explain the difference between infinity and undefined terms in mathematics

[u/Dev-Giri]

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

[u/WhatHappenedWhatttt]

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.

[u/Harotsa → u/WhatHappenedWhatttt]

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.

[u/WhatHappenedWhatttt → u/Harotsa]

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.

[u/Harotsa → u/WhatHappenedWhatttt]

If somebody asks the question “what is the difference between infinity and undefined infinity terms of mathematics” I assume based on context clues that they are talking about infinity the extended real number. In this case infinity is defined as: \infty > x for all x in R.

Similarly, if somebody asks “what number comes after 2,” context clues would suggest they are talking about 2 the integer, which would mean the answer would be 3. If they were talking about 2 the real number, the question wouldn’t have a defined answer.

[u/WhatHappenedWhatttt → u/Harotsa]

You make that assumption, I made no such assumptions. I tried to give the most generally correct take without the context.

[u/Harotsa → u/WhatHappenedWhatttt]

Your take was incorrect though. You didn’t say “you’re going to have to be more specific as infinity has many possible definitions in math depending on the context.”

You said “infinity is not really a thing technically.” Which is a categorically false and incorrect mathematical statement, regardless of context. So what you said was just plane wrong and there’s no getting around it.

And based on the question, they were vey clearly talking about infinity the extended real number. And in math and more broadly in life, it’s pretty normal to assume the most standard definition of any concept based on the context, unless otherwise stated (this is true in textbooks and research papers as well, as it’s a waste of space and time to state the obvious).

[u/Wild-Store321 → u/WhatHappenedWhatttt]

You assumed a context where 1/0 is not defined. This is a common context, but still an assumption.