In this version of "epic maths" we'll go you v a five-part introduction to the ide infinity. Thankfully, we have actually some muppets and a most charts to aid us follow me the way.
Spoiler! The answer"s no infinity plus one. Heck, it"s not even infinity times infinity. (Yes, I"m sad to say that ad with the guy in the fit sitting through the kidsis lying come you.)

But over there are ways infinity can be bigger than infinity. And also actually, we can describe this idea making use of logicthose youngsters sitting through that man would understand. So buckle up, below are five huge ideas you need to recognize to wow your friends with epic math skillzat that next dinner party:

I. Collection Theory:

Before children learn to count, they find out to team stuff. Mathematician Steve Strogatz, in his excellent series on numbers because that The brand-new York Times, recalled a sketch from "Sesame Street" where the slow-witted Humphrey take away an bespeak of fish by trying no to count them:"Go call the kitchen fish, fish, fish, fish, fish fish," Humphrey tells an overwhelmed assistant. Mercifully, Ernie interjects, telling Humphrey it"s a lot simpler to simply countthe fish and tell the kitchen a number. Humphrey can"t save his amazement as he realizes exactly how "this counting thing have the right to really conserve a human being a most trouble."


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Essentially, what Humphrey was doing was collection theory. He took a team of stuff and put it right into a characterized set: 6 fish. As Humphrey is excited to discover, that set could just have easily been a collection of number 0, 1, 2, 3, 4, 5, 6 ... , or a set of spark plugs, or cinnamon buns.

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Humphrey learns the doesn"t issue what particular items are in a set. All that matters is the there is ingredient in the set. And also that"s set theory. Currently that you"ve mastered that idea - you"re more than likely wondering: just how does this assist me understand infinity?

II. Correspondence:

Correspondence is a concept in set theory that functions by relating each individual items in one collection to an individual item in an additional set. Those items might be things prefer Humphrey"s six fish or they could be number 1, 2, 3, 4, 5, 6 .... Through correspondence, the abstractions don"t issue - all the matters is that each article in one set can be compared to things in an additional set. For example, one fish to "1," 2 fish to "2," three fish to "3," and so on ...

In theory, Humphrey"s collection of fish can go ~ above forever. This is referred to as "cardinality" and also it leader to the smallest kind of infinity, i m sorry we prefer to speak to ...

III. Welcome come "Aleph-Null"

Before we go on, I need to stop because that a second and say this article is heavily indebted come Alasdair Wilkins" exceptional essay, "A Brief development to Infinity."

Let"s get ago to the math. Ours question: how have the right to one type of infinity be smaller sized than another? To understand that, let"s take a basic example using two different sets the "counting" numbers (the an intricate term because that these numbers is "natural") and imagine those set both extend forever. (I know there"s some debate about zero gift natural, however I"m a radio producer that studied history in college, so cut me part slack and also let"s simply say it is.)

SET A: 0, 1, 2, 3, 4, 5 ...

SET B: 1, 2, 3 , 4, 5, 6, ...


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As you have the right to see, also though set B starts one number greater than collection A, both sets contain the very same amount the "stuff," which method they represent equal types of infinities. Said another way, every number in set A can be synchronized to an additional number in set B that is one value greater 0 coincides to 1, 1 come 2, 2 come 3, etc... . You might keep increase the one-to-one correspondence forever.

Congratulations! You"ve simply stumbled top top the smallest form of infinity (fancy term: "aleph-null").

IV. Why Infinity to add One Isn"t Bigger than Infinity

So let"s get ago to why the commercial through the man in the suit was wrong. To the casual observer, it would certainly stand to reason that "aleph-null add to one" would certainly be larger than plain oldaleph-null, however when we use the logic of collection correspondence, we discover that"s no actually the case.

Let"s take an additional example. I occupational at a radio station, so let"s usage the instance of a huge juicy microphone together our "plus one." We"ll revisitSET A and set B again, yet this time, collection A has actually one extra thing in it:

SET A: microphone, 1, 2, 3, 4, 5 ...

SET B: 1, 2, 3, 4, 5, 6 ...

Using correspondence, we"ll complement the microphone come 1, 1 to 2, 3 to 4, and also so on. Visualized this way, you"ll see it"s possible to save up this one-to-one correspondence in between our set forever, which way infinity and also infinity to add one space actually equal. Georg Cantor, the mathematician who pioneered the work-related on infinity, stated this logical contradiction basically blew his mind. And as Alasdair Wilkins notes, "it it s okay weirder."


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Imagine you developed a set using only natural numbers ending in zero 0, 10, 20, 30, etc... , and also you contrasted that come a collection using every the natural numbers 0, 1, 2, 3, etc.... You"d think the second set would be ten times larger than the an initial set, but since both sets never ever end, as much as set theory is concerned, the two room equal. Crazy, right?

V. An Infinity Bigger than Infinity

Well, if that"s the case, girlfriend may find yourself questioning howany infinity could ever it is in bigger than one more infinity. Enter the human being of real numbers. (A actual number is any number representing a quantity along a continuous line. Forty-two, 2.335436643, the portion 5/6, -5, and pi are all genuine numbers.)

Now imagine 2 points on a line. Together an actuarial friend of mine placed it come me last night, "Between any kind of two limited points, there is an infinite and also uncountable collection of numbers in between those two endpoints."

That"s a reasonable idea. Expressed another way, you can say, "Infinity is the political parties to a circle." and that provides logical feeling as well.But due to the fact that were handle with set theory, let"s put these two examples to the test using collection theory.

So far, all of the sets we"ve encountered have to be "countable," which method that every the terms deserve to be associated with a natural number 0, 1, 2, 3, 4, 5, and so on ... To get to a enlarge infinity, we must come up v something that is uncountablyinfinite (i.e. The sides to a circle).

Alasdair Williams spells this the end brilliantly by informing us come imagine a binary number system, in which all the digits in every set are one of two people zero or one. He then illustrates those sets as decimal expressions of real numbers.

SET A: 0, 0, 0, 0, 0, 0 .000000 ...

SET B: 1, 1, 1, 1, 1, 1 .111111 ...

SET C: 0, 1, 0, 1, 0, 1 .010101 ...

SET D: 1, 0, 1, 0, 1, 0 .101010 ...

SET E: 0, 0, 1, 1, 0, 0 .001100 ...

Williams tells us to imagine those sets expand forever. He climate poses a question: can the numbers of any one collection be rearranged in together a way that they produce an entirely new set not included within the original boundless set?

To attend to this, mathematician Georg Cantor proposed using what he referred to as "diagonals." Namely, relocate through the to adjust diagonally, taking the train station of each number and creating an entirely new set. It sound complicated, for this reason let"s visualize it:

SET A: 0, 0, 0, 0, 0, 0 .000000 ...

SET B: 1, 1, 1, 1, 1, 1 .111111 ...

SET C: 0, 1, 0, 1, 0, 1 .010101 ...

SET D: 1, 0, 1, 0, 1, 0 .101010 ...

SET E: 1, 1, 0, 0, 1, 1 .110011 ...

We now have an entirely new set 0, 1, 0, 0, 1 ... or .01001, which we recognize to it is in a actual number.

Now let"s take the inverse of the sequence diagonal line 1, 0, 1, 1, 0 .... or .10110. Is that number component of any of thesets we just created? that can"t it is in a part of set A due to the fact that the 1 is the opposite the 0, right? it can"t be part of collection B due to the fact that the 0 is opposite the 1. It can"t be part of collection C because the 1 is the opposite the zero. You see where this is going. The contradiction expand forever come infinity.

Conclusion: It"s impossible to develop a one-to-one correspondence between counting numbers and also every possible real number. Thus, we"ve stumbled top top an infinity that is logically bigger 보다 the infinity alpeh-null. Hooray! making use of sets we"ve proven the idea that "between any kind of two finite points there is one infinite and uncountable collection of numbers between those two endpoints."


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Take a couple of minutes to conference your mind and be sure to lug a document and pencil to your following dinner party. As soon as infinity comes up, start doodling sets and you"re sure to it is in the resident expert who wows every the various other guests v your l337knowledge that epicmaths. (I wouldn"t recommend this as a sound method to choose up a date, but hey ... Possibly it will work.) great luck!

SOURCES/FURTHER READING:

As noted over and also over, this post is heavily indebted to AlasdairWilkins" wonderful article for Gizmodo, "A Brief development to Infinity." continue reading there if you want to go deeper right into infinity v his an overview of "the continuum." I supplied a lot of Williams" collection examples for this blog article (and as Colin and I all set for critical week"s show).

I"d likewise recommend "The Hilbert Hotel" by Steve Strogatz, i beg your pardon outlines the occupational of mathematician Georg Cantor in a method much more eloquent than anything I would certainly dare summarize.

This video clip breaking down the different species of numbers is fun.

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If you"re looking for an expedition of infinity in time, check out this explanation that "Planck Time."

Information around "Humphrey" was sourced, together always, indigenous Muppet Wiki. And last, however certainly no least, inspect outThe Colin McEnroe Show infinity edition with Steve Strogatz and astrophysicist Ron Mallett for an ext on sequences and infinity in the cosmos.