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“So infinity +1 has no additional meaning.”

If the infinities in both cases are equal, then yes it does. Different size infinites are one of the fundamental principals of calculus. It's also one of the most difficult to understand.

Don't think in terms of sets where each number can only be used once. That's a purely artificial restriction that has absolutely no meaning whatsoever in this context. If you have a set of all reals, for example, you have a set with infinite members, none of which has a value of infinity. We are actually talking about infinite series. Not sets.

Lets say we have the following summation,

S1 = 1+2+3+4 + · · ·

We can all agree that the above goes to infinity. Now lets compare it with the following series,

S2 = 2+4+6+8 + · · ·

so that each number in the series is 2*n. Both are infinite. However, S2/S1 is exactly 2.

Because each term in S2 is divisible by 2, it is absolutely correct to write,

S2 = 2*(1+2+3+4 + · · ·)

In S2/S1 the series cancel out, their value (infinite or otherwise) is completely irrelevant.

S2, despite being infinite, is *exactly* twice S1, which is also infinite.

Whether a number has already been used is irrelevant. It's a sum. I can add the same value any number of times.

1+1+2+3+4 + · · · > 1+2+3+4 + · · ·

I think that's tripping some people up is treating "infinity" as though it were a discrete number that can be compared. In discrete calculus, we were always careful to say a sequence could "approach" infinity faster than another sequence, which is both valid and fairly easy to understand.

The moment you treat "infinity" like a discrete number and manipulate it with discrete operators like comparison, you break the concept of infinity. Nothing is bigger than infinity. Infinity plus one isn't a discrete number, neither is infinity minus one. Sequences do not "equal" infinity because the transient property of equality would imply that all sequences approaching infinity are equal, which they're not.

One might be tempted to say infinity minus infinity is zero, but that's not semantically valid because infinity isn't a discrete number.

S1=1+2+3+4...

S2=2*S1

Both sequences are infinite, but neither are equal, nor do they "equal infinity". S2-S1 doesn't equal zero, it equals S1.

Member since:

2010-09-13

Scenario:

Two 10 year old boys are playing 'name the largest number'

In the game of name the largest number, the child that names the largest number wins.

The largest 'set' has absolutely no relevance in this game and never has.

Child 1: says 100

Child 2: says 1,000

Child 3: says 1 million

Child 1: says 1 billion

Child 2: says infinity

Child 3: says infinity+1

In this game, child 1 has won. They said 1 billion, the largest number stated in the game.

infinity is accepted as an answer in the real world, but is not in actuality a number, but a concept.

As stated earlier infinity+1 has no importance, because the game was never about having the largest set. However, infinity+1 is not, necessarily a larger set.

An example of a set:

The unique numbers in (1,2,3,4,5)

Now lets add the "number" 1 to this set:

The unique numbers in: (1,2,3,4,5,1)

The set size has not changed.

Lets do a more creative set:

I define this set as: all numbers in the world, real or imagined, that can exist, or cannot exist. I specifically define as already being included in this set, any numbers you attempt to add to it later, and it contains all the numbers in any set you compare to it.

That's my definition, and just add salt to the wound, in this intellectual exercise, this set is labeled "infinity" doesn't matter if you like it or not.

Now what??? Bring it on bitch.

Look, I don't get to decide the parameters - the children's game of name the highest number - infinity is accepted as an answer - it's accepted as being 'whatever the highest number is in theory' -

So infinity +1 has no additional meaning.

The last paragraph is controversial, you may argue against it. Please do.

One thing we can all agree on, the very fact that I wrote this nonsense at 3:34am in the morning, is a sign of mental - well lets be polite, of being mental.

So, yeah, I lose. Later guys!