GROUPING (Serial and Parallel)

"A set is Several Several that can be considered One" (Cantor).

"A set is a simultaneous consideration of entities" (Bertrand Russell).

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Serial Grouping (Sequence)
Semantics and Syntax

The serial grouping (also called "sequence") of the expressions x1, x2, ... , xn is a new higher-order expression in which its components are arranged in sequential order and which are evaluated serially.

There are two types of sequences:

1. Normal. The syntax is:
   (x1 x2 ... xn)
   
   
2. Holgadas. The syntax is:
   ( x1 x2 ... xn )

The series of expressions is delimited by curved parentheses, but in the case of baggy sequences there is an additional blank space at each end.

Both types of sequences are equivalent, except when the sequence is composed of a single component that is itself a sequence. For example, the sequence 123, which represents the normal three-component sequence (1 2 3), is not the same as the loose sequence ( 123 ), which has only one component, which is 123.


Justification

Grouping, in general, is a basic mechanism of consciousness that allows us to synthesize, that is, to consider a collection of individual elements as a unit, both spatially and temporally.

• At the spatial level, the components of a sequence are considered to occupy consecutive positions (spatial sequence) in abstract space.
  
  
• At the temporal level, the components are evaluated in sequence, one after the other (temporal sequence).

Examples

1. (123 xy a+b)
   
   
2. (a (b c) d)
   
   
3. (a a a a b b b c)
   
   
4. x+2+3 // equals the sequence (x + 2 + 3) ev. x+5
   
   
5. (x = 1)
(x x+1 x+2 x+3) // ev. 1234
   
   
6. ((x = 3) (y = 5) x+y x*y) // ev. (x=3 y=5 8 15)
   
   
7. (⟨(f(x) = x+7)⟩ f(3)) // ev. (⟨(f(x) = x+7)⟩ 10)
   (sequence of two elements, the first being the definition of a function)
   
   
8. (a α b)
   (sequence consisting of a, an expression and b)
   
   
9. (Ω a)
   (sequence consisting of Ω (representing all expressions) and a

Remarks

• A sequence is also called a "serial expression".
  
  
• A sequence can contain expressions of any type, including other sequences.
  
  
• The empty sequence is (). The sequence ( ) (sequence with one or more blanks) is equal to the empty sequence: ( ( ) = () ).
  
  
• A sequence may contain equal components.
  
  
• In a sequence the order of the components is important, so, for example, ab and ba are not equivalent.
  
  
• Several pairs of consecutive parentheses are replaced, when evaluated, by a single one. For example,
  
  ((((a b)))) // ev. (a b) ev. ab

(a ((((b c))))) d) // ev. (a (a (b c)) d) ev. a(bc)d

Sequence with atomic components

If the components of a serial expression are atomic, they can be specified together, thus saving parentheses. Examples:

(1 2 3 4) can be written 1234

(a b c) can be written abc

(a+b+c) can be written a+b+c

(a (b c) d) can be written a(b c)d  o  a(bc)d  o  (a bc d)

In fact, these expressions are evaluated towards their simplest and most compact form, eliminating intermediate targets. Examples:

1. (1 2 3 4) // ev. 1234
   
   
2. (a + b + c) // ev. a+b+c
(a (b c) d) // ev. (a bc d)

Single-component sequence

As we have said in the definition of sequence, it may happen that a sequence has only one component and that component is, in turn, a sequence. To "force" the existence of the main or higher sequence (of length 1), then separator targets must be inserted at the ends. Examples:

(a) // ev. a

(123) // ev. 123 eq. (1 2 3)

( 123 ) // eq. ( (1 2 3) ) (sequence of one component)

( 123 )# // ev. 1 (sequence of length 1)

123# // ev. 3 (sequence of length 3)


Names

A name is a sequence of letters (uppercase or lowercase) or digits, but always beginning with a letter. For example,

a31 Pepe k

Pseudo-sequences

These are sequences in which not all components are defined, i.e., there are null components. Example:

(x/40 = a) // the 40th component of x is a

(x/768 = b) // the 768 component of x is b

x is a pseudo-sequence, since its components 1 to 39 and 41 to 767 are not defined.


Text

A sequence of characters (letters, digits, symbols or special characters, including white) delimited by double quotes. The component characters are never evaluated separately, the text being a single unit. If you want to include the "double quotation mark" character, you must specify it in duplicate.

Examples:

"abc" // self-assesses.
"a+b*c" // self-assesses.
"ab""cd" // eq. ab "cd


Axioms

1. ⟨(((x θ) = x)⟩ // By the definition of null expression
   
   
2. ⟨((θ x) = x)⟩ // id.
   
   
3. ( (θ) = () ) // empty sequence
   
   
4. ⟨( ((( x ) = x) ← ( ( x# = 1) )⟩ // sequence of an atomic component

Parallel Grouping (Set)
Semantics

The parallel grouping (also called "set") of the expressions x1, x2, ... , xn is a new higher-order expression in which its components are evaluated in parallel, so that if there were equal components, the repeated ones are eliminated.


Syntax

{x1 x2 ... xn}

The series of expressions is delimited by braces.


Justification

Set is the dual concept of sequence:

• At the spatial level, the components of a set are considered to occupy the same place (spatial concurrency) in abstract space.
  
  
• At the temporal level, the components of a set are evaluated simultaneously (temporal concurrency).

Examples

1. {a b c d}
   
   
2. {a a b} // ev. {a b}
   
   
3. {ab ab ab} // ev. {ab}
   
   
4. {123 xy a+b}
   
   
5. {123 {xy a+b} abc}
   
   
6. {2*a a=b ccc}
   
   
7. {a/b a+b 123.5 xxx}
   
   
8. {a b b b c c c c} // ev. {a b c}
   
   
9. (x = 1) (y = 1).
{x y} // ev. {1}
   
   
10. (y = 4).
{(x = 3) x*y} // ev. {x=3 x*4}
    
    
11. {a α b}
    (set consisting of a, an undefined expression and b)

Remarks

• A set is also called a "parallel expression".
  
  
• A set can contain expressions of any type, including other sets.
  
  
• The empty set is {}, which is also symbolized by ∅.
  
  
• In a set, the order of the components is indifferent, so the following expressions are equivalent:
  
  {a b c} {a c b} {b a c} {b c a} {c a b} {c b a}
  
  
• The expression {a b c} cannot be written {abc}, since it would be interpreted {(a b c)}, that is, as a set formed by the sequence abc.
  
  
• There do not exist, as in sequences, loose sets, since for example the expressions {123} and { 123 } are identical: they specify a set of a single component, which is the sequence 123.
  
  { 123 } // ev. {123} (blanks are removed)
  
  
• Several consecutive braces do not, unlike curly brackets, equal one brace. For example,
  
  {{{a b}}} // is not equivalent to {a b}}
(a {{{b c}}}} d) // not equivalent to (a {{b c}} d)

Axioms

1. ⟨({x x} = {x}]⟩ //by the definition of set
   
   
2. ⟨({x y} ≡ {y x}]})⟩ // id.
   
   
3. ⟨({x θ} = {x})⟩ // by definition of null expression
   
   
4. ⟨({x Ω} = {Ω})⟩ // by the definition of universal expression
   
   
5. ( {θ} = {} ) // empty set
   
   
6. (∅ =: {} ) // ∅ is the name of the empty set