The paradoxes currently associated with infinity are only products of applying finite logic to describe infinity. Trying to understand infinity by analogies within the finite word are a bit like asking ‘what flavour is Buckingham Palace’? It’s absurd. To think about infinity one has to think within an infinite term of reference, as only this can lead to a sensible understanding of what infinity means. Once the true nature of infinity is understood, the paradoxes disappear but the very existence of infinity becomes questionable. Infinity and zero begin to look remarkably similar, although so far I have in no way managed to demonstrate the impossibility of infinity; it just seems a bit shaky from what I have so far thought. Further exploration by greater mind would be most appreciated – Prof Doron Zeilberger very kindly suggested I was on the right track, but had no time to comment further. Hey ho.

Here are some examples of infinitely erroneous logic.

1) ∞ + 1 = ∞

This is an absurd notion using finite logic to describe infinity.

Imagine the Grand Hotel, with an infinite number of rooms, each of which is occupied. This means that there must be an infinite number of people in the hotel. What happens when an additional person checks into the hotel? Does everyone just move up one room to accommodate them? After all, with infinite rooms there must always be more space to accomodate people, no?

The answer is don’t be stupid: nobody else can check in. Any infinite set of objects necessarily contains every single one of those objects that could possibly exist. Now, the hotel contains an infinite number of people, which means that every single person who can possibly exist MUST already be in the hotel. There can be nobody left outside the hotel to check in.

This is not just fatuous nit-picking and semantics, it is fundamental to the possible nature of infinity. It demonstrates that the only way to add to an infinite set of objects is to add an entirely different type of object. This would lead to the type of mathematical question ‘what do you get if you divide Buckingham Palace by a tomato?’ It’s meaningless.

∞ + 1 = a ridiculous idea

2) ∞ and ‘endless’ are the same thing

The Grand Hotel concept confuses ideas of ‘endless’ with infinity. Imagine that a person in Room 10 of the hotel is a notorious gangster and is tipped-off that the police are coming to get him. He has been told that the hotel is infinite, so runs out of his room and sprints down the corridor. He knows that in an infinite hotel he can keep running & running and never get caught until the end of time (assuming the police are not faster at running, of course). However, in his panic to get a head-start on the police, he turns the wrong way and quickly runs into the foyer, out the front door and into the arms of the police. What a nasty surprise!

The hotel is not infinite: as our gangster demonstrates the hotel comes to an end. It might go on for ever in one direction – ascending room numbers – but it ends at Room 1. The hotel would only be infinite if the rooms carried on under the numbers 0, -1, -2, -3…

It is possible that the corridor of rooms would loop back to the other side of the foyer, so that our gangster would run past Room 1 and straight to Room ∞, but the latter would have to be at the position of Room 0. ∞ & 0 are looking rather similar if ∞ is to make any kind of practical sense.

3) Different ∞s are different sizes

Cantor’s Diagonals apparently show that an infinite list of integers will be smaller than an infinite list of decimals. However, this is not possible and actually refers to large or endless sets of numbers, not infinite. The diagonal logic is fine apart from the beginning and end points. If the integers start at 1 and carry on for ever, they are not infinite as they also finish at 1. To be truly infinite, the list of integers MUST contain every conceivable integer including 0, -1, -2, -3…These numbers exist (I’ve just used them, after all) so they have to be in the list.

It is therefore impossible to pick a diagonal that has no integers (and corresponding endless decimal number) above it, so it is impossible to be sure that the new number made from such a diagonal is not represented above it. In fact, in an infinite list it MUST be represented.

The idea is:

-forever

.

.

-100 0.8979878565657…

-99 0.9768543456788…

-98 0.3456789987654…

-97 0.089764390005…

.

.

1 0.4647448847464…

2 0.5775858493933…

3 0.4647438387464…

4 0.3758594837636…

.

.

+forever

0.4748…. can be changed to 0.5657….. and this number is unique for all decimals from 1+. However, there must be this decimal somewhere between 0 and –forever. There is no way to create a new diagonal that does not exist above the point at which the original diagonal starts, in this case ‘1’.

Therefore ∞=∞. All ∞s are equal.

It is interesting to note that the standard Cantor Diagonal concept has a non-infinite but endless number of integers, each one of which is paired with a decimal number, the expression of which can is infinite. They are simply numbers so have no dimension associated with them – no start or finish, just value. If the expression of this value involves an endless list of numbers, the lack of dimension makes this endlessness be the same as infinite (the expression of the number, not its value).

4) ∞ universe.

The idea of there being an infinite number of infinite universes comes from finite logic being used for infinity. The universe has direction, not just value, so to be infinite it has to be endless in all directions. This does not appear to be the case, as the universe appears to have started at the big bang. Or possible was just very finite at that point.

If the big bang was the start, a real singularity, then the universe cannot be infinite: it started at the big bang, which means it has limits in at least one of its directions and so is finite. Big I’ll grant you, but not infinite. I’m not sure it the finite nature of the big bang means that the universe is now finite, or if the universe could have changed from finite to infinite in the last few billion years – one to work on!

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