My basic understanding of set theory

[mr3000]

TL;DR

I’m wondering if this intuition I have is valid regarding set theory

If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities, and... (infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and...) continues forever. and that continues forever. and that continues forever. and that continues forever. and that continues forever. and.....(…)…

Is this different from standard cardinality?

 

[FactChecker]

That's a good observation. There are multiple ways of having infinities within infinities within ...
Cantor developed that idea formally. If there is a one-to-one correspondence between two sets, they have the same cardinality. This applies whether the two sets are finite or infinite. There are different sizes of infinity. It can be shown that the rational numbers are the same size (cardinality) as the natural numbers, but that the irrational numbers are a much larger set (see Cantor's diagonal argument).

 

[jedishrfu]

Your observation is encoded in Cantor’s notion of aleph numbers, which measure different sizes of infinity.

The natural numbers have cardinality $\aleph_0$, the cardinality of the countable infinity.

The real numbers have cardinality $2^{\aleph_0}$, known as the continuum.

Whether this cardinal equals $\aleph_1$ is the Continuum Hypothesis, which is independent of standard set theory: it can neither be proved nor disproved.

 

[Hornbein]

Right. Take any two unequal rational numbers x and y. The open set (x,y) -- all rationals between x and y -- is infinite.

 

[Office_Shredder]

A countable union of countable sets is countable.

This means when you say hey I've got infinity infinities, that's gotta be a bigger infinity. Turns out in all your examples here with rational numbers, they are not.

 

[mr3000]

The definition of infinity is that it is something that is larger than any natural number

 

[PeroK]

The definition of infinity is that it is something that is larger than any natural number

Define "larger".

 

[PeroK]

The definition of infinity is that it is something that is larger than any natural number

To understand the concept of infinity in mathematics, you could start here:

https://en.wikipedia.org/wiki/Infinity

 

[FactChecker]

The definition of infinity is that it is something that is larger than any natural number

On the subject of set sizes, it would benefit you to read about what Cantor did and how he approached the subject. IMHO, it has been very well thought out and is interesting.

 

[mr3000]

To understand the concept of infinity in mathematics, you could start here:

https://en.wikipedia.org/wiki/Infinity

https://en.wikipedia.org/w/index.php?title=Infinity&oldid=1325158009

Up until a few weeks ago, this was the definition as given in the opening sentence.

“Infinity is something that is boundless, limitless, endless, or larger than any natural number”

 

[Hornbein]

https://en.wikipedia.org/w/index.php?title=Infinity&oldid=1325158009

Up until a few weeks ago, this was the definition as given in the opening sentence.

“Infinity is something that is boundless, limitless, endless, or larger than any natural number”

We pedants would prefer "an infinite set is larger than any bounded set of natural numbers." But clearly that is what it means. The "problem" is that it sort of implies infinity is a number.

 

[PeroK]

https://en.wikipedia.org/w/index.php?title=Infinity&oldid=1325158009

Up until a few weeks ago, this was the definition as given in the opening sentence.

“Infinity is something that is boundless, limitless, endless, or larger than any natural number”

That's a description, not a definition.

There is no single concept of infinity in mathematics. We might first encounter infinity in the context of an infinite set. In this case, we can define infinite as "not finite". That can be made more rigorous.

In this context, the set of natural numbers, $\mathbb N = \{1, 2, 3 \dots \}$ is an infinite set. And, we say specifically that it is countably infinite. The set of rational numbers is also countably infinite. The set of real numbers is also an infinite set, but has a greater cardinality- it is not countably infinite. The idea of having a greater cardinality can be made rigorous.

The second context might be in terms of a limit. As in $\lim_{x \to \infty} \frac 1 x = 0$. In this case, limits can be rigorously defined in a number of ways, usually by what is called standard real analysis.

Third, there is the concept of distance between two points in mathematics. An infinite set can be bounded (which means it has finite size - note the difference between size and cardinality.) For example:

The set of natural numbers is countably infinite and unbounded (it has a lower bound, but is unbounded above.)

The set of real numbers from 0 to 1 is uncountably infinite, but is bounded.

The point is that there are many different contexts in which infinity is used in mathematics. A final example is that we talk about a function being infinitely differentiable. That's another use of infinity in mathematics.

 

[Hornbein]

Infinity literally means "without end."

 

[PeroK]

Infinity literally means "without end."

That has no relevance to mathematics. The meaning of infinity depends on how it is formally defined.

 

[Hill]

Just to add to examples in the post #12, there are "points at infinity" in projective geometry, e.g.,

 

[FactChecker]

IMO, this discussion should focus on the OP, which specified the counting size of sets. Otherwise, it can become very unfocused.

 

[mr3000]

IMO, this discussion should focus on the OP, which specified the counting size of sets. Otherwise, it can become very unfocused.

Agreed

 

[FactChecker]

Agreed

Then you MUST study what Cantor did. You will find it beautifully logical. The set of Natural Numbers are countably infinite and any set that can be put into 1-1 correspondence with them is also countably infinite. The set of Rational Numbers is countably infinite. Any countably infinite union of countably infinite sets is countably infinite. Cantor's approach leads to some surprising results.

Many people hated Cantor's ideas at the time. It implies that the set of even numbers { 2, 4, 6, ...} is the same size as the set of Natural Numbers { 1, 2, 3, ...} even though it is a subset. But in Cantor's view, the set 2*{1, 2, 3, ...} is the same size as {1, 2, 3, ...}. There are sets that are larger than the Natural Numbers. The set of Irrational Numbers is larger. In fact, given a set, S, of any size of infinity, one can construct a larger set which S can not match in a 1-1 correspondence.

 

[PeroK]

IMO, this discussion should focus on the OP, which specified the counting size of sets. Otherwise, it can become very unfocused.

The basics are important. Things in mathematics are what they are defined to be. There is no single concept of "infinity" in mathematics. Cantor set theory is a quite advanced topic. You can't approach it with woolly ideas and definitions from the Oxford Engish Dictionary!

 

[mr3000]

https://en.wikipedia.org/wiki/Axiom_of_infinity

This is the best definition of infinity in my opinion, and it is mathematically rigorous

 

[MidgetDwarf]

I would read chapter 1, then flip to the last chapter. It will walk you through.

https://richardhammack.github.io/BookOfProof/Main.pdf

The book above is open source.