Some infinities are bigger than others meaning. Some infinities are bigger than others 2022-11-01

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In mathematics, the concept of infinity refers to something that is endless or boundless. However, not all infinities are created equal. Some infinities are bigger than others, and this concept can be a bit confusing to grasp at first.

One way to understand this idea is to consider the concept of a "countably infinite" set versus an "uncountably infinite" set. A countably infinite set is one that can be put into a one-to-one correspondence with the natural numbers (1, 2, 3, etc.). This means that each element in the set can be paired with a unique natural number, and the set is considered to be the same size as the set of natural numbers. An example of a countably infinite set is the set of all even numbers.

On the other hand, an uncountably infinite set is one that cannot be put into a one-to-one correspondence with the natural numbers. This means that there are more elements in the set than there are natural numbers, and the set is considered to be larger than the set of natural numbers. An example of an uncountably infinite set is the set of all real numbers.

So, in this sense, we can say that the set of all real numbers is larger than the set of all even numbers, because there are more real numbers than there are even numbers. In other words, some infinities are bigger than others.

This concept becomes even more complex when you consider the fact that there are different levels of infinity. For example, the set of all natural numbers is considered to be "countably infinite," but it is also considered to be a "small" infinity compared to the set of all real numbers. The set of all real numbers, in turn, is considered to be a "small" infinity compared to the set of all points on a plane, which is an "uncountably infinite" set that is even larger than the set of all real numbers.

In conclusion, the concept of infinity can be confusing, but it is important to understand that not all infinities are the same size. Some infinities are larger than others, and this is a fundamental aspect of mathematics that helps us understand and describe the world around us.

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Humans come up with the rules of math. Consequently, it remains unclear whether sets exist that are both larger than the natural numbers and smaller than the real numbers. This is hardly surprising because counting is extremely useful in terms of evolution. However, we do know that Augustus fears oblivion. So we began to wonder if all sets with an endless amount of objects were the same size. Personal attacks, slurs, bigotry, etc.

Some Infinities Are Bigger Than Others But There’s No Biggest One

So imagine we have two spaceships, the Real spaceship and the Natural spaceship. . } you can map it to a unique real number and if you did that for all the natural numbers, you would account for all the real numbers. The concept of infinity is used to describe a quantity that is larger than any finite number. For all practical purposes and for most of the math you will ever encounter, this is correct. But P X grows rapidly for larger X. The most common ones are denoted using Aleph and Beth either with a subscript , the first two letters of the Hebrew alphabet.

Then P L is a strictly larger uncountably infinite set than L, P P L is strictly larger than that, and so on for as many Ps as we want to stack up. If you wish to compare sets with numerous but finitely many elements, there are two well-established methods. A list is just a way to assign a real number to each integer. Calculating the area of more complicated subsets of the plane sometimes requires other tools, such as the integral calculus taught in school. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size.

Strange but True: Infinity Comes in Different Sizes

The fact that each series forms an infinite set means the sets of numbers are the same size, even though one set is contained within the other! Some infinities are bigger than other infinities, and everybody wants forever within the numbered days. Math is a human invention so all of its rules and definitions are whatever humans have come up with. If the points on a line were the same infinite size as the set of counting numbers, then we should find such a one-to-one correspondence. Regular resistors only have a real part. Whatever it is you do, do it in such a way that others can benefit. I believe this is why laypeople like myself often have difficulty accepting this statement some infinities are larger than others , because it's assuming that we are referring to cardinality which most people don't have that as their default interpretation of size. Several additional inequalities have been shown to hold between the 12 infinite cardinals we just defined.

Some Infinities Are Bigger than Other Infinities, and Some Are Just the Same Size

Now create p following Cantor's construction: the digit in the first decimal place should not be equal to that in the first decimal place of r 1, which is 4. There are ways to make this into a true mathematical statement by precisely interpreting what's going on using mathematical definitions, but you will be using different definitions than the standard argument that there is a larger infinity of reals than the infinity of naturals. I don't disagree with any of this. With a countable set you could list the elements 1 by 1 on and on forever and that process will in theory list every single element of the set. So we have decided that, in math, we can have things called sets.

A Deep Math Dive into Why Some Infinities Are Bigger Than Others

A first course in real analysis should have disabused you of your perspective. That there would always be and endless amount of real numbers left over. But what about the size of sets that have an endless amount of things? But the real numbers are a strictly bigger set. Infinity plus one is still infinity. This happens because there are far too many real numbers.

ELI5: How are some infinities bigger than others? : explainlikeimfive

Could, for example, all the independent entries be simultaneously different? Remember, this point on the line is irrational and can go on forever. Similarly, A is defined to be less than or equal to B if there is a mapping from A to B that uses each element of B once at most. This number does not appear on the list. It would take forever to count them all. So I wouldn't put too much thoughts into it. So is the number of integers, and somewhat surprisingly the number of rational numbers. Don't sweat it if it takes while before it starts making sense though.

You're actually better off reading wikipedia. If you've seen Cantor's diagonal argument or many other mathematical examples, you'll see that they didn't play or meddle directly with infinity but a finite portion of it and expanded their ideas to infinity. For example, the rational numbers within the real line are a null set even though there are infinitely many of them. If you want to discuss that argument then you need to read and address that argument, not just guess or philosophically ponder what it might mean. Infinity is not just a value that you can toss around and do operations on it. Hence, 27 is a real number, as is π, or 3.

Are Some Infinities Actually Bigger Than Others? I'm not convinced. : math

What you do with it. Doesn't change their value, it is a cosmetic change. So I really don't think your apple example helps to much. This is Reddit, not a scientific journal. Same thing with time: will it go on for all eternity, and does it stretch back infinitely far into the past? The rules to add and multiply them are slightly more tricky that Cardinals, but they can absolutely be defined.

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A set is infinite when any finite set can not accommodate its full nature. Hazel and Augustus knew that, and they made the most of their time. No then you'd have missed 0. The relation between Aleph and Beth numbers is pretty subtle dealing with infinity gets weird fast. This gives will give a number starting with : 0. But a smaller number of null sets though they would not be one-element sets could also satisfy the requirements. Once we give proper definitions things clear out.