Infinity

As HTML doesn't (yet) support equations, Greek symbols for numbers are written out in English, and exponentiation is written using the ^ symbol. Thus a squared is written a^2, b to the power a as b^a, etc.

Transfinite Numbers

I. for an ordinal a, there is a least ordinal greater than a, called a+1;
II. for some definite sequence of increasing ordinals there is a least ordinal greater than all the a's, called lim(a).

So if you start with 0 as the first ordinal, you can apply principle I repeatedly to get the ordinal numbers 0, 1, 2, ... . To get past the infinite sequence of finite ordinals use principle II to get the limit lim(n), which is usually called omega, or w.

omega can be imagined in the following ways.

1. Imagine you want to get to a pub two kilometres away. You walk the first kilometre in one hour, then 0.5 km in half an hour, then 0.25 km in a quarter of an hour, 0.125 km in 7.5 minutes, and so on. Eventually, after 2 hours (since 1 + 0.5 + 0.25 + 0.125 + ... sums to 2) you have travelled 2 km, but you have done it in w stages.
2. Imagine climbing a mountain that is stepped like a pyramid, or the Incas' terraces, with each step or cliff of equal height. However, the mountain is higher than infinity. To (attempt to) get to the top you climb the first cliff in one hour, then double your climbing speed and climb the next cliff in half an hour, double your speed again and climb the next cliff in 15 minutes, etc. After two hours you have climbed w cliffs and are travelling at w miles an hour.

We can go further (of course). Applying principle I to omega gives w+1, w+2, ... , and eventually, using principle II, w+w which is also denoted by w.2

Now, when numbers get this big, they have their own rules of addition and multiplication. We know that 1 + 2 = 2 + 1 = 3. However:
i) 1 + w = w
ii) w + 1 = w + 1
Case (i) tells us that if you take 1, and add infinity to it, then you just stick at infinity. However, case (ii) says that starting off with infinity and adding 1 to it, gets you one step further. So when one school kid tells another that he's got infinity brain cells and the other responds by saying well she's got infinity plus one, then she's basing her argument on the non-commutivity of transfinite ordinals, and is actually cleverer.

Multiplication follows the same track. Two times omega (2.w) is omega lots of two, which is just omega; omega times two (w.2) is two lots of omega and is twice the size. Extending this using principles I & II to find the limit lim(w.n) we get omega-squared.

Just using these principles gets you to omega-to-the-power-of-omega, and further still to omega-tetrated-to-the-omega (which is omega-to-the-power- of-omega-to-the-power-of-omega-to-the-power-of- ...). This last number is usually called epsilon-nought. But what about epsilon-nought-tetrated-to-the-epsilon-nought ...? Mindfuck.

So you can see that you can go on defining new schemes for combining omegas in many exotic ways, but as long as you stick to addition (remembering that multiplication, powering, tetration, etc, are all based on addition) you'll always be beating around the lower foothills of the Absolutely Infinite mountain.

Cardinal Numbers

This might seem blatantly obvious, but consider this. It can be proved that while w is a cardinal number, w+1 is not. Neither is w+w, nor w^2, and not even epsilon-nought is a cardinal number. The cardinality of omega is the same as the cardinality of all these combinations of omega. What this implies is that given w rooms in a hotel, you can quite happily fit w+1, or even epsilon-nought guests into them, at one guest per room. Bizarre.

So it appears that all cardinal numbers are ordinals, but not all ordinal numbers are cardinals. This fact will be useful in shedding the baggage when we get to the really big numbers.

The Absolute Infinite

Mathematics can help us get an inkling as to what Omega is all about. To do this we use the Reflection Principle:
For any conceivable property, P, if Omega has property P, then there are Omega ordinals less than Omega that also have property P. For otherwise, Omega could be conceived of as the ath ordinal with property P for some ordinal a, thus being describable, which it can't be.

Or simply, that in order to be indescribable, Omega cannot be the only ordinal with such-and-such a mathematical property. It cannot be the 12th this or the 342nd that either, else we could ask, well what's the 343rd this? Omega must share all its properties with the Absolutely Infinite number of ordinals less than it, so that it is not unique. The Reflection Principle is the mathematicians way of saying something is indescribable, and we will use it later on.

The Alefs

We can begin to understand alef-one mathematically in several stages.

Return to Principle I at the top of the page. Hidden in this statement is the fact that no ordinal is less than itself. For if there were an ordinal a such that a < a, and there were also an ordinal b such that a < b, then we would have a < a < b, meaning that there is an ordinal between a and b: b is never the least ordinal greater than a. As b can be any number (greater than a), this implies that there is no ordinal greater than a, contradicting Principle I, which won't do.

Now we can combine Principles I & II to form Principle III:

III. for every set A of ordinals, there is a least ordinal greater than every member of A, called sup A.

The next stage concerns a little set theory.

Let On be the collection of all ordinals. Let's assume that On is a set. By Principle III, there is an ordinal Omega = sup On. But this is impossible, for if Omega is an ordinal, then Omega is an element of the collection On of all ordinals, implying that Omega < sup On = Omega. Now remember from above that no ordinal can be less than itself. Therefore, we must take our first assumption, that On is a set, as false. So, by Principle III, we have deduced that no set of ordinals stretches all the way out to Omega. This principle infers that, given any set A of ordinals, we can always find some ordinal that is bigger than every member of A but less than Omega.

The collection On (a collection is mathematically different from a set), is identified with Omega. A set is a form or thought that can be known in an objective way: it is a "Many which allows itself to be thought of as a One". But here we have shown that On is not a set, which implies that the Absolute Infinite cannot be conceived objectively. On is a collection of all possible sets, including the observer, and as such can only be known by complete immersion, or surrender, into On.

As a final step towards understanding alef-one, we consider what makes a number countable. An ordinal is countable if it can be reached by successive applications of Principles I & II. We have seen that a first application of these gives us omega, while subsequent applications give us w+w, w^2, w^w, etc. All these numbers are countable since, while infinite, they can be reached by applications of Principles I & II.

Now, alef-one. We have already mentioned that alef-one is the next cardinal (though not, of course, ordinal) number after alef-null (omega), so you should have an inkling of what we're dealing with here.

Let us define a number, x = sup(Principles I & II), that is, a least ordinal greater than every number that can be generated by an endless amount of applications of Principles I & II. This number is alef-one. Alef-one lies beyond any countable sum of ordinals less than itself. We can only reach alef-one by adding together alef-one ordinals. In this respect, alef-one is not countable: it is very hard to reach alef-one from below.

So we can re-define countability: a set S is countable if and only if its cardinality is no greater than the cardinality of omega. So S is countable if S is empty, S is finite, or Card(S)=Card(w).

Some ways of contemplating alef-one are:

1. The limit to the number of guests you can fit into a hotel with omega rooms is alef-one. You might be able to fit epsilon-nought-tetrated-to-the- epsilon-nought-times-omega-squared-plus-one guests into the rooms, but not alef-one guests. This arises from the definition of cardinality, in that there is no one-to-one mapping from alef-one to omega.
2. In climbing the infinite mountain, it is possible to scale a transfinite number of cliffs in a finite time, by continually doubling your climbing speed. However, you would never reach the alef-oneth cliff: you would need to travel at alef-one miles per hour to get there.

There are, naturally, an infinite number of cardinals: alef-two, alef-omega, alef-alef-one, alef-alef-omega, ...

As a last note, an ordinal a is regular if it cannot be represented as the sum of less than a ordinals less than a. Of all the ordinals less than alef-one, only 0, 1, 2, w and alef-one are regular. 0 is regular since it is not the sum of less than 0 ordinals less than 0 (which doesn't even make sense). 1 is not the sum of no ordinals less than one, since 1 cannot be obtained by adding no zeros together. 2 is not the sum of 1 ordinal less than 2. w is regular since it cannot be obtained by adding together finitely many finite numbers (though, paradoxically, it is countable). And alef-one is regular since to add together less than alef-one ordinals less than alef-one, is to add together countably many countable ordinals, which always just gives another countable ordinal less than alef-one. Conversely, 3 is not regular, as it is the sum of 1 and 2; etc.

We might also add that Omega is regular too. Think about it ...

Large Cardinals

Reflection Principle Revisited

The corollaries from this definition are:
1) if an ordinal has property P, then Omega must also have property P, as it can't be defined as not having property P;
2) if Omega has property P, then there is at least one ordinal less than Omega that also has property P (slightly different wording but still faithful to the original);
3) if an ordinal a has a "fixed-point property" P, then there are a ordinals less than a that also have the fixed-point property P. (This is the "Fixed-Point Reflection Principle", and is weaker than the full Reflection Principle in that it only applies to fixed-point properties rather than all conceivable properties. Examples of fixed-point properties are given following.)

Inaccessible Cardinals

A cardinal theta is inaccessible if:
i) it is regular; and
ii) for any K less than theta, K+ is as well.

Now, even though 0 is regular, there is no K less than zero (we don't count negative numbers here), so inaccessibility cannot be defined. 1 is not inaccessible, since even though it is regular, 0+ = 1 which is not less than 1. And similarly for 2, 3, 4, ...

However, alef-null (omega) is regular, and K+ for any K < w is just K+1 which is still less than w; therefore alef-null is the first inaccessible cardinal. Alef-one might be regular, but (alef-null)+ = alef-one, so alef-one is not inaccessible as it doesn't fulfil the second criterion.

Now, Omega is also an inaccessible cardinal, since by the Reflection Principle, it must share the properties of all the ordinals less than it. Furthermore, also by the Reflection Principle, there must be more than one inaccessible cardinal (since Omega can't be the second ...). The first inaccessible cardinal after alef-null is called theta. It is extremely hard to get to theta from below: theta cannot be reached by taking the lim of less than theta cardinals.

We can think of theta with the following analogy:

So, just as omega is the first infinite cardinal, so theta is the first large cardinal. The passage from alef-one to theta is like trying to get from 2 to omega.

In actual fact, application of the Reflection Principle leads us to conclude that there are actually Omega inaccessible cardinals less than Omega.

Hyperinaccessible Cardinals

Another way of defining nu is to say that nu is hyperinaccessible if (i) nu is inaccessible, and (ii) whenever K is less than nu, then the first inaccessible cardinal greater than K is less than nu as well. A hyperinaccessible cannot be reached from below by taking the sup of any smaller set of ordinals (since it is regular), nor can it be reached by jumping from cardinal to cardinal (since it is a limit cardinal), nor can it be reached by jumping from inaccessible to inaccessible cardinal (since it is the limit of inaccessibles).

Using the Reflection Principle is is possible to go further, defining hyper-hyper-inaccessibles, super-hyper-inaccessibles, hyper^a-inaccessibles (where a itself may be an inaccessible cardinal!).

Mahlo Cardinals

The first Mahlo cardinal, rho, can be understood by analogy. Going from the first inaccessible cardinal, theta, to rho is like trying to get from omega (alef-null) to alef-one. Recall how alef-one could not be reached by adding together any amount of ordinals less than alef-one.