SetBang (or S∈tBang) is a language based on the foundation of all mathematics: *set theory*. It’s esoteric by design, so I wouldn’t use it for anything where performance, readability, or programmer sanity were requirements. However, it’s probably great for job security.

**Getting SetBang**

You’ll need Leiningen (Clojure’s package manager) to run SetBang. SetBang lives on this repository.

**“Hello, World!” in SetBang.**

The first thing one does in any programming language is write “Hello, World!”. In SetBang, that can achieved as follows.

- Open a new file called
`helloworld.sbg`

. - In that file, type in the following short program:
`6^#~8[2>'2>\_#]_!5^#[2>'2>\_#]_~5[2>'2>\_#]_!~9''' [2>'2>\_#]_~~!!3[2>'2>\_#]_~!5^#~9'''[2>'2>\_#]_!!~8 ]`

`[2>'2>\_#]_!~!~3[2>'2>\_#]_!~3[2>\_#2>\_#]_!9''[2>\_#2>\_#]_!5^#'!9'!___`

- Run
`./setbang helloworld.sbg`

at the command line.

Your output should be:

`Hello, world!`

If you don’t feel like typing that, the program is provided at `resources/hello_nomacro.sbg`

.

It may not be clear, from the code above, what SetBang is doing. In fact, that particular program doesn’t use any set theory at all. Don’t worry: both the language and the theory behind it will become clearer over time.

**The Stack**

Every SetBang program runs with a *stack*. This is the most accessible form of state in SetBang. To see how it works, run `./setbang`

at the command line. Turn `:numeric`

(a configuration option) off like so:

`S∈tBang> :numeric off`

Stack:

Right now, the stack is empty. We introduce the first fundamental operator of SetBang: `0`

. It’s a program that pushes a value, the empty set, on the stack.

`S∈tBang> 0`

Stack: {}

S∈tBang> 000

Stack: {} {} {} {}

We’ll describe this as `0`

having the behavior `... -> ... 0`

. The program `000`

has the behavior `... -> ... {} {} {}`

. It puts 3 empty sets on the stack.

Empty sets aren’t terribly exciting, so we’ll use another command, `/`

, whose behavior is `... X e -> ... (X ∪ {e})`

.

`S∈tBang>`

/

Stack: {} {} {{}}

S∈tBang>/

Stack: {} {{{}}}

S∈tBang>/

Stack: {{{{}}}}

If you want to drop the top-of-stack (“TOS”) use `_`

, whose behavior is `... X -> ...`

. You can duplicate TOS with `~`

, whose behavior is `... X -> ... X X`

.

`S∈tBang> _`

`Stack:`

`S∈tBang>0~`

`Stack: {} {}`

`S∈tBang> /`

`Stack: {{}}`

There’s an additional command `'`

, which behaves identically to `~/`

. Its behavior is `... X -> ... X ∪ {X} `

. It’s not strictly necessary, but it can make SetBang coding easier, so it’s included.

`S∈tBang> _`

`Stack:`

`S∈tBang> 0'`

`Stack: {{}}`

`S∈tBang> ~/`

`Stack: {{}{{}}}`

`S∈tBang> '`

`Stack: {{}{{}}{{}{{}}}}`

One might be tempted to ask: so what are these things sets *of*? That’s the beauty of SetBang: just sets! The only values in SetBang are sets. It’s the ultimate unityped language, as there are no urelements at all. SetBang lives strictly within the Von Neumann universe.

Of course, this can get unwieldy. For example, execute the program `''' `

on the stack above, and you’ll get:

`S∈tBang>`

'''

`Stack: {{}{{}}{{}{{}}{{}{{}}{{}{{}}}}{{}{{}}}}{{}{{}}{{}{{}}}}{{}{{}}}{{}{{}}{{}{{}}{{}{{}}{{}{{}}}}{{}{{}}}}{{}{{}}{{}{{}}}}{{}{{}}}}}`

That’s not very human-readable. It turns out that a certain class of sets get a lot of use. They’re the *natural numbers*, defined like so:

- {} is a natural number and often called “0”.
- if n is a natural number, then n ∪ {n}, also known as n’ (or S(n) or n+1), is a natural number.

If you want your natural numbers visualized in the more familiar notation, use `:numeric on`

.

`S∈tBang>`

:numeric on

`Stack: 6`

`S∈tBang>`

'''

`Stack: 9`

Because SetBang is unityped, we’re going to use **1 = {0} = {{}}** as synonymous with “True” and **0 = {}** as synonymous with “False”.

There are commands `1`

, `2`

, … , `9 `

for putting some of the more commonly used natural numbers on the stack. However, don’t be fooled into thinking of them as *values*. Values don’t exist in SetBang, and there are no literals. They’re *commands* that do specific things, and `10`

does not place the number 10 on the stack, because it’s a `1 `

followed by a `0 `

. Instead, it places a {{}} (“1”) and a {} (“0”) on the stack. We’ll discuss tools for creating large numbers later.

`S∈tBang> 10`

`Stack: 9 1 0`

The other numerical commands aren’t necessary to write SetBang, of course. One can always use `0'''`

instead of `3`

, for example.

Another operator of interest is the cardinality operator, `#`

, whose behavior is `... X -> ... #X `

where `#X `

is the size of `X`

. Let’s build up the set {2, 3, 5, 7}:

`S∈tBang> 02/3/5/7/`

`Stack: 9 1 0 {7, 3, 2, 5}`

We use `#`

and we get back 4. Then, to show that it’s identical to {0, 1, 2, 3}, we use `?`

, whose behavior is `... X e -> (1 if e ∈ X, 0 if else)`

. That tells us that 2 ∈ 4, as we expect.

`S∈tBang> #`

`Stack: 9 1 0 4`

`S∈tBang> 2?`

`Stack: 9 1 0 1`

Crucial in stack manipulation are the `>`

and `<`

commands, which behave as follows.

`> : ... X1 X2 ... Xk K -> ... Xk X1 ... Xk-1 where k = #K (i.e. K's cardinality)`

.

`< : ... X1 X2 ... Xk K -> ... X2 X3 ... Xk X1`

Thus, for example `>`

would have the behaviors:

`... X Y 2 -> ... Y X`

`... X Y Z 3 -> ... Z X Y`

`... X Y Z W 4 -> ... W X Y Z`

The `2>`

pattern is commonly used to swap the top two values on the stack and, of course, `2<`

has the same effect.

`S∈tBang> 2>`

`Stack: 9 1 1 0`

`S∈tBang> 3<`

`Stack: 9 1 0 1`

`S∈tBang> 4<`

`Stack: 1 0 1 9`

`S∈tBang> 7 3`

`Stack: 1 0 1 9 7 3`

`S∈tBang> 6>6>5<`

`Stack: 7 1 0 1 9 3`

`S∈tBang> >`

`Stack: 7 1 9 0 1`

There’s a command `;`

whose behavior is `... X Y -> ... Y`

. If it weren’t provided, it could be simulated with `2>_`

.

For one last thing before getting into the theory that motivated the language, I’ll introduce conditional execution. The block `(code)`

executes the program in `code`

if and only if TOS is non-empty, and `(then,else)`

will execute the program `then`

if TOS is non-empty and `else`

if TOS is an empty set.

`S∈tBang> (___34,9)`

`Stack: 7 1 3 4`

`S∈tBang> 0(9,___)`

`Stack: 7 1`

`S∈tBang> (80)(9)`

`Stack: 7 1 8 0`

How much can we do with what we have so far? Surprisingly, a lot. However, let’s motivate the language with some set theory by introducing more operators, those justified by the ZFC system of set theory.

**ZFC and SetBang**

I won’t reiterate the axioms, but I’ll explain what they mean and what they contribute to SetBang.

*Extensionality*

The *Axiom of Extensionality* defines what it is for sets to be the same. Two sets are equal if they contain the same elements. If they are equal, they’re members of the same sets. This also implies the nonexistence of urelements by stating that there’s only one thing (namely, the empty set) that has no members.

This is realized in SetBang with the operator `=`

. Its behavior is:

`... X Y -> ... (1 if X = Y, 0 if X /= Y)`

There is one caveat: equality checking can take a *long* time. Specifically, it can take infinitely long, as we’ll see when we deal with infinite sets. It behaves well on finite sets, though.

`S∈tBang> 02/3/5/7/ 4`

`Stack: {7, 3, 2, 5} 4`

`S∈tBang> =`

`Stack: 0`

Of course, {7, 3, 2, 5} is *not* equal to 4, so this returns false/0. Equality is independent of ordering, hence:

`S∈tBang> _`

`Stack:`

`S∈tBang> 02/3/5/7/ 07/3/2/5/`

`Stack: {7, 3, 2, 5} {7, 3, 2, 5}`

`S∈tBang> =`

`Stack: 1`

*Pairing*

Pairing is the quintessential building axiom in ZFC set theory. It says that for any sets x, y, there is also a set {x, y} containing exactly them (and no other elements). Pairing is implemented using the `+`

command, with behavior:

`... X Y -> ... {X, Y}`

.

For user convenience, we have some related commands:

`" : ... X -> {X}`

`% : ... X Y -> {{X}, {X, Y}}`

Both of these can be defined in terms of `+`

:

`" is ~+`

`% is 2>~~+3>++`

The latter of these may be hard to believe. One can work it out at the REPL.

`S∈tBang> 3 8 %`

`Stack: {{3}, {3, 8}}`

`S∈tBang> 3 8 2>~~+3>++`

`Stack: {{3}, {3, 8}} {{3}, {3, 8}}`

`S∈tBang> =`

`Stack: 1`

For more assurance, one can try it out using QuickCheck-like generative testing, via the `:test`

directive.

`S∈tBang> :test % 2>~~+3>++`

`............... All tests passed.`

`Stack: 1`

This gives us a high degree of confidence that the programs behave identically. To see a case of failure, let’s do one that ought to fail:

`S∈tBang> :test #62>? _1`

`.........`

`Test #9 FAILED!`

`Stack was: Stack: {{{0, 3}, 2}, 4} {4, {0, 5}} {7, 2} 9`

`Program #1: #62>?`

`Result #1 Stack: {{{0, 3}, 2}, 4} {4, {0, 5}} {7, 2} 0`

`Program #2: _1`

`Result #2 Stack: {{{0, 3}, 2}, 4} {4, {0, 5}} {7, 2} 1`

Noting that `2>`

is a swap and that `?`

is comparison on natural numbers, the behavior of `6#2>?`

is:

`... X -> ... (1 if #X < 6, 0 if #X > 6)`

This is why the first nine tests (numbering starts at zero, of course) pass– that program is identical to `_1`

in the simpler cases– but #9 fails. Although the complexity parameter is hard-coded right now, I plan to change that.

If you want to test something against the empty string, use`:test`

with one argument, like so.

`S∈tBang> :test ~_`

`............... All tests passed.`

Note that `~_`

is a no-op; it duplicates TOS, then drops it.

*Infinity*

The Axiom of Infinity provides the ZFC system with an infinite set, ω, defined such that:

- 0 = {} ∈ ω.
- if x ∈ ω, then x’ = x ∪ {x}.

In SetBang, you can put infinite sets on the stack. The `$`

command achieves that. It places, on the stack, the set ω = {0, 1, 2, 3, …}.

`S∈tBang> $`

`Stack: {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, ...}`

`S∈tBang> $`

`Stack: {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, ...} {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, ...}`

Much larger infinities can be approached in SetBang. Of course, computers are finite, so doing any work that requires the entire set is impossible. We can use it, still, because it’s realized lazily. Let’s generate the natural number 512 using `9^#`

(to be explained later) and do a membership check in that infinite set.

`S∈tBang> $`

`Stack: {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, ...}`

`S∈tBang> 9^#`

`Stack: {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, ...} 512`

`S∈tBang> ?`

`Stack: 1`

It’s there. Now, here’s a bit of bad news. In SetBang, `?`

and `=`

are extremely “dumb”. If you use `?`

to check for membership in an infinite set and what you’re looking for isn’t there, SetBang will search forever. (It exits quickly in the “true” case). If you’re comparing sets for equality, SetBang will keep working until it finds a difference, and that can only happen if one set is finite. SetBang will quickly resolve `$0=`

(they’re not equal) but `$$=`

, while clearly “true” (ω = ω) will never terminate.

Why is this? Determining equality of sets, at the infinite level, is undecidable in the general case. There are tricks that could be used to make sets “smarter”, but I’ve chosen to make SetBang (designed to be an esoteric language, and not a practical tool) predictably dumb instead of unpredictably smart. We’ll show ways to get around this, later on.

*Specification, Replacement, and Union*

The Axiom Schema of Specification says that any describable subset of a set is also a set. As specified, it refers to logic formulas (free in one variable) like

(∃y)(∃z) (y∈x)⋀(z∈x)⋀~(y=z)

which is free in x and corresponds to “x has at least 2 elements.” SetBang doesn’t use symbolic logic *per se*, but it uses code objects (to be described) in their stead. Since it allows the pulling of subsets, it’s the *filter* (as in map, filter, and reduce) of set theory.

The Axiom Schema of Replacement says that the image of a set under a function of a set is also a set. This is the *map* of set theory.

Finally, the Axiom of Union allows one to take the union of a set (the set of all elements of its elements) and call that a set. This is an unordered, limited *reduce* that exists in set theory.

In short:

`S is a set, p is a predicate and f is a function.`

`Comprehension: {s : s ∈ S and p(s)} is a set.`

`Replacement: {f(s) : s ∈ S} is a set.`

`Union: {x : x ∈ s for some s ∈ S} is a set.`

We’re able to unify all three of these with the `{}`

comprehension (or set comprehension). If `code`

is a program whose behavior is `... x -> ... f(x)`

, then `{code}`

has the behavior:

`Y1 ... YN X -> ... Union(f(x) for all x ∈ X)`

.

where `Y1 Y2 ... YN`

is the stack at the time the `{}`

is encountered.

Set comprehensions may be lazy, and will have to be so if applied to infinite sets. The stack used (the `Yi`

above) will be the stack *as it was when the lazy set was created*, which may be a different time from when its elements are realized. For this reason, it’s best that code sequences put inside `{} `

comprehensions not contain side effects, and *we make it a rule that they’re not allowed to*. Side effects on the stack will be ignored, and I/O becomes unpredictable.

In a language like Haskell or Lisp, you can simulate `map`

, `concat`

and `filter`

using the more general `mapcat`

, like so:

`map f xs = mapcat (\x -> [f x]) xs`

`filter f xs = mapcat (\x -> if f x then [x] else []) xs`

`concat xs = mapcat identity xs`

We use the same principles to implement replacement (map), specification (filter), and union in SetBang. If `F `

is code whose behavior `... x -> f(x) `

then

`map F --> {F"}`

(Remember that the behavior of `" `

is `... X -> ... {X}`

. For example:

`Stack: {{15, 5, 10}, {7, 13, 3, 2, 11, 5}, {0, 1, 4, 9}}`

`S∈tBang> ~{#"}`

`Stack: {{15, 5, 10}, {7, 13, 3, 2, 11, 5}, {0, 1, 4, 9}}`

{6, 3, 4}

and

`filter F --> {~F(_",__0}}`

like so with `F = 5?`

, a program returning `1`

(true) if 5 ∈ X, where X is TOS.

`Stack: {{0, 1, 4, 9}, {15, 5, 10}, {7, 13, 3, 2, 11, 5}}`

`S∈tBang> ~{~5?(_",__0)}`

`Stack: {...}`

{{15, 5, 10}, {7, 13, 3, 2, 11, 5}}

Finally, the union is the easiest of all to write:

`union --> {}`

`Stack: {{0, 1, 4, 9}, {15, 5, 10}, {7, 13, 3, 2, 11, 5}}`

`S∈tBang> {}`

`Stack:`

{0, 7, 1, 4, 15, 13, 3, 2, 11, 9, 5, 10}

SetBang comes with built-in operators for the binary union, intersection, difference, and exclusive or of two sets:

`| : ... X Y -> ... (X ∪ Y)`

`& : ... X Y -> ... (X ∩ Y)`

`- : ... X Y -> ... (X - Y)`

`. : ... X Y -> ... (X - Y) ∪ (Y - X)`

If, however, these didn’t exist, they could be written (with one caveat, explained below) using set comprehensions:

`| <--> +{}`

`& <--> {2>~3<~3>?(_",__0)};`

`- <--> 2>{2>~3<~3>?(__0,_")};`

`. <--> 2>~3<~3>2>{2>~3<~3>?(__0,_")};3>2>2>{2>~3<~3>?(__0,_")};+{}`

In truth, it’s much better to use the built-ins, and here’s why. `{}`

-comprehensions become strange when infinite sets are involved. One will get identical behavior from `0$& `

and `$0&`

, but the same is not true of the intersection program written above. Let’s look at this using the macro system (which will be featured heavily in future posts).

`S∈tBang> :macro inter {2>~3<~3>?(_",__0)};`

`Stack:`

`S∈tBang> $ 0 :inter:`

`Stack: 0`

`S∈tBang> 0 $`

`Stack: 0 0 {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, ...}`

`S∈tBang> :inter:`

`.... hangs forever!`

As I said, SetBang prefers to be predictably dumb in handling infinite sets, rather than unpredictably smart. However, the implementation of `&`

is smart enough to know that set intersections commute and to avoid looping over an infinite set when possible. Set comprehensions don’t do this, so `0 $ :inter: `

triggers an infinite loop (over `$`

).

Laziness can usually rescue us, but the REPL demands elements. The problem with the set computed by `0 $ :inter:`

is that, while empty, it produces no elements. It’s equivalent to filtering, with `\_ -> False`

, from an infinite stream. The result is an empty stream, but the computer is not usually smart enough to figure it out (and, in fact, the general problem is mathematically undecidable). We call this ⊥ (pronounced “bot”). It’s equal to the empty set, but the computer will never know that it’s equal to the empty set: it will try, fruitlessly, to find elements forever.

The `&`

command is smart enough to reorder arguments when it reduces the likelihood of producing a ⊥, while set comprehensions aren’t.

*Power Set*

The Axiom of the Power Set allows one to produce, for a given set X, the set of all of its subsets. The command for this is `^`

, with behavior `... X -> ... P(X)`

.

`S∈tBang> 3`

`Stack: 3`

`S∈tBang> ^`

`Stack: {0, 1, {2}, 3, 2, {1}, {1, 2}, {0, 2}}`

These are the 8 subsets of 3 = {0, 1, 2, 3}.

Laziness comes into play at a certain size (currently, around 2^16 elements) because power sets can be very big.

`S∈tBang> 5^#`

`Stack: 32`

`S∈tBang> ^`

`Stack: {0,1,{1},2,{2},{0, 2},{1, 2},3,{3},{0, 3},{1, 3},{0, 1, 3},{3, 2},{0, 3, 2},{1, 3, 2},4, ...}`

`S∈tBang> _$`

`Stack: {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, ...}`

`S∈tBang> ^`

`Stack: {0,1,{1},2,{2},{0, 2},{1, 2},3,{3},{0, 3},{1, 3},{0, 1, 3},{3, 2},{0, 3, 2},{1, 3, 2},4, ...}`

`S∈tBang> ^`

`Stack: {0,1,{1},2,{{1}},{0, {1}},{1, {1}},{0, 1, {1}},{2},{0, 2},{1, 2},3,{2, {1}},{0, 2, {1}},{1, 2, {1}},{0, 1, 2, {1}}, ...}`

In theory, we can place huge sets on the stack. `$^`

isn’t only infinite but uncountably infinite. That said, SetBang can only generate a finite number of values, so most elements of that set P(ω)– namely, infinite subsets of ω– will never be generated.

*Choice*

One historically controversial axiom in set theory is the Axiom of Choice, which says that there exists a “function” f : **Set** -> **Set** such that f(X) ∈ X for all non-empty X.

Our choice command `\ `

has the behavior:

`... {} -> ... {} {}`

`... X -> ... (X - {e}) e where e ∈ X `

Which element does it pick? This is where SetBang gets weird and hilariously impractical: in this implementation, it’s *nondeterministic*. (It is not, however, *random*.) You can’t, in general, rely on elements being chosen in the same order. Why is it this way? Due to our needs as finite creatures, *how a set is computed* determines what we can observe. (This is the difference between {} and ⊥. The latter takes an infinite amount of time for one to figure out that it’s empty.) The easiest way to impose a choice function on sets is to create an ordering on them and, as a matter of rule, choose the *least* element. However, as implemented, infinite sets are processes that generate elements and one may never know when the least element has been generated.

I might remark that this presents a difference between an idealistic notion of choice (a function that exists platonically) and a practical desire, when possible, to produce elements quickly. SetBang favors the latter. Iterated choice will produce new elements in *some* order (until the set is empty).

For convenience, there’s also a choose-many command ```

whose behavior is:

`` : ... X K -> (X - Y) Y where #Y = min(#K, #X) and Y ⊆ X`

It’s used like so, to choose 9 elements from 65536 = {0, …, 65535}:

`S∈tBang> 4^#^#`

`Stack: 65536`

`S∈tBang> 9`;`

`Stack: {0, 60363, 20083, 25535, 39036, 10859, 41408, 65065, 50300}`

The mathematical Axiom of Choice is not actually necessary for SetBang’s choice operator to be allowed, insofar as only finitely many choices can actually be made, computationally speaking, and Choice isn’t required for that. So, to be pedantic, it’s technically `{\"}`

that requires the Axiom.

*Regularity*

The Axiom of Regularity is one of my favorite set-theoretic axioms. The others all define ways to construct sets, while Regularity imposes an upper bound on what can be sets. Strictly speaking, it says that every non-empty set contains an element that is disjoint from it. This prevents cycles in the membership relation (e.g. a ∈ a or b ∈ c ∈ d ∈ b) that would raise paradox-inducing questions around “set-ness”. Luckily, we don’t have to worry about that in SetBang: there is no way to produce a set that violates the Axiom of Regularity.

**The rest of the language**

We’re already Turing complete. We can express general recursion in `{} `

comprehensions, and we can get things out of sets, but so far we’ve been writing programs at the REPL that don’t actually do anything. If you run a file consisting of `00/ `

at the command line, you will *compute* the set {0}, but not actually do anything with it.

SetBang, therefore, comes with two I/O operators: `! `

and `@`

. Their behavior is as follows:

`! : ... X -> prints byte valued at min(#X, 255) to stdout`

`@ : ... -> ... C -> where C is the natural number corresponding to one byte read from stdin`

It might be tempting to use this in conjunction with set comprehensions, but that’s not a good idea, because set comprehensions are lazy, making it unpredictable when that I/O will happen. (In fact, it’s not legal SetBang to do I/O in a set comprehension.)

Imperative, eager loops can be realized using square brackets (`[]`

). The behavior of `is, if TOS is empty, to skip the block entirely. If TOS is not empty, then the program `

`code`

is executed and TOS is checked again, with execution repeating until TOS is empty.

Thus, this program is an “echo” that will take one character from stdin and echo it to stdout. It never terminates.

`1[@!]`

.

This program will put the bytes 10, 9, 8, …, 1 to the console.

`9'[~!\_#]`

The behavior of `\_#`

, please note, is comparable to decrement (except on 0, when it returns 0 because there is no -1).

**What else?**

There are 42 characters that can serve as SetBang operators. We’ve seen 41 of them.

`0`

for putting empty sets on the stack. Also:`1`

,`2`

, … ,`9`

as macro-like replacements for`0'`

,`0''`

and so on.`=`

and`?`

for equality and membership testing, returning`1`

on true and`0`

on false.- e.g.
`... {2, 4} 4 -> ... 1`

under`?`

since 2 ∈ 4 = {0, 1, 2, 3}.

- e.g.
`+`

for pairing (`... X Y -> ... {X, Y}`

).`'`

for the increment operator (`... X -> ... X ∪ {X}`

).`"`

for “quoting” a value into a set, (`... X -> ... {X}`

).- Equivalent to
`~+`

.

- Equivalent to
`_`

to pop from the stack (`... X -> ...`

).`~`

to duplicate TOS (`... X -> ... X X`

).`<`

and`>`

for rotations on the top of the stack,- e.g.
`3>`

is a program whose behavior is (`... X Y Z -> ... Z X Y`

).

- e.g.
`;`

as shorthand for`2>_`

(“swap-drop”) with behavior`... X Y -> ... Y`

.`{}`

for set comprehensions, which build sets based on existing sets.`&`

,`|`

,`-`

, and`.`

are for intersections, unions, differences, and exclusive-or respectively.`\`

(“choice”) and its (right-)inverse`/`

and the iterated choice```

.`\ : ... X -> ... (X - {e}) {e}`

`/ : ... X Y -> (X ∪ {Y})`

- and then, e.g.,
`... {2, 3, 5, 7} 2 -> {2, 11, 5} {7, 3}`

under```

.

`^`

which creates the power set (e.g.`... X -> ... P(X)`

).`#`

for replacing TOS with natural number according to cardinality.`... X -> #X`

works reliably when X is finite.- when X may be infinite, you’ll get a lazy possibly-infinite set.
`#`

does not handle transfinite cardinalities. It doesn’t detect uncountability, for example.

`()`

and`,`

for conditional execution based on TOS.`(^'#)`

executes the program`^'#`

if TOS is non-empty.`(^'#,_1)`

is similar but executes`_1`

if TOS is empty.

`$`

to push an infinite set {0, 1, 2, …} on to the stack.-
`!`

for output and`@`

for input. Both convert bytes to natural numbers between 0 and 255 inclusive. `[]`

for looping.`[\_#]`

will execute the program`\_#`

until TOS is empty.`1[]`

will loop forever (doing nothing).

`:`

for macro invocation.- Canonical example is
`:macro swap 2>`

. Then`:swap:_`

->`2>_`

.

- Canonical example is
`%`

for `… X Y -> {{X}, {X, Y}}, a sort of special pairing.- Could be implemented as
`2>~~+3>++`

.

- Could be implemented as

There’s one more (using the character `*`

) which, although useful, is also not necessary as a primitive of the language. If we wanted to build it, we could represent it as:

`(~#1=(_{}{}~,_~\2>\2>_.2>\2>\2>_&{}2>{}),0)`

For now, I’ll leave this “mystery operator” unexplored, but we’ll come back to it in the next installment. It turns out to be very useful.

**Going forward**

So, that’s the language. There are many questions to ask about it, though, such as:

- is it possible to make infinite sets “smarter”? For example, the program
`${'"}0`

searches for 0 in the set {1, 2, 3, …}. Because SetBang is deliberately dumb in interpreting infinite sets, that program will search forever. Is it possible to make it smarter? - Full SetBang has a lot of redundant commands. Is it possible to strip some of them out of the language? What’s the minimum possible SetBang?
- How should we handle error conditions (e.g. stack underflow)?
- Can we get better at handling larger ordinals like ω+1, ω*2, ω^ω and the like?
- What do useful programs in SetBang actually look like?
- SetBang is slow. Is it possible to make it faster?
- Is this language Turing complete; that is, as powerful as any other programming language? It turns out that the answer is
**Yes**, and I’ll prove it in Part 2.