1) what is infinity? an endless sequence?

1) what is infinity? an endless sequence?
2)what is the difference between infinite whole numbers and infinite irrational numbers? wouldn't they both sum up endlessly?
3) why can't we say 1/0 is infinity if endless division means the dividend is equal to 0?
4) why is infinity this symbol
5) is infinity properly understood or do we have room for improvement?

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Huge numbers. Irrational and rational infinite numbers have infinite digits

how do you prove a number can't be enumerated finitely?

>5) is infinity properly understood or do we have room for improvement?

This is the faulty premise that you need to erase from your mind.

"Infinity" isn't a reality for us to understand. It's a defined mathematical concept that has precise meaning in specific contexts.

It's just a symbol that corresponds to "too big to count" broadly speaking.

then where does the idea of cardinality come from? can something be even harder to count than 1+2+3...∞?

>then where does the idea of cardinality come from? can something be even harder to count than 1+2+3...∞?

It's made up.

You can categorize kinds of infinities by looking at their cardinality.

There's no reason that YOU have to accept that categorization, but if you do, the mathematics that goes along with it has been developed.

why? what for? what do you mean by harder? Infinity isn't about being hard, it shows that one number is so big that the other one compared to it can be neglected. At least that's how i understand that.

Infinity can't exist.

so ∞ is just a number so far away that it's like all other numbers are 0 relative to an ACTUAL number?

what's \frac{∞}{∞}

[latex]\frac{∞}{∞}[\latex]

alot of physical models uses infinity that way and it works very well.

1) Infinity is just an unbounded limit, at least in the reals.
2) When you think about real numbers, you can think about them as convergent limits of infinite sequences (this is part of why we end up with .999...=1). I don't think you really ever see infinite whole numbers because those wouldn't converge, so this question doesn't really have meaning.
3) Because division by zero, at least in the reals, isn't defined. If you try to define it and have everything else work properly you'll end up with some kind of contradiction like 0=1.
4) That's more a question for Google than Veeky Forums. There's definitely some historical reason for it.
5) Most if not all mathematicians have a pretty thorough understanding of infinity.

Also I'm just a petty undergrad so take my words with a grain of salt.

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1) The weight of your mom.

2) If you created a set of whole numbers (I assume you mean the natural numbers here) and a set of irrational numbers the set containing whole numbers would be infinitely larger than the irrational set. But both would be infinite.

3) Division by 0 is not defined. However the limit x->0 of 1/x approaches infinity.

4) Noone knows. Most likely its taken from the roman symbol of 1000, CIƆ, or CƆ. Personally I like to think of it as a Möbius strip (a strip with only one surface).

5) Yes and yes. We have a good understanding of infinity. There is always room for improvement.

[math]\infty = 10^200[/math]

fuck, rookie mistake
im too mindfucked by wildberger's lecture
[math]\infty = 10^{200}[/math]

But the limit x->0 of -1/x equals negative infinity
Checkmate

>3) why can't we say 1/0 is infinity if endless division means the dividend is equal to 0?
if 1/0=inf then 1=inf*0
it does't make sense.

the real question is why does it matter
all words are just close deletions being filled by context
inf is just what fits in the blank. it doesnt have to be anything beyond that

1) There are different concepts that may be called "infinity." The [math]\infty[/math] symbol usually means "a value greater than any element of the set," but isn't an element of the set itself. For example, in real analysis, when we say that a limit of a sequence [math](a_n)[/math] "equals" [math]\infty[/math], what we mean is that [math]\forall_M \exists_N \forall_{n \geq N} a_n > M[/math]. [math]\infty[/math] isn't a number, it's never reached by the sequence, it's just a symbol representing the behavior of the sequence.

Another concept where infinity comes up is cardinality. Both [math]\mathbb{N}[/math] and [math]\mathbb{R}[/math] are infinite sets, but [math]|\mathbb{N}| = \aleph_0 < \mathfrak{C} = |\mathbb{R}|[/math].

2) Not sure what you mean. If you mean cardinalities, look up Cantor's diagonal proof. If you mean "the largest possible whole number" vs. "the largest possible irrational number," then the question doesn't make sense, since those objects don't exist.

3) We don't define division by 0 because it leads to contradictions.

4) >That's more a question for Google than Veeky Forums. There's definitely some historical reason for it.

5)