Infinity #1

One principle of infinity the most people would have actually encountered in a mathematics class  is the infinity that limits. V limits, us can try to recognize 2∞ as follows:


The infinity price is provided twice here: very first time to stand for “as x grows”, and a 2nd to time to represent “2x eventually permanently exceeds any details bound”.

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If we usage the notation a little loosely, we might “simplify” the limit above as follows:


This would imply that the answer come the concern in the location is “No”, however as will be obvious shortly, using infinity notation loose is no a an excellent idea.

Infinity #2

In addition to limits, there is another place in mathematics where infinity is important: in collection theory.

Set theory recognizes infinities of lot of “sizes”, the smallest of i beg your pardon is the set of confident integers: 1, 2, 3, … . A set whose size is equal to the size of optimistic integer set is dubbed countably infinite.

“Countable infinity add to one” If we include another aspect (say 0) to the collection of optimistic integers, is the brand-new set any type of larger? To see that it can not be larger, you can look at the problem differently: in collection 0, 1, 2, … each element is just smaller by one, contrasted to the set 1, 2, 3, … . So, even though we included an aspect to the limitless set, us really simply “relabeled” the facets by decrementing every value.

“Two times countable infinity” Now, let’s “double” the collection of positive integers by including values 0.5, 1.5, 2.5, … The brand-new set might seem larger, because it contains an infinite number of brand-new values. However again, you have the right to say that the sets room the very same size, just each facet is fifty percent the size:
“Countable infinity squared” come “square” countable infinity, us can type a set that will certainly contain every integer pairs, such together <1,1>, <1,2>, <2,2> and so on. By pairing up every integer v every integer, we are successfully squaring the size of the creature set.Can bag of integers additionally be basically just relabeled through integers? Yes, they can, and so the set of integer bag is no larger than the set of integers. The diagram listed below shows how integer pairs can be “relabeled” with ordinary integers (e.g., pair <2,2> is labeled as 5):

“Two come the strength of countable infinity” The set of integers contains a countable infinity of elements, and also so the collection of all integer subsets should – loosely speaking – contain two come the strength of countable infinity elements. So, is the number of integer subsets equal to the number of integers? It transforms out that the “relabeling” cheat we offered in the an initial three instances does not occupational here, and also so it shows up that there are more essence subsets than there are integers. Let’s look at the fourth instance in an ext detail to recognize why the is so fundamentally different from the an initial three. You deserve to think that an essence subset as a binary number with an infinite sequence of digits: i-th digit is 1 if i is had in the subset and 0 if i is excluded. So, a usual integer subset is a succession of ones and also zeros walking forever and ever, v no sample emerging.

And now we are gaining to the crucial difference. Every integer, half-integer, or essence pair can be explained using a finite variety of bits. It is why we can squint at the set of integer pairs and see the it yes, really is simply a collection of integers. Each integer pair deserve to be easily converted to an integer and back.

However, an integer subset is one infinite sequence of bits. The is impossible to explain a general scheme for converting an limitless sequence of bits right into a limited sequence without info loss. That is why it is impossible to squint in ~ the set of essence subsets and also argue the it really is just a set of integers.

The diagram below shows instances of infinite sets the three various sizes:


So, in set theory, there are multiple infinities. The the smallest infinity is the “countable” infinity, 0, the matches the number of integers. A larger infinity is

1 that matches the number of real number or integer subsets. And also there are even larger and also larger unlimited sets.

Since over there are much more integer subsets 보다 there space integers, it must not be surprising that the math formula listed below holds (you can find the formula in the Wikipedia article on continuum Hypothesis):


And due to the fact that 0 denotes infinity (the smallest kind), it seems that it would not be much of a large to write this:


… and also now it appears that the answer to the concern from the title need to be “Yes”.

The answer

So, is the true the that 2∞ > ∞? The answer relies on which id of infinity we use. The infinity of borders has no dimension concept, and also the formula would certainly be false. The infinity of set theory does have a dimension concept and also the formula would be type of true.

Technically, explain 2∞ > ∞ is neither true no one false. As result of the ambiguous notation, that is impossible to phone call which ide of infinity is being used, and also consequently which rules apply.

Who cares?

OK… yet why would certainly anyone treatment that there space two various notions the infinity? that is easy to get the impression the the conversation is just an pundit exercise v no valuable implications.

On the contrary, rsnucongo.orgus expertise of the two kinds of infinity has actually been really important. After properly understanding the an initial kind of infinity, Isaac Newton to be able to build calculus, followed by the concept of gravity. And, the second kind of infinity was a pre-requisite for Alan Turing to define computability (see my post on Numbers that cannot be computed) and also Kurt Gödel to prove Gödel’s Incompleteness Theorem.

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So, expertise both kinds of infinity has lead to necessary insights and also practical advancements.