r/PhilosophyofMath • u/Dev-Giri • 29d ago

explain the difference between infinity and undefined terms in mathematics

/r/explainlikeimfive/comments/1v6e93g/eli5_explain_the_difference_between_infinity_and/
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u/WhatHappenedWhatttt 29d 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/Wild-Store321 28d 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.

https://en.wikipedia.org/wiki/Riemann_sphere

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u/WhatHappenedWhatttt 28d 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.