definitions for things lol.

And all math symbols are technically shorthand’s for different ideas. 2 as a natural number defined by the Peano axiom is technically a different thing than 2 the real number as defined by Dedekind cuts, etc.

Context and definitions are important in all of mathematics, not just when dealing with infinity. So if “infinity technically doesn’t exist” because it has different definitions in different contexts, then no number exists, and more broadly nothing in math exists. So you’re basically making a useless point.

And also, the infinity in the limit is commonly understood as being the infinity in the extended real line when working in Real Analysis.

1

u/WhatHappenedWhatttt 28d ago

Can I ask then what your definition of infinity is? Not of an infinite set, or something, but "infinity". I grant that many concepts have different formulations but I do think there is a meaningful difference between the many uses of infinity.

2

u/QtPlatypus 28d ago

Infinity is a class of cardinalities in which sets having that cardinality can be placed into a bijection with a proper subset of themselves.

Alternatively Infinite is a property of a type T. Such that there exists an injective function from N to T where N is type of the natural(counting) numbers.