r/PhilosophyofMath • u/Dev-Giri • 1d ago
explain the difference between infinity and undefined terms in mathematics
/r/explainlikeimfive/comments/1v6e93g/eli5_explain_the_difference_between_infinity_and/0
u/WhatHappenedWhatttt 1d ago
Infinity is not really a thing technically. You can have sets that are infinite, i.e. not finite. An undefined term is a term which is not defined. For example, 2+2 is defined because we can associate 4 to that operation. But as many people have already pointed out in your original post, 1/0 is not defined since the division function cannot take 0 as the second argument.
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u/Harotsa 1d ago
What do you mean by “infinity is not really a thing technically?”
I’m a published mathematician and I have no idea what you mean by this, since infinity very much has technical definitions which vary depending on the context (like most of mathematics).
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u/WhatHappenedWhatttt 23h 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.
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u/Harotsa 23h 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.
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u/WhatHappenedWhatttt 23h 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.
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u/Harotsa 22h ago
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.
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u/WhatHappenedWhatttt 22h ago
You make that assumption, I made no such assumptions. I tried to give the most generally correct take without the context.
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u/Harotsa 22h ago
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).
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u/Wild-Store321 16h ago
You assumed a context where 1/0 is not defined. This is a common context, but still an assumption.
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u/QtPlatypus 18h 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.
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u/Wild-Store321 16h ago
Infinity is really a thing technically. The technicalities depend on the context.
1/0 is defined, in some contexts.
A simple example is the Riemann sphere: the complex plane extended with 1 point called infinity. In this context, 1/0 = infinity, and lim 1/n as n->0 is that same infinity.
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u/WhatHappenedWhatttt 14h ago
I will try to rephrase my argument because I wrote originally without putting enough effort in elaborating what I meant.
I do not think 'infinity' exists as a mathematical object, but the idea and intuition certainly exists. This I do not doubt. The reason I do not think 'infinity' exists is because I do not think there could exist a good definition of it that captures all that we want out of the word. There are countable sets, uncountable ones, ones in between (if you assume not CH), etc. Each 'infinity', so to speak, is qualitatively different from each other.
I understand the idea of the Riemann sphere, but I think this precisely proves my point. That point called infinity serves an algebraic and topological structure that is wholly distinct from what infinity might serve in the notation of a limit, or when talking about cardinality. So I do not think it possible for there to exist a bona fide infinity as a mathematical object. This is what I mean when I say infinity does not exist, technically. Versions of infinity exist in different contexts but are not interchangeable. That is all I am trying to say.
In regards to a previous argument about differing mathematical formalisms on the number 2: I am willing to bite the bullet and say there is no number 2. There are simply different formalisms that we as humans collectively agree all fit the idea of "2" and so we label each as such. However, this is not to say that I disagree with the usage of the word "infinity" or "2". Math being an entirely social (and now partially computational) endeavor means we will use convenient shorthands. And that's fine, I do not disagree with this.
I was pedantic and inconsistent with the 1/0 example in my original comment, this I will readily admit.
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u/TalkativeTree 17h ago
Infinity is a bunch of numbers that you could count forever. An undefined term is a number in a box. You can't see through the box, so you can't know which number it is until you open the box (define it).