Who gave me more money?

[Tiberius]

I'm considering bribes from two people.

Person A gives me an infinite number of envelopes. The first envelope has $1, the second has $2, and so on, with the nth envelope containing n dollars.

Person B also gives me an infinite number of envelopes. The first envelope has $2, the second has $4, and so on, with the nth envelope containing 2n dollars.

Person B says they gave me more money, since for each envelope n, his has more money than A's envelope.

But Person A says they've given me more, since his bribe contains values of every integer size, but the odd numbered amounts are missing from B's.

So who gave me more money?

 

[stvdv]

I'm considering bribes from two people.

Person A gives me an infinite number of envelopes. The first envelope has $1, the second has $2, and so on, with the nth envelope containing n dollars.

Person B also gives me an infinite number of envelopes. The first envelope has $2, the second has $4, and so on, with the nth envelope containing 2n dollars.

Person B says they gave me more money, since for each envelope n, his has more money than A's envelope.

But Person A says they've given me more, since his bribe contains values of every integer size, but the odd numbered amounts are missing from B's.

So who gave me more money?

B

 

[Brickjectivity]

Its a mistake to assume the numbers of envelopes for A and B are the same.

If there are an infinite number of envelopes then that means the number of envelopes isn't truly the same or cannot be proven to be the same. Since the amounts increase linearly, the totals are effectively in the same range. The 'Cardinality' is the same. You can presume that B would be more money, but you can't show it. Both amounts approach infinity, but infinity does not equal infinity. Infinity is not a number.

It would be different if we were talking about finite numbers of envelopes.

 

[Tiberius]

Its a mistake to assume the numbers of envelopes for A and B are the same.

If there are an infinite number of envelopes then that means the number of envelopes isn't truly the same or cannot be proven to be the same. Since the amounts increase linearly, the totals are effectively in the same range. The 'Cardinality' is the same. You can presume that B would be more money, but you can't show it. Both amounts approach infinity, but infinity does not equal infinity. Infinity is not a number.

It would be different if we were talking about finite numbers of envelopes.

I wouldn't say they "approach" infinity, I'd say they ARE infinity.

 

[Tiberius]

B

How did you reach that conclusion?

 

[Watchmen]

Person A if the dollars in each envelope doubles the prior envelope. That’s how I understood it.

 

[Yerda]

Imagine having to lick an infinite number of envelopes.

How do you determine which infinite amount is "more"?

If A - B = x, and B - B = 0, where x is infinite (or any positive value) then I'd say A > B.

@Polymath257

 

[PureX]

They are both giving the same amount ... infinite dollars.

 

[stvdv]

How did you reach that conclusion?

I rather give others the chance to solve it themselves

 

[Jedster]

I'm considering bribes from two people.

Person A gives me an infinite number of envelopes. The first envelope has $1, the second has $2, and so on, with the nth envelope containing n dollars.

Person B also gives me an infinite number of envelopes. The first envelope has $2, the second has $4, and so on, with the nth envelope containing 2n dollars.

Person B says they gave me more money, since for each envelope n, his has more money than A's envelope.

But Person A says they've given me more, since his bribe contains values of every integer size, but the odd numbered amounts are missing from B's.

So who gave me more money?

B


After n envelopes are given

A's total is At = n(n+1)/2 (sum of a linear series)

B's total is Bt = 2n(2n+1)/2

Bt-At = (2n*n + n) - (n*n/2+n/2)
= (3/2)n*n -(n*n/2) + n/2
= n*n +n/2


So Bs total is always greater than A's, no matter what the size of n.


ETA(Both persons(A & B) know that you won't live forever and that will stop after sometime, so it is reasonable to accept that you will stop after a large number of envelopes are opened. i.e. you won't reach infinity. i.e you will stop after sometime and check the amounts.)

 

[Audie]

How did you reach that conclusion?

Infinity times two is two infinities?

 

[PureX]

B


After n envelopes are given

A's total is At = n(n+1)/2 (sum of a linear series)

B's total is Bt = 2n(2n+1)/2

Bt-At = (2n*n + n) - (n*n/2+n/2)
= (3/2)n*n -(n*n/2) + n/2
= n*n +n/2


So Bs total is always greater than A's, no matter what the size of n.


ETA(Both persons(A & B) know that you won't live forever and that will stop after sometime, so it is reasonable to accept that you will stop after a large number of envelopes are opened. i.e. you won't reach infinity. i.e you will stop after sometime and check the amounts.)

But until person A or B dies, the money will always be available. So they both will spend, or have available to spend, the same endless amount regardless of the number of envelopes it comes in.

 

[Stevicus]

They are both giving the same amount ... infinite dollars.

Yeah, that's what it seems like to me.

I wonder what infinite dollars would do to the inflation rate. What good is a bribe in infinite dollars if all you can do is paper your wall with it?

 

[Yerda]

This is to expand on my earlier post as it probably makes no sense to anyone other than me.

I'm not sure if this is entirely valid but it makes sense to me.

 

[Audie]

Yeah, that's what it seems like to me.

I wonder what infinite dollars would do to the inflation rate. What good is a bribe in infinite dollars if all you can do is paper your wall with it?

It would be an infinite source of biofuel.
For a while.
But quickly the pile of money gets out of hand, the only possible limit being when the ball of money stops expanding because it cannot grow faster than the speed of light.

We see a black hole in the future, one that will eat the universe.

Neither a bribest nor a bribee be
For lo
Thou knowest not the perils of infinity.

 

[Polymath257]

They give you exactly the same amount. There is a one-to-one correspondence between the dollars one gives and the dollars the other gives.

It is a mistake to assume that infinite quantities behave the same way as finite ones. In particular, twice an infinite cardinal is the same infinite cardinal, and and cardinalities cannot be canceled in equations.

 

[Polymath257]

B


After n envelopes are given

A's total is At = n(n+1)/2 (sum of a linear series)

B's total is Bt = 2n(2n+1)/2

Bt-At = (2n*n + n) - (n*n/2+n/2)
= (3/2)n*n -(n*n/2) + n/2
= n*n +n/2


So Bs total is always greater than A's, no matter what the size of n.


ETA(Both persons(A & B) know that you won't live forever and that will stop after sometime, so it is reasonable to accept that you will stop after a large number of envelopes are opened. i.e. you won't reach infinity. i.e you will stop after sometime and check the amounts.)

Given two sequences, a_n and b_n, the fact that a_n < b_n for all n does NOT imply that their limits a, and b, also satisfy a<b. All you get to conclude is a<=b.

 

[bobhikes]

I'm considering bribes from two people.

Person A gives me an infinite number of envelopes. The first envelope has $1, the second has $2, and so on, with the nth envelope containing n dollars.

Person B also gives me an infinite number of envelopes. The first envelope has $2, the second has $4, and so on, with the nth envelope containing 2n dollars.

Person B says they gave me more money, since for each envelope n, his has more money than A's envelope.

But Person A says they've given me more, since his bribe contains values of every integer size, but the odd numbered amounts are missing from B's.

So who gave me more money?

Person B

Because while the envelops are infinite your life is not if you opened them alternately throughout your life Person B will always have given you more. Honestly I would just open Person's B envelopes and leave person A's alone.

 

[Stevicus]

Person B

Because while the envelops are infinite your life is not if you opened them alternately throughout your life Person B will always have given you more. Honestly I would just open Person's B envelopes and leave person A's alone.

Yeah, I'm still kind of wondering what one would do with an "infinite" pile of money. It would cover the whole Earth, the entire solar system, galaxy, and universe.

Sounds like an interesting premise for a horror movie. Something along the lines of "The Blob," but instead of a blob, it's a big pile of money which keeps growing and growing and devouring people.

 

[PureX]

Yeah, that's what it seems like to me.

I wonder what infinite dollars would do to the inflation rate. What good is a bribe in infinite dollars if all you can do is paper your wall with it?

If the bribe is only to person A and B, it won't do anything to the inflation rate.