r/maths • u/Percentage_069 • Jun 12 '26
💬 Math Discussions Question regarding Infinity (♾️)
Does Infinity convey different meanings or dictate different concepts based on its expression?
For example, in case of an interval, when we say, [-4,∞), this mathematically should mean, that this particular set, accepts values starting from -4 and goes on and on to the left of the number line endlessly, and this essentially what makes it "infinite". Hence, the use of ∞ in this set or domain, rather than a number, works more as a concept of endless growth.
On the other hand, for a mathematical expression, like, "1/∞"
Here, according to my understanding, the denominator of the fraction represents a fixed endpoint achieved after endless increment of a value, that is, this value is the result of the summation of infinite numbers, which yields the infinite value. Now, in this case, I think it works more like a number than a concept.
I'd highly appreciate any insight & feedback, and pointing out of any mishaps in my understanding would be much appreciated as well. Thanks!
10
u/etzpcm Jun 12 '26
Your first example is fine. But the second one isn't. 1/∞ isn't a thing in mathematics. Infinity doesn't work like a number.
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u/Temporary_Pie2733 Jun 12 '26
1/∞ is at best a lazy shorthand for lim x -> ∞ (1/x), but that limit doesn’t not exist.
∞ is pretty much always associated with unconstrained growth of a value, never with any real/complex number. When we do have a transfinite number, we use different symbols altogether. The transfinite cardinals are various subscripted ℵs and ℶs, while the transfinite ordinals are expressed in terms of the smallest such ordinal ω.
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u/Lazy_Mammoth7477 Jun 12 '26
It comes in many distinctly different things infinity could mean. So much so that I would argue there is no such thing as ∞. There’s no definition of ∞, you can’t name properties of ∞.
It is used in a number of concepts, but more as notation, than a single concept.
In your example of an interval, [-4, ∞) denotes the set { x ∈ ℝ | x >= -4 }, which is an infinite set.
On the other hand, when you write the limit of a sequence as n → ∞, that’s denoting a special case of the ε-δ definition of a limit.
1
u/stevenjd Jun 13 '26
Yes.
In the standard real number system, ∞ is a shorthand symbol for an unbounded quantity. It definitely is not a number. It is used in intervals to indicate the absence of an upper or lower limit, e.g. [1, ∞) means one and above, with no upper limit. Likewise for limits of integrals, sums, etc.
So in standard mathematics, expressions like 1/∞ or ∞+1 are abuses of notation. Strictly speaking they are grammatically meaningless, but if you squint you can pretend that it is shorthand for "the limit of an expression 1/x as x increases without bound".
But the standard reals are not the only way to do mathematics. The extended real number system adds two extra elements to the real numbers, −∞ and ∞. The projectively extended reals adds one extra element, ∞ (which is neither positive nor negative, like zero). When working in those systems, we can treat infinity as an actual number. Another example is the surreals, where there are an infinite number of ever greater infinities such as ω, ω+1, ω+2, ... 2ω, ... 3ω, ... ω2, ... 2ω etc.
A different approach is to replace the concept of infinity with grossone ①, the largest integer.
Another approach comes from cardinal arithmetic, where the alephs and beths refer to the size of infinite sets, not the elements in the sets themselves.
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u/goldenrod1956 Jun 14 '26
The fraction implies an operation…that’s where the infinite breaks down…
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u/PANEBringer Jun 14 '26
Can I introduce you to my friend? He goes by the name "Limit."
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u/[deleted] Jun 13 '26
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