Set theory

Firstly, we must begin with sets. Put simply, a set is a collection of objects. Sets are notated as {a,b,c,d….} where a, b, c, d… are numbers (all numbers are shown as sets in set theory)  (Example: {3,2,5}).

Imagine a set to be a cardboard box, when the box contains nothing, we can show it as { }, now imagine placing a single apple into the box, now our set is {apple}, now if we add a single pear, we get {apple,pear}. Sets can even contain other sets, {green,{red,blue}} for example is a valid set, it contains the object green and the object {red, blue}.

In mathematics, sets are used to contain numbers (which are just sets), for example {3,2,6,1}. A single object in a set is called an element. In the set {2,6}, 2 is an element of the set, and 6 is an element of the set. A set can’t contain 2 distinct copies of the same elements, for example {3,7,3} is not a valid set as it contains two 3s.

## Part 2.2: Ordinals and Cardinals

So, why is this relevant? Sets can be used to represent 2 kinds of numbers, ordinals and cardinals. We can then use these sets to represent numbers greater than the standard infinity, but we will get to that in the next section.

Whereas cardinals describe the size of a set, ordinals describe the position of the set on the “numberline”. Cardinals are amounts like “4 things” or “5002 things” or “0 things”, whereas ordinals are “the 4th thing” or “the 5002th thing“. Cardinals are amounts and ordinals are positions.

A set can be an ordinal (keep in mind that sets can both be an ordinal and a cardinal) if it is the set of all smaller ordinals, where 0 = { } is the least ordinal. For example, {0,1} can describe the ordinal 2, {0,1,2} can describe the ordinal 3, and so on. {2,5,6} is not an ordinal, as it misses out 0, 1, 3, 4. It’s not necessary for the sets of ordinals to be arranged in a sequence like this, but it helps display it clearer, it’s more important that the set can be rearranged into an increasing sequence starting at 0 and missing out no gaps.

The ordinals go like this:

{ } = 0

{0} = 1

{0, 1} = 2

{0, 1, 2} = 3

And so on, you will notice that this corresponds exactly to the cardinality of these sets, at finite stages the ordinals and cardinals match.

## Part 2.3: Basics of The Transfinite

Transfinite sets are sets that go beyond the finite, the usual notion of “infinity” is broken down and expanded upon using sets, transfinite simply means “beyond finite“.

The first transfinite set is the set containing all natural numbers, including 0, this can be represented as {0, 1, 2, 3…..}, where “......” shows that this set continues this pattern infinitely. The cardinality of this set, shown as |{0, 1, 2, 3…...}| is known as Aleph\_0, it is the smallest infinite cardinal. The size of Aleph\_0 is equivalent to the amount of finite numbers. The ordinal corresponding to {0, 1, 2, 3……} is known as ω, ω is the first number greater than any finite number, but we can still go further.

Although ω and Aleph\_0 are “equal” in the same way that the ordinal 4 and the cardinal 4 are equal, the relationship between cardinals and ordinals starts to split past ω/Aleph\_0.

ω+1 is the first ordinal coming after ω, the set corresponding to ω+1 is

{0, 1, 2, 3…… ω}. Ordinals contain all ordinals less than themselves, so if the ordinal {0, 1, 2, 3…… ω} contains ω, it implies that ω is less than it, this shows that we can have ω+1 be a greater ordinal than ω, and in fact if ω+1 was not greater than ω, that would break the rule of ordinals. Cardinals do not have to comply to this since they don’t have the requirement of having to contain all lesser cardinals.

The cardinality of the set {0, 1, 2, 3……. ω} is however still not greater than Aleph\_0, despite it being a higher ordinal, this is because we can re-order the set to match the pattern of natural numbers:  {ω, 0, 1, 2, 3……}, and if we can match the elements of 2 sets that means that they have the same amount of elements.

ω would match to 0

0  would match to 1

1  would match to 2

…

and we see that we still somehow have the same amount of elements. The ordinal that the set {0, 1, 2, 3…… ω} describes is greater than ω, but it’s cardinality is still not greater than Aleph\_0, as this is where ordinals and cardinals split

We can keep going with higher and higher ordinals, for example ω+2, which set is {0, 1, 2, 3…… ω, ω+1}, and ω+3, which set is {0, 1, 2, 3…… ω, ω+1, ω+2}. We can generalize addition this way, and reach ω+4, ω+5, and so on, until we eventually reach ω+ω, or ω\*2. Ordinal constructed by arithmetical operations continue on, we reach ω\*3, ω\*4, ω\*5, and so on, then eventually ω\*ω, ω^3, ω^4 all the way to ω^ω, and we can still go further to ω^ω^ω, ω^ω^ω^ω and so on. The limit of ω^ω^ω^ω……. is notated as ε0, a diverse world of further ordinals beyond ε0 exists, but it won’t be important for the focus of the document, so now we must move on.

So far, although we have reached higher and higher ordinals, the cardinality of all these sets have been the same, the cardinality of the set describing ω is the same as the cardinality of the set describing ω^ω^ω^ω…. even if one is a higher ordinal than the other. Aleph\_1 is the first cardinal larger than Aleph\_0, the ordinal equivalent of Aleph\_1 is called ω\_1, an ordinal which is vastly greater than all extensions and functions on ω, such as ω^ω^ω and ε0.

Aleph\_1 has a larger cardinality than Aleph\_0, it's an amount of things bigger than an infinite amount of things. Aleph\_1 can also be described as the cardinality of the set containing all countable ordinals, a countable ordinal being one that only has a cardinality of ω or below, ω^ω^ω…… for example is still a countable ordinal, as its cardinality is still the same as ω.

And there’s no reason we have to stop here, for ω\_1 we can still add one to go further, since with ordinals we can always add one and find the next in the sequence, ω\_1+1, ω\_1+2 and so on, but as with before, these all still have the cardinality of Aleph\_1. The next cardinal after Aleph\_1 is called Aleph\_2, and again, Aleph\_2 actually has a higher cardinality than Aleph\_1, the ordinal version of Aleph\_2 is known as ω\_2, and again the set ω\_2 has a greater cardinality than ω\_1.

We can continue on, and the pattern continues on this way, we can get Aleph\_3, ω\_3, Aleph\_4, ω\_4. We can even get to Aleph\_ω (or in terms of ordinals, ω\_ω) . Eventually we reach a point where the next number is not so obvious, Aleph\_ω\_ω\_ω……., an infinite chain, this is called the Aleph fixed point, or AFP for short. The ordinal version of this is ω\_ω\_ω……., known as the Omega Fixed Point, or OFP for short. We can still go further than AFP and then, further still into the realm of so-called Large Cardinals.

## Part 2.4: Large Cardinals

Warning, this provides an extremely informal explanation of these concepts, aimed purely at beginners.

AFP and OFP are not the end, we can still create further “fixed points” of Aleph and Omega. OFP is an ordinal, so we know that we can find the next ordinal by simply doing OFP+1. We know that ω\_OFP is just OFP, as that would simply just be adding another “ω\_” on top of the infinite chain of “ω\_”s, the pattern of the of “ω\_”s would be the exact same.

However, ω\_(OFP+1) is bigger than OFP, we can continue on and get ω\_ω\_(OFP+1), ω\_ω\_ω\_(OFP+1) and so on until we eventually reach the limit of this, ω\_ω\_ω…..ω\_ω\_(OFP+1), this is the 2nd Omega Fixed Point, shortened to 1OFP. ω\_ (1OFP) is obviously again just 1OFP. OFP is the first smallest instance of a number big enough that adding “ω\_” on top, does not change it, 1OFP is the second smallest instance of a number big enough that adding another “ω\_” does not change it. We can continue on with the 3rd smallest instance of this, which is 2OFP, then we can get 3OFP, 4OFP, ωOFP, OFPOFP, OFPOFP….. And so on, creating even infinite sequences of OFP itself.

Now, we have to jump a huge gap, to find the The Inaccessible Cardinal, shown as Ι. The Inaccessible Cardinal is the first cardinal large enough such that it cannot be reached from below using standard cardinal arithmetic, things such as addition, multiplication, and the successor function (the successor function is the function that finds the next cardinal after, for example the successor of Aleph\_0 is Aleph\_1). Essentially, no amount of addition, multiplication, “get next set” operation, or anything else used on things less than I could ever “reach” I. Note, there is an ordinal version of I, there is an ordinal version for any cardinal, but we will purely be focusing on cardinals for this section.

As Ι is unreachable from cardinal arithmetic and functions based on cardinal arithmetic used on things below I, it is well above the OFPs, the first α=αOFP, the limit of various extensions to this system, and so on. In a way it is similar to ω. ω itself can not be reached below by iterating powerful functions upon themselves either. No amount of functions and recursions  could ever hope to “reach” ω. In that sense, ω is to finite numbers as Ι is to all the large cardinals we have discussed so far, such as OFP.

Yet, we can still go beyond by finding the next cardinal after Ι, there is nothing stopping us from doing that, every cardinal has a larger cardinal after it. We can find the next cardinal after it, which is Aleph\_(I+1), the next OFP after it, which is (Ι+1)OFP, where again we are using the ordinal Ι, and so on. After that there are more inaccessible cardinals, where the second inaccessible cardinal is similar to I but also includes arithmetic on I. Eventually we reach our next major milestone, Ι(1,0), this is the first thing inaccessible even if we add inaccessible based functions and their extensions into the list of permissible operations. And still we can go further and expand on Ι(1,0), using all the same methods we used to surpass Ι, Ι(2,0) is inaccessible even if we consider “next 1-inaccessible cardinal” as a permissible operation. And we can use the same pattern and create Ι(3,0), Ι(4,0), Ι(ω,0)…. even Ι(Ι(1,0),0), Ι(1,0,0), Ι(1,0,0,0)… of which the limit of creating levels of inacessiblity is a Mahlo cardinal. Mahlo cardinal is not simply “inacessibly inacessible”, as that is only 1-Inacessible, Mahlo cardinal is so large that no amount of finding higher levels of inacessibility could ever reach it.

Things don’t stop here, but things start to become more technical and complicated, there are large cardinals unfathomably larger than even Mahlos, it really is only the beginning. The world of large cardinals is extremely rich and interesting, but it’s not the focus of this document, so we will have to jump ahead to a concept larger than all of this… The Absolutely Infinite.