Infinity*infinity (vs) Infinity^infinity

[TheUnshakableOne]

Is infinity times infinity the same as infinity to the power of infinity?

Which one is bigger?

I'm sorry if this is a dumb question..

 

[ActuallySpaceMan42]

Infinity * Infinity = Infinity + Infinity + Infinity + Infinity + Infinity -->, which is simply Infinity.

Infinity ^ Infinity = Infinity * Infinity * Infinity * Infinity * Infinity -->, which is Uncountable Infinity (Aleph-1)

 

[Enter_Bluey]

Okay I was curious if the site still treated this as uncountable infinite, good to know it still does. And yeah, like explained above

 

[TartaChocholate]

Infinity * Infinity = Infinity + Infinity + Infinity + Infinity + Infinity -->, which is simply Infinity.

Infinity ^ Infinity = Infinity * Infinity * Infinity * Infinity * Infinity -->, which is Uncountable Infinity (Aleph-1)

Do you have an example of each one? Curious because I think I've actually heard about the second case.

 

[GreatInfernal]

They are all calculated by math. Dunno if latex works here so
Infinity*infinity=a_0*a_0=a_0
Infinity^infinity=a_0^(a_0)>2^(a_0)=a_n for any n in N.

 

[ActuallySpaceMan42]

Do you have an example of each one? Curious because I think I've actually heard about the second case.

What do you mean?

 

[Lewis]

1.None of this proves that infinity ^infinity =aleph1, You should at least use the power set (P) to describe the existence of aleph1 /P{aleph0}=2^{aleph0}
2. You should not use arithmetic operation with infinity, which is clearly explained.

 

[Lewis]

Axiom of infinity - Wikipedia

en.m.wikipedia.org

 

[Lewis]

This video will help you understand more.

 

[TartaChocholate]

What do you mean?

Ah, my bad. I meant do you know of a verse that has a multiverse of each case? IIRC I read around that Fire Emblem Heroes has a Infinity ^ Infinity Multiverse, but didn't quite catch how it achieved that. And from what I've seen (though not yet confirmed) Final Fantasy has infinite different universes and each of them have infinite futures and possibilities, but don't know how that could be classified.

 

[Flashlight237]

Infinity*Infinity is simply Infinity^2, which is more like two layers of infinity than anything else.

Infinity^Infinity, on the other hand... Yeah, that'll throw you up a few tiers beyond Low 2-C.

 

[TartaChocholate]

Ah, dangit, I was mistaking infinity with infite worlds and such. Derp.

 

[ActuallySpaceMan42]

Ah, my bad. I meant do you know of a verse that has a multiverse of each case? IIRC I read around that Fire Emblem Heroes has a Infinity ^ Infinity Multiverse, but didn't quite catch how it achieved that. And from what I've seen (though not yet confirmed) Final Fantasy has infinite different universes and each of them have infinite futures and possibilities, but don't know how that could be classified.

The Blue and White Series has a pretty clear example;

Crossing timelines to avoid Victor was something that 'Lan Mu' had done a long time ago. His actions formed a kind of history, giving rise to all kinds of possibilities, and countless timelines.

Similar bifurcations were everywhere, and there were more and more variables.

One variable had N possibilities. When two variables were superimposed, it was N to the N power. If N variables were superimposed, it would be impossible to calculate.

Other than that, 001 was also creating new timelines, grafting other possibilities and combining them. This way, there would be new possibilities.

Just like natural numbers, there were countless permutations and combinations.

Therefore, there were too many timelines, so many that they could be expressed as infinity, or even infinity.

 

[SlendVeny]

This video will help you understand more.

Just change hotels

 

[Shadow_Somnius]

The Blue and White Series has a pretty clear example;

Is this like a verse mechanic or is it based on some real world logic. Shouldn't N variables each of N possibilities produce N^2 instead of N^N?

 

[ActuallySpaceMan42]

Is this like a verse mechanic or is it based on some real world logic. Shouldn't N variables each of N possibilities produce N^2 instead of N^N?

Verse mechinic.

 

[sen_]

Is this like a verse mechanic or is it based on some real world logic. Shouldn't N variables each of N possibilities produce N^2 instead of N^N?

well, no, n variables each of n possibilities do indeed produce n^n, here's a quick proof of this fact:
let's label the possibilities (p₀, p₁, p₂, p₃, ..., pₙ). Now let's put the same value of n possibilities again, how many possibilities do we have?
Well, we have p₀(p₀ + p₁ + p₂ + ...) + p₁(p₀ + p₁ + p₂ + ...) and so on until pₙ, which is just (p₀, p₁, p₂, p₃, ..., pₙ)². Now if we have n variables this becomes
(p₀, p₁, p₂, p₃, ..., pₙ)ⁿ.

 

[Shadow_Somnius]

well, no, n variables each of n possibilities do indeed produce n^n, here's a quick proof of this fact:
let's label the possibilities (p₀, p₁, p₂, p₃, ..., pₙ). Now let's put the same value of n possibilities again, how many possibilities do we have?
Well, we have p₀(p₀ + p₁ + p₂ + ...) + p₁(p₀ + p₁ + p₂ + ...) and so on until pₙ, which is just (p₀, p₁, p₂, p₃, ..., pₙ)². Now if we have n variables this becomes
(p₀, p₁, p₂, p₃, ..., pₙ)ⁿ.

Wait, I don't think I quite grasp the concept. Could you give another analogy? What sort of math does it fall under?

 

[sen_]

Wait, I don't think I quite grasp the concept. Could you give another analogy? What sort of math does it fall under?

Sure, i can give you another analogy, consider a 2-digit password where you can only use binary digits 0 and 1, the list of possible 2-digit passwords is:
00, 01, 10, 11.
Which are 2 · 2 = 2² = 4 possible 2-digit passwords with binary digits, this is known as the counting principle and it falls under a branch of mathematics called combinatorics.

 

[Shadow_Somnius]

Sure, i can give you another analogy, consider a 2-digit password where you can only use binary digits 0 and 1, the list of possible 2-digit passwords is:
00, 01, 10, 11.
Which are 2 · 2 = 2² = 4 possible 2-digit passwords with binary digits, this is known as the counting principle and it falls under a branch of mathematics called combinatorics.

Yeah, I still don't get it. The example you gave falls under basic permutation.

Using your same example, If I add one more number to the binary digits ie. 2. Then the no. of possible 2-digit passwords would be 3^2.
If its a 3-digit password, it would be 3^3, no?

So, going by that line of logic, if their are n numbers, then the possible 2-digit permutation would be NxN or N^2. The only way it becomes N^N would be if there were N number of digits for the numbers to fill in.

 

[sen_]

Yeah, I still don't get it. The example you gave falls under basic permutation.

Correct, it falls under basic examples of the counting principle.

Using your same example, If I add one more number to the binary digits ie. 2. Then the no. of possible 2-digit passwords would be 3^2.
If its a 3-digit password, it would be 3^3, no?

yes, if you used the digits 0, 1, 2 the possible 2-digit passwords would be: 00, 01, 02, 10, 11, 12, 20, 21, 22 which are 3² = 9. If they were 3-digits passwords then the answer would be 3³ = 27 and so on.

So, going by that line of logic, if their are n numbers, then the possible 2-digit permutation would be NxN or N^2. The only way it becomes N^N would be if there were N number of digits for the numbers to fill in.

Precisely, but your question was "Shouldn't N variables each of N possibilities produce N^2 instead of N^N?", to which the answer is no. Because 2 variables each of N possibilities generate N² as you said here.

 

[Shadow_Somnius]

Correct, it falls under basic examples of the counting principle.

yes, if you used the digits 0, 1, 2 the possible 2-digit passwords would be: 00, 01, 02, 10, 11, 12, 20, 21, 22 which are 3² = 9. If they were 3-digits passwords then the answer would be 3³ = 27 and so on.

Precisely, but your question was "Shouldn't N variables each of N possibilities produce N^2 instead of N^N?", to which the answer is no. Because 2 variables each of N possibilities generate N² as you said here.

Oh I am so sorry. I was pointing out how weird it was for the verse to give rise to N to the N power of possibilities from two variables of N possibilities each. I didn't realize I misworded it. Forgive me for wasting your time.

 

[sen_]

Oh I am so sorry. I was pointing out how weird it was for the verse to give rise to N to the N power of possibilities from two variables of N possibilities each. I didn't realize I misworded it. Forgive me for wasting your time.

No problem at all.