Although this is a wiki mainly for fictional numbers I would like to develop a foundational theory that may be used to talk about *sizes* beyond those in conventional Set Theory in a way that maintains some semblance of logic and structure. Currently there does not appear to be any theoretical foundation on which to base anything on.

As our basic **foundation** I propose that we begin with a one sort naive version of set theory that does not presuppose classes, nor any type of objects other than sets, but is otherwise unrestricted. I will call this **Urset Theory**. The primary object of study in Urset Theory are Ursets. In this blog I will give an account of some elementary results of this theory. This blog post presupposes the reader has some basic familiarity with set theory.

Let's begin. First we define an urset. In order to define it we first define the operation of selection. A **selection** is the operation of choosing an element from a given set. If the element selected is also a set we may apply selection to it. A **selection-chain** is a repeated process of selecting an element then an element of that element, then an element of that, and so on. We can now define an urset. An **Urset** is a set such that all selection-chains either end when there is no elements to select from in the last element of the chain, or continue without end. This can only happen if we never encounter anything other than a set. In other words, an Urset is a set such that there does not exist a selection-chain ending in an urelement (non-set element).

Let the elements of a set be encapsulated by the opening and closing curly braces respectively: { }. If all the elements are themselves sets we may omit separating commas. It follows from this that all ursets can be formed solely from the "{" and "}" symbols.

The empty-set is the set of no elements: { }

**Theorem I.** The Empty-set is an urset.

**Proof**. The empty-set contains only one selection-chain. When we attempt to select an element there is no element to select. Therefore no selection-chain ends in a urelement. QED.

**Theorem II.** A set of Ursets is an Urset

**Proof.** Let A be a set of ursets. We may select any element of A, and by definition its an urset. Say we select an element, called B. Since B is an urset we know that all selection chains are infinitely descending or end with no element to select. For all such chains add the additional selection, select B. For all chains which terminate this increases their length by 1. For all non-terminating chains they still do not terminate. Since all elements of the set are ursets, this same argument applies to all of them. Hence all selection-chains of A are either terminate without an element to select or descend indefinitely, which proves it's a urset. QED.

**Theorem III.** The set of only itself is an urset

**Proof.** Let A be a set only containing itself. This means A = {A}. The only selection we can perform on A is select A. Therefore we have only one choice at each selection in the chain and no matter how many selections perform we return A. Thus there is a single non-terminating selection-chain, therefore A is an urset. QED.

**Theorem IV.** A non-empty Urset must contain only ursets.

**Proof.** Let A be a non-empty urset. Assume it contains an element, B, which is not an urset. If B is not an urset it must either be a urelement or set which is not an urset. If B is an urelement, we may select B from A to show A is not a urset. If B is a set which is not an urset then there exists a chain starting with B that ends in an urelement, which means there is also a chain reaching this urelement from A, which means A is again not an urset. Therefore if A is an urset there can exist no B element of A which is not an urset.

**Theorem V.** A set of ursets and itself is an urset.

**Proof.** By theorem 2 a set of ursets is an urset, but the set itself is included which may or may not be an urset. Call the set of ursets and itself A. If A is an urset then A is an urset (by theorem 2). If A is not an urset then A is not an urset (by theorem 4). To prove A is a urset we will begin by assuming it's not an urset. This means some chain exists which ends in an urelement. When we begin at A if we select any other element besides A we know there can not be a selection-chain ending in an urelement since these are ursets by definition. Therefore we must select A to find one. However this brings us back into the same set of choices. So we would be forced to select A indefinitely, which is an infinite-descending selection-chain. Therefore there can not exist a selection-chain ending in a urelement. Therefore A is an urset.

**Theorem VI.** The union of two ursets is an urset.

**Proof**. Let A and B both be ursets from theorem 4 we know that all the elements of A and B are ursets. Therefore AUB contains only ursets. By Theorem 2 a set of ursets is an urset. Therefore the union of two ursets is a urset.

**Theorem VII.** The set of all Ursets is an urset and contains itself and is consistent.

**Proof.** The set of all ursets, the **URset**, also called the **URverse** (**ж**), contains only ursets, so by theorem 2 is also an urset. The URset is therefore a member of itself. Is the URset an urset? Yes. By theorem 5 a set of ursets and itself is an urset. Thus again it is still an urset.

We will now extend this theory further. We first will define urordinality. For two ursets, A and B, we will impose an order on them as follows:

A < B <--> A ∈ B

We will associate with every urset a unique urordinal. Note that with ursets it is entirely possible (though not necessary) that they contain themselves. Thus it is entirely possible here for an urordinal to be greater than itself. It is also possible for urordinals to be greater than each other. For example:

A = { { } , B } , B = { { { } } , A }

It follows that A > B and B > A. Note that this property this not transitive like normal. So even though { } < A and A < B, it does not follow that that { } < B. This is actually false since the only elements of B are { { } } and A. We can however say { } < { { } }.

We will also define urcardinality. Two sets have the same urcardinality if and only if they are the same set. If A is a proper subset of B, than the urcardinality of A is less than the urcardinality of B. It is entirely possible for two ursets to be incomparable.

In addition to these concepts the usual notions of ordinals and cardinals still apply as well.

Lastly let it be noted that all ordinals are ursets. An **ordinal** is the empty-set or a set of all ordinals less than itself. The empty-set is an urset. Furthermore in standard set theory a property of ordinals is that there is no infinite monotonically decreasing sequence of ordinals. Since this is synonymous with selection-chains this implies that all selection-chains within an ordinal must eventually end by trying to select an ordinal from the empty-set, the least ordinal. So all ordinals are ursets.

We briefly note that every member of the Von Neumann Universe is also an urset. This can be seen because the rank of a member of the Von Neumann universe is the smallest ordinal rank greater than all the ranks of all its members. So if we make a selection, regardless of the choice we make the rank decreases. Thus all selection-chains will again reduce us eventually to the selection of no elements.

Note however that the urcardinality of the URset is greater than the urcardinality of V, the Von Neumann Universe. This is because every member of V is an urset but not every urset is in V. All members of V do not contain infinitely descending selection chains, yet the members of UR do. Thus UR is a superset of V. That means UR is at least as large.

For the same reason UR's urcardinal is greater than Ord's urcardinal.

Now notice that we may form the urset containing all ordinals, which we will call O. Is O to be included in O or not. If we do not include it in itself, then O is a set of smaller ordinals and thus O is an ordinal by definition and must be included. But if we do include it it is no longer and ordinal by that definition since ordinals can not contain themselves.

How can we fix this? Let's define a SelfSet as an urset which contains itself and possibly other things. Now define the SelfSet containing all ordinals, SO. SO is not an ordinal since it contains itself ... therefore it should not be included as an ordinal. However SO doesn't only include the ordinals but itself as well. So it is included, just not as an ordinal. This resolves the paradox.

Now notice that SO is urordinally larger than any given ordinal, since every ordinal is an element of it. However it is not the supremum of the ordinals since it is not itself an ordinal (since it contains itself).

I will call this transordinality. We can say that SO is the first transordinal number. This is exactly analogous to transfinite. The transfinite numbers transcend the finite, and the **transordinal numbers** transcend the ordinals. Note that Cantor described the set of all ordinals as an *inconsistent absolutely infinite set.* In other words, we have found a set whose urordinality is greater than all the ordinals and therefore transcends an *absolutely infinite set*. We can therefore think of transordinal numbers as being at or above absolute infinity (in a certain technical sense). That makes transordinal numbers Class 2, according to this wiki's classification system. I would suggest however that all the finite and transfinite numbers form a foundation that should be treated as a single Category, which I will call The Foundation. Post-Foundation numbers, such as the transordinals, are the first to go beyond The Foundation of Set Theory.

There is certainly more to explore and discover here I think. But we will save that for a later blog perhaps.

I will finish by coining the URcardinal. **The URcardinal**, also called the **URsize** (**ж#**), is the cardinality of the UR set. It is not entirely clear to me how it compares to Absolute Infinity, but it has to be at least as large as Ord is a proper subset of it. On the other hand UR seems to contain almost everything.

I therefore conjecture that The URcardinal lies between Absolute Infinity and Absolutely Everything. That is:

Ω <= UR <= AE