How an infinity can be bigger than another if they're both infinite? I just don't get it

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# How an infinity can be bigger than another if they're both infinite? I just don't get it

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How an infinity can be bigger than another if they're both infinite? I just don't get it

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learn what a bijection is

Mathematicians use a private definition of "greater than", which is not how anyone else uses the term.

One infinity can have more stuff in it than other. For example there are an infinite number of integers AND an infinite number of even integers, but since there are twice as many integers as there are even integers, the first infinity is twice is big as the second

Good example. Rationals are even greater in number. N < Z < Q.

They all have the same cardinality

So one set can be bigger than another set with the same cardinality

You are right. Infinities are not bigger than others. Mathematicians redefined the word "bigger" to mean something else completely.

While this is obviously true, it's not true that N, Z, Q have different cardinalities. In fact, their cardinalities are the same - it's possible to construct a bijection between each pair.

However, having the same cardinality is not the same as having the same size.

I'm pretty sure all mathematical claims are falsifiable. A falsification in this context just means a proof to the contrary, which is always in principle possible given a well-formed mathematical statement (a condition that is algorithmically verifiable).

What you probably wanted to say is that it doesn't have an computable interpretation, which is true, because you can only encode stuff on a computer that you understand (and mathematicians don't understand infinities).

Axioms are neither empirically derived nor empirically inspired. Axioms are a post-hoc rationalization and compression of a collection of observed mathematical facts. Those facts however, which come prior to the axioms, are often empirically inspired. So many fruitful mathematical conjectures (which later became proofs) were created by observing a natural mathematical pattern by computing examples.

Which infinite sets are mathematicians confused about?

Plane segments of any size contain curves of indefinite length

think of the infinite amount of decimals between 0 and 1, then the infinite amount of decimals between 0 and 2. but yeah it's silly and pointless

unfalsifiable nonsense that has no practical consequences for society

It's not an empirical claim. I'm not sure how falsifiability is relevant.

> It's not an empirical claim

then it’s bullshit and irrelevant. Actual math is based on empirically derived axioms and can theoretically be experimentally verified. The Pythagorean theorem, for example, is falsifiable. So don’t even try to start saying that math is not falsifiable or empirical, because it is, and it certainly can lead to disastrous consequences if you misuse math. But with actual infinities? It makes no difference. Cope and seethe, platonist troony

this is your brain on scientism

Axioms are not empirically derived, they are empirically INSPIRED. We still teach the pythagorean theorem in school even though it has been "falsified" by general relativity.

homie how the FRICK do you discover math without OBSERVING the world? Axioms are just generalizations based on past experience, it’s not magic. And by the way, here is a pro tip to help make your religion sound a little more reasonable: drop the idea of “bigger” infinities, and instead talk about “denser” infinities. The integers are denser than the evens, the rationals are denser than the integers, and the reals are denser than the rationals. But they’re all infinite so comparing their absolute size is moronic. There, so much better. You’re welcome.

both of you are metaphysically moronic

>>>/x/

Humans don't have the ability to conceive the infinite to begin with, you're not supposed to "get it"

These are forces beyond what most humans have the capacity to properly calculate

Just think about infinites as rates of change, it naturally lends itself to the idea of different sizes of infinity

Infinity isn't "big" to begin with.

We need a century-long moratorium on use of the word "infinite".

It cannot, endless is a synonym for infinite and there is no 2 sizes of endless

It's just a larp

Wrong, "even more endless"

>muh more nonstop process

infinity is a continuum, some are bigger than others.

There's basically two concepts.

The first is correspondence. If all of the elements in one set can be made to uniquely correspond to elements in another set, that other set is either the same size or bigger. As an example, we know that all of the rationals are "countable" because it is possible to establish a 1-1 correspondence between the rationals and the natural numbers.

The other concept is coverage. If a set "covers" another set in the sense that every element in the inner set is also guaranteed to be within the outer set, we say the outer set is at least as large as the inner set. in this case, we see the rationals cover the integers (because every single is in between two rationals of finite size). In this sense, the rationals can be argued to be larger than the integers, because it not only covers the integers, but has an impossibly large collection of elements between every two integers.

Because infinity is not a number, it's a different concept. More like a direction, or a vector with no specified magnitude: "Just keep going that way... forever." Or a vector with infinite magnitude, but that's a circular definition, which is dumb. Don't be dumb.

Anyway you can still make comparisons: There are infinite integers, but there's twice as many integers as there are odd integers.

Just don't think it's a number or you'll be like Georg Cantor and you'll go nuts.

A line and a plane are both infinite.

But the plane is bigger when viewed from our higher 3D.