III: Typographical Number Theory
================================
In this section, we define Hofstadter's TNT [with some subtle modifications].
[TNT is PA]
Language of TNT
---------------
Alphabet:
0 S + * ( ) =
~ /\ \/ =)
a b c d e x y z '
A E
(We no longer have propositional variables.)
Variables:
a, b, c, d, e, x, y, z are variables
If v is a variable, so is v'.
Terms:
any variable is a term;
0 is a term;
if t and s are terms, then so are
St, (t+s), (t*s);
nothing else is a term.
wffs:
if t and s are terms, then t=s is a wff (an _atomic formula_);
if \phi and \psi are wffs, so are
~\phi, <\phi /\ \psi>, <\phi \/ \psi>, <\phi =) \psi>;
if \phi is a wff and v is a variable, then
Av:\phi and Ev:\phi
are wffs;
nothing else is a wff.
Remark:
We have unique parse trees.
Bound and free variables:
An _occurrence of a variable_ v in a wff is a location in the wff where
the variable appears, where appearances in substrings of the form "Av:" or
"Ev:" do not count.
(e.g. there are *no* occurrences of y in "Ay:y'=y'", but two of y')
An occurrence of a variable v in a wff is _bound_ if it occurs within a
substring of the form "Av:\phi" or "Ev:\phi" (\phi a wff). Else, the
occurrence is _free_.
The _free variables_ of a wff are those variables which occur free in the
wff.
The standard interpretation:
Variables stand for natural numbers.
Call a choice of natural number for each variable, i.e. a map
f : [variables] -> N, a "variable assignment".
Given a variable assignment f, we evaluate terms as natural numbers:
eval_f(v) = f(v)
eval_f(0) = 0
eval_f(St) = eval_f(t) + 1 ("Successor")
eval_f((t+s)) = eval_f(t) + eval_f(s)
eval_f((t*s)) = eval_f(t) * eval_f(s)
Now we determine truth of a wff wrt a variable assignment f:
* An atomic formula "t=s" is true wrt f iff eval_f(t) = eval_f(s).
* "<\phi /\ \psi>" is true wrt f iff \phi and \psi are both true wrt f.
* Similarly for ~, \/, =), as in propositional logic.
* Av:\phi is true wrt f iff \phi is true for any
variable assignment f' which is the same as f except possibly on v.
(i.e. f'(w)=f(w) if w!=v)
[ with notation:
Av:\phi ^ f = T iff \forall f'. \forall w\in Variables.
((w!=v -> f'(w) = f(w)) -> \phi ^ f' = T)
]
* Ev:\phi is true wrt f iff \phi is true for some
variable assignment f' which is the same as f except possibly on v.
Clearly, whether a wff \phi is true wrt a variable assignment f depends
only on the values of f at the free variables of \phi.
A wff with no free variables is a _sentence_, and is just true or false.
A wff with 1 free variable expresses a _property_ of a natural number
(e.g. primeness, oddness...).
A wff with n free variables expresses an _n-ary relation_ (aka _predicate_)
(1-ary == unary, 2-ary == binary etc)
(e.g. "x is less than y"; "x is between y and z").
Examples:
Ax: Ey: Sx=y
(first think what Ey: Sx=y says about x
(first think what Sx=y says about x,y))
Ex: Ay: Sx=y
(remark: cf ambiguity of english
"every number is the predecessor of some number")
Ax: Ey: x=Sy
Ax:
Ey: (y+y)=x
Ey: S(y+y)=x
Ax:
Ax:
Ax: Ey: <(y+y)=x =) ~(x*x)=x>
Ez: x=(y+z)
Ax: Ez: (x*x)=(x+z)
Ez: x=(y+Sz)
Ax: Ez: (x*x)=(x+Sz)
<~x=0 /\ <~x=S0 /\ Ay:Az:<(y*z)=x -> >>>
Euclid:
Ax: Ey: Ez:
Fermat (n=3):
~Ex:Ey:Ez:(x*(x*x))+(y*(y*y))=(z*(z*z))
Goldbach:
Ax:Ey:Ez:<<~Ey':Ez':y=(SSy'*SSz') /\ ~Ey':Ez':z=(SSy'*SSz')> /\ (y+z)=(x+x)>
Non-standard interpretations:
A _structure in the language of arithmetic_ consists of
a set N';
an element 0' \in N';
a unary function S' : N' -> N';
binary functions +', *' : N'^2 -> N'.
We denote the structure by , or just N'.
The _standard_ arithmetic structure is , i.e. the set of
natural numbers N, with usual 0, successor, addition and
multiplication.
An assignment of variables for N' assigns an element _of N'_ to each
variable; terms evaluate to elements of N' using 0', S', +', and *';
wffs evaluate, wrt a variable assignment, to True/False as above (so "Ax:"
now means "for all x in N'").
For a sentence \sigma, we write
N' |= \sigma
to mean that \sigma is true when interpreted in N'.
Example: the integers Z with usual zero, successor, addition, and
multiplication is a structure in the language of arithmetic.
"Ex:Sx=0" is true in Z but not in N.
(Z |= Ex:Sx=0, but N |/= Ex:Sx=0)
PRED
----
Axioms:
Axiom 0: Ax:x=x
Rules:
Rules of the propositional calculus; premises of fantasies may now be
arbitrary *TNT*-wffs.
Generalisation: \phi |-> Av:\phi
where v is a variable.
RESTRICTION: v must not occur free in any premise of \phi.
Specification: Av:\phi |-> \phi[t/v]
where \phi[t/v] is the result of replacing each free occurrence of the
variable v in \phi with the term t.
RESTRICTION: any new occurrences of variables resulting from the
substitution must be free.
Interchange: XAv:~Y <-|-> X~Ev:Y
X~Av:Y <-|-> XEv:~Y
where v is a variable;
i.e. whenever "Av:~" occurs within a wff, it may be rewritten as
"~Ev:", and vice-versa.
Existence: \phi[t/v] |-> Ev:\phi
where \phi is a wff, v is a variable, t is a term, and \phi[t/v] is
the result of replacing each free occurrence of v in \phi with t.
RESTRICTION: the substitution must meet the restriction imposed in the
specification rule: any occurrences of variables created in
passing from \phi to \phi[t/v] must be free.
Symmetry: t = s |-> s = t
Transitivity: (t = s, s = r) |-> t = r
Congruence:
t=s |-> St = Ss
(t_1=s_1, t_2=s_2) |-> (t_1+t_2) = (s_1+s_2)
(t_1=s_1, t_2=s_2) |-> (t_1*t_2) = (s_1*s_2)
where t,s,r,t_i,s_i are terms.
Notation:
|- \phi means \phi is a PRED-theorem
(note \phi can have free variables)
Examples:
|- :
[
Ax:Ay:x=y
Ay:Sy=y
]
|- :
[
Ax:Ay:x=y
Ay:Sy=y
Sx=x
Ax:Sx=x
]
|- Ax: =) x=SSx>:
[
~x=SSx
[
x=Sy
y=Sx
Sy=SSx
x=SSx
]
< =) x=SSx>
<~x=SSx =) ~>
~
Ay:~
~Ey:
]
<~x=SSx =) ~Ey:>
=) x=SSx>
Ax: =) x=SSx>
Non-examples (demonstrating necessity of the restrictions):
[
x=0
Ax:x=0
]
Ax:
Uhoh!
[
Ax:Ey:~x=y
Ey:~y=y
]
Uhoh!
Ax:x=x
Ey:Ax:y=x
Uhoh!
[
Ax:x=(x*S0)
Ex:Ax:x=(x*x)
]
Uhoh!
[
Ax:~x=Sx
~x=Sx
Ex:~x=x (existence, t:=Sx)
]
Uhoh!
Definition:
A _TNT-tautology_ is a substitution instance of a propositional tautology,
obtained by replacing propositional variables with TNT-wffs.
Remark:
By completeness of PROP, any TNT-tautology is a PRED-theorem.
(Make the substitution in a PROP-derivation, yielding a PRED-derivation)
Example:
|- :
[
Ex:~x=x
~Ax:x=x (interchange)
Ax:x=x (axiom 0)
(joining)
[...lines proving following tautology omitted...]
< =) (SS0+SS0)=SSSSS0>
(SS0+SS0)=SSSSS0 (detachment)
]
(fantasy rule)
For convenience, we add all TNT-tautologies to PRED as axioms.
[ omitting this for now; might put it in later if it surviving without it is
too annoying:
Lemma:
Suppose |- <<\phi =) \phi'> /\ <\phi' =) \phi>>,
and suppose \theta is a formula in which \phi occurs as a subformula, and
\theta' is the result of replacing that subformula with \phi'.
Then |- <<\theta =) \theta'> /\ <\theta' =) \theta>>.
Proof:
[omitted, but straightforward by induction on length of \theta, and using
the previous remark]
For convenience, we add as a rule to PRED:
Substitution: \theta |-> \theta'
whenever \theta and \theta' are as in the previous lemma.
]
Remark:
The existence rule can be deduced from the specification rule and
interchange:
[
~Ev:\phi
Av:~\phi
~\phi[t/v]
]
<~Ev:\phi =) ~\phi[t/v]>
<\phi[t/v] =) Ev:\phi>
Soundness and completeness
--------------------------
Notation:
For \Sigma a set of sentences and \tau a sentence, write
\Sigma |- \tau
if \tau is a theorem of the system PRED+\Sigma obtained by adding \Sigma
as axioms to PRED, and
\Sigma |= \tau
if \tau is satisfied by every structure in the language of arithmetic
which satisfies all the sentences in \Sigma; i.e.
\Sigma |= \tau <=> for all N': N'|=\Sigma => N'|=\tau
Fact (Soundness):
If \Sigma |- \tau, then \Sigma |= \tau
Fact (Gödel's Completeness Theorem):
If \Sigma |= \tau, then \Sigma |- \tau
(So \Sigma |= \tau <=> \Sigma |- \tau)
// Allaying possible confusion concerning the term 'complete':
Definition:
A system in the language of TNT is _complete for the standard
interpretation_, abbreviated "N-complete" or "complete for N", if every
TNT-sentence which is true in the standard interpretation is a theorem.
It is _negation complete_ if for every TNT-sentence \sigma, at least one
of \sigma and ~\sigma is a theorem.
Remark:
N-complete => negation complete.
PRED is not negation complete!
e.g. neither 0+0=0 nor ~0+0=0 is a theorem!
GIT1 proves negation incompleteness, hence N-incompleteness.
TNT'
----
Definition:
TNT' is the system obtained by adding the following axioms to
PRED:
Axiom 1: Ax:~Sx=0
Axiom 2: Ax:(x+0)=x
Axiom 3: Ax:Ay:(x+Sy)=S(x+y)
Axiom 4: Ax:(x*0)=0
Axiom 5: Ax:Ay:(x*Sy)=((x*y)+x)
Axiom 6: Ax:Ay:
We will also write "TNT'" to refer to the set of these 6 axioms, so
TNT' |- \sigma
means that \sigma is a TNT'-theorem.
Remark:
Hofstadter has the rule
Drop S: Sx=Sy |-> x=y
in place of Axiom 6; this makes no real difference - the two systems prove
the same theorems.
[ Remark for the initiated: TNT' is basically Robinson's Q, although we're
missing the axiom that only 0 has no predecessor (which is ok for our
purposes, as this is implied by the induction axioms)]
Note N |= TNT', so by soundness if TNT' |- \sigma then N |= \sigma.
Example:
TNT' |- S0+S0=SS0:
Ax:Ay:(x+Sy)=S(x+y)
Ay:(S0+Sy)=S(S0+y)
(S0+S0)=S(S0+0)
Ax:(x+0)=x
S0+0=S0
S(S0+0)=SS0
(S0+S0)=SS0
Fact:
TNT' can prove every sentence which is true in N of the form t=s,
where t and s are terms.
Example:
TNT' |- Ax:(x*(S0+S0))=((x*S0)+x):
(S0+S0)=SS0 (shown above)
Ax:x=x
x=x
(x*(S0+S0))=(x*SS0)
Ax:Ay:(x*Sy)=((x*y)+x)
Ay:(x*Sy)=((x*y)+x)
(x*SS0)=((x*S0)+x)
(x*(S0+S0))=((x*S0)+x)
Ax:(x*(S0+S0))=((x*S0)+x)
TNT' is still not N-complete!
In other words, there are "non-standard" structures in the language of
arithmetic which satisfy axioms 1-6, but satisfy sentences N does not.
Example:
Let Mat_2(N) be the structure in the language of arithmetic
consisting of 2x2 matrices with natural number entries, with matrix
addition and matrix multiplication and with S(M) := M+I, where I is the
identity matrix.
Then Mat_2(N) |= TNT'.
Now let \sigma be the sentence Ax:.
Clearly N |= \sigma.
But M |= ~\sigma, since
2
(01) = (00)
(00) (00)
So by soundness of PRED, {Ax 1-6} |/- \sigma.
However, by the Fact above, whenever t is a numeral
(i.e. one of 0, S0, SS0, ...),
<(t*t)=0 =) t=0>
*is* a TNT'-theorem!
Similarly, the following are *not* TNT'-theorems:
Ax:(0+x)=x
Ax:Ay:(x+y)=(y+x)
Ax:Ay:(x*y)=(y*x).
[ Remark: Mat_2(\N) |/= Q, though ]
TNT
===
What's missing?
In N, if \phi[0/v] and \phi[S0/v] and \phi[SS0/v] and so on
all hold, then so does Av:\phi.
But e.g. if \phi := <(x*x)=0 =) x=0>, then \phi[0/x] and \phi[S0/x] and
\phi[SS0/x] and so on are all theorems, but Av:\phi is not. Similarly with
\phi := (0+x)=x.
Proposed "Rule of All":
(\phi[0/v], \phi[S0/v], \phi[SS0/v], ...) |-> Av:\phi
BUT rules of formal systems have *finitely* many inputs, this has *infinitely*
many. You could never use this rule as part of a finite derivation!
Consider how we prove statements of the form "for all n" in everyday
mathematics...
Induction rule:
( \phi[0/v], Av:<\phi =) \phi[Sv/v]> ) |-> Av:\phi
where v is a variable and \phi is a wff, and \phi[t/v] is the result
of replacing each free occurence of v in \phi with the term t).
// But let's use axioms rather than a rule, so we have access to soundness and
// completeness.
Definition:
TNT is the system obtained by adding to TNT' the following infinite set of
axioms:
Induction axioms: for each wff \phi with one free variable v, the axiom
<<\phi[0/v] /\ Av:<\phi =) \phi[Sv/v]>> =) Av:\phi> .
(Again, we will also use "TNT" to refer to the set of axioms)
Remark:
TNT is more commonly known as PA ("(first-order) Peano arithmetic")
Example: Ax:(0+x)=x is a TNT-theorem:
1. Ax:(x+0)=0
2. (0+0)=0
3. [
4. (0+x)=x
5. Ax:Ay:(x+Sy)=S(x+y)
5. Ay:(0+Sy)=S(0+y) (spec x->0)
6. (0+Sx)=S(0+x) (spec y->x)
7. S(0+x)=Sx
8. (0+Sx)=Sx
9. ]
10. <(0+x)=x =) (0+Sx)=Sx>
11. Ax:<(0+x)=x =) (0+Sx)=Sx> (gen)
12. Ax:(0+x)=x (induction: lines 2, 11)
Remark:
N |= TNT, so any TNT-theorem is true in N (i.e. TNT is sound for N).
Question:
Does the converse hold? i.e. is TNT complete for N?
We will answer this presently!
Related question:
If a structure N'= satisfies TNT, must every element of N' be
one of 0', S'0', S'S'0', ...?
Answer: no! However, there aren't any easily described examples like
Mat_2(N). [See Tennenbaum's theorem]
Fact:
There is a Post formal system, FormalTNT, such that a wff is a theorem of
TNT iff it is a theorem of FormalTNT.
Appendices:
===========
A: deviations from Hofstadter
-----------------------------
PRED is set up so as to satisfy Gödel's completeness theorem - this meant
adding Ax:x=x as an axiom, and inserting the congruence rules. I also added
the other form of interchange, because surviving without it is painful (though
possible).
The existence rule got rewritten. Here's an equivalent version which looks
more like Hofstadter's:
Existence: \phi |-> Ev:\phi'
where \phi is a wff, v is a variable, t is a term, and \phi' is the
result of replacing one or more occurrences of t in \phi with v.
RESTRICTION: no bound occurrences of variables may be created or
destroyed in passing from \phi to \phi', and there may be no
occurrences of v in \phi' other than those introduced through
replacing occurrences of t in \phi with v.
"Drop S" became Axiom 6, and the induction rule became a set of axioms, to
ensure that TNT is of the form PRED+\Sigma.
B: FormalTNT
------------
We can implement TNT in a Post formal system.
It's more than a little ugly! But conceptually it's straight-forward.
[Again, I'm omitting this from the lectures, but including it here for the
curious.]
(note that 'x', 'y' and 'z' are now in our alphabet, so we use 'X', 'Y', 'Z',
'X1', 'Z37' and so on for variables when giving production rules.
Let's simplify things by removing E from our formal system, considering "Ev:"
to be just an abbreviation for "~Av:~".
Alphabet: as above, but add new symbols |- ? |, and all the roman alphabet in
lower case and in upper case, except X Y and Z.
Axioms and Production rules:
Var|x
Var|y
Var|z
Var|X |-> Var|X'
Var|X |-> Term|X
VarNeq|x|y
VarNeq|y|z
VarNeq|z|x
VarNeq|X|Y |-> VarNeq|Y|X
(Var|X, Var|XY') |-> VarNeq|X|XY'
Term:0
Term|X |-> Term|SX
(Term|X, Term|Y) |-> Term|(X+Y)
(Term|X, Term|Y) |-> Term|(X*Y)
(Term|X, Term|Y) |-> WFF|X=Y
WFF|X |-> WFF|~X
(WFF|X, WFF|Y) |-> WFF|
(WFF|X, WFF|Y) |-> WFF|
(WFF|X, WFF|Y) |-> WFF|
// NoFree|Z|Y : variable Z doesn't appear free in wff Y
// NoFreeT|Z|Y : variable Z doesn't appear in term Y
VarNeq|Z|Y |-> NoFreeT|Z|Y
NoFreeT|Z|0
NoFreeT|Z|X |-> NoFreeT|Z|SX
(NoFreeT|Z|X, NoFreeT|Z|Y) |-> NoFreeT|Z|(X+Y)
(NoFreeT|Z|X, NoFreeT|Z|Y) |-> NoFreeT|Z|(X*Y)
(NoFreeT|Z|X, NoFreeT|Z|Y) |-> NoFree|Z|X=Y
(NoFree|Z|X, NoFree|Z|Y) |-> NoFree|Z|
(NoFree|Z|X, NoFree|Z|Y) |-> NoFree|Z|
(NoFree|Z|X, NoFree|Z|Y) |-> NoFree|Z|
NoFree|Z|X |-> NoFree|Z|~X
(Var|Y, NoFree|Z|X) |-> NoFree|Z|AY:X
WFF|X |-> NoFree|Z|AZ:X
// NoFreePrems|Z|X : variable Z doesn't appear in any of the wffs in the
// ?-separated list X
Var|Z |-> NoFreePrems|Z|
(NoFree|Z|X, NoFreePrems|Z|Y) |-> NoFreePrems|Z|Y?X
(X|-Y, NoFreePrems|Z|X) |-> X|-AZ:Y // (generalisation)
// Sub|X|Z|Z1|Y : Y is the result of validly substituting all free
// occurrences of the variable Z in the wff X with the term Z1, where
// "valid" means that no variable occurring in Z1 gets put in somewhere it
// gets bound.
// SubT|X|Z|Z1|Y : same, but X is a term.
(Var|Z, Term|Z1) |-> SubT|Z|Z|Z1|Z1
(Var|Z, Term|Z1) |-> SubT|0|Z|Z1|0
SubT|X|Z|Z1|Y |-> SubT|SX|Z|Z1|SY
(SubT|X|Z|Z1|Y, SubT|X1|Z|Z1|Y1) |-> SubT|(X+X1)|Z|Z1|(Y+Y1)
(SubT|X|Z|Z1|Y, SubT|X1|Z|Z1|Y1) |-> SubT|(X*X1)|Z|Z1|(Y*Y1)
(SubT|X|Z|Z1|Y, SubT|X1|Z|Z1|Y1) |-> Sub|X=X1|Z|Z1|Y=Y1
(Sub|X|Z|Z1|Y, Sub|X1|Z|Z1|Y1) |-> Sub||Z|Z1|
(Sub|X|Z|Z1|Y, Sub|X1|Z|Z1|Y1) |-> Sub||Z|Z1|
(Sub|X|Z|Z1|Y, Sub|X1|Z|Z1|Y1) |-> Sub||Z|Z1|
Sub|X|Z|Z1|Y |-> Sub|~X|Z|Z1|~Y
(VarNeq|Z|Z2, NoFreeT|Z2|Z1, Sub|X|Z|Z1|Y) |-> Sub|AZ2:X|Z|Z1|AZ2:Y
(VarNeq|Z|Z2, Sub|X|Z|Z1|X) |-> Sub|AZ2:X|Z|Z1|AZ2:X
(Var:Z, WFF:X) |-> Sub|AZ:X|Z|Z1|AZ:X
(X|-AZ:Y, Term|Z1, Sub|Y|Z|Z1|Y1) |-> X|->Y1 // (specification)
(Term|X, Term|Y, Z|-X=Y) |-> Z|-Y=X (symmetry)
// other equality rules similar and omitted
// (the following is the same as for PROP)
|-
(X|-Y, WFF:Z) |-> X?Z|-Z // (pushing into a fantasy)
(X|-Y, WFF:Z) |-> X?Z|-Y // (carry-over)
(X?Y|-Z, WFF:Y) |-> X|- // (popping out of a fantasy)
X|- |-> X|-Y
X|- |-> X|-Z
(X|-Y, X|-Z) |-> X|-
// and so on for the other deduction rules of PROP
|-Ax:x=x
// and similarly for axioms 1-6
// Induction axiom scheme:
(Sub|X|Z|0|Y, Sub|X|Z|SZ|Y1) |-> |-<>=)AZ:X>
|-X |-> X // (deriving wffs)