Typographical Number Theory (also known as TNT) is a formal axiomatic system describing the natural numbers that appears in Douglas Hofstadter's book Gödel, Escher, Bach. It is an implementation of Peano arithmetic. In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Douglas Richard Hofstadter (born February 15, 1945) is an American academic. ... GEB cover Gödel, Escher, Bach: an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize-winning book by Douglas Hofstadter, published in 1979 by Basic Books. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ...

Like any system implementing the Peano axioms, Typographical Number Theory is capable of referring to itself (it is self-referential). A self-reference occurs when an object refers to itself. ...

Numerals

In Typographical Number Theory we do not have a distinct symbol for each natural number. Instead we use a simple, uniform way of giving a compound symbol to each natural number: In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ...

zero	0
one	S0
two	SS0
three	SSS0
four	SSSS0
five	SSSSS0

The symbol S can be interpreted as "the successor of", or "the number after". Since this is, however, a number theory, such interpretations are useful, but not strict. We cannot say that because four is the successor of three that four is SSSS0, but rather that since three is the successor of two, which is the successor of one, which is the successor of zero, which we have descibed as 0, four can be "proved" to be SSSS0. Typographical Number Theory is designed such that everything must be proved before it can be said to be true. This is its true power, and to undermine it would be to undermine its very usefulness.

Variables

We need a way of refering to unspecified terms, or variables. In TNT, there exist five variables. These are In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ...

a, b, c, d, e.

More variables can be constructed by adding the prime symbol after them, so This article is not about the symbol for the set of prime numbers, â„™. The prime (′, Unicode U+2032, ′) is a symbol with many mathematical uses: A complement in set theory: A′ is the complement of the set A A point related to another (e. ...

a', b', c', a'', a'''

are all variables.

In the stricter version of TNT, known as "austere" version of TNT, only

a', a'', a''' etc.

exist.

Operators

Addition and multiplication of numerals

In Typographical Number Theory, the usual symbols of "+" for additions, and "·" for multiplications are used. Thus to write "b plus c", we write

(b+c)

and "a times d" is written as

(a·d) .

The parentheses are required. Any laxness would violate TNT's formation system. Also only two terms can be operated on at once. Therefore to write "a plus b plus c", we must write either

((a+b)+c)

or

(a+(b+c)) .

Equivalency

The "Equals" operator is used to denote equivalence. It is defined by the symbol "=", and takes roughly the same meaning as it usually does in mathematics. For instance,

SSS0+SSS0=SSSSSS0

is a true statement in TNT, with the meaning "3 plus 3 equals 6".

Negation

In Typographical Number Theory, negation, i.e. the turning of a statement to its opposite, is denoted by the "~" or negation operator. For instance, Negation (i. ...

~(SSS0+SSS0)=SSSSSSS0

is a true statement in TNT.

Quantifiers

There are two quantifiers used: ∀ and ∃.

• ∃ means "There exists"
• ∀ means "For every"
• The symbol : is used to separate a quantifier from other quantifiers or from the rest of the formula.

For example:

∀a:∀b:a+b=b+a

("For every number a and every number b, a plus b equals b plus a", or more figuratively, "Addition is commutative.")

~∃c:Sc=0

("There does not exist a number c such that c plus one equals zero", or more figuratively, "Zero is not the successor of any (natural) number.")

Atoms and propositional statements

All the symbols of propositional calculus are used in Typographical Number Theory, and they retain their interpretations. In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables. ...

Atoms are here defined as strings which amount to statements of equality, such as

~S0=SS0 .

2 plus 3 equals four:

(SS0+SSS0)=SSSS0

2 plus 2 is not equal to 3:

~(SS0+SS0)=SSS0