Löb’s theorem

Löb’s theorem

Löb’s

theorem

In this short article we will provide an example where one’s intuition can be faulty, and that strict formalization is always needed when one does mathematics in order to avoid possible hidden intuitive inconsistencies. A striking example is the erroneous but intuitively obvious thesis: ”if a

formal system

F

proves that there exists proof of

formula

A

in

F

, then

F

has proved

A

” or slightly differently version is this one: ”it must be always provable in all formal systems that from formula saying that there exists proof of

A

in

F

, indeed follows

A

in

F

”.
Both formulations of this naive intuition concerning basics of math logic severely fail. We will show that a

consistent

F

cannot prove for arbitrary formula

ϕ

n

that from formula that asserts “exists proof
of

ϕ

n

” follows

ϕ

n

. This is a major ingredient of a theorem proved by Löb in 1955.

Löb’s theorem: If

F

⊢

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

⇒

ϕ

n

, where

x

is the Gödel number of the proof of the formula with Gödel number

n

, and

n

ˇ

is the numeral of the Gödel number of
the formula

ϕ

n

, then

F

⊢

ϕ

n

.

The proof presented here follows Karlis Podnieks (2006):

Suppose

F

⊢

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

⇒

ϕ

n

.
Let

F

*

be the new

proof system

that is equal to

F

plus the new axiom

¬

⁢

ϕ

n

. By

modus tollens

F

*

⊢

¬

⁢

ϕ

n

⇒

¬

⁢

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

.
Since

F

*

has as axiom

¬

⁢

ϕ

n

by

modus ponens

F

*

⊢

¬

⁢

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

.
In this case

F

*

proves that

ϕ

n

is not provable in

F

(and also that

F

is consistent).
This however leads to proof that

F

*

is also consistent because it contains only

F

and

¬

⁢

ϕ

n

, and we already know both that

F

is consistent and that

F

does not prove

ϕ

n

. Thus

F

*

proves its own

consistency

. According to Gödel’s second theorem however

F

*

cannot prove
its own consistency unless being inconsistent. Therefore

F

+

¬

⁢

ϕ

n

must be always inconsistent theory, and we conclude that

F

⊢

ϕ

n

.

Now see that the intuitive notion that

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

⇒

ϕ

n

is not provable within arbitrary formal system. If the formal system is inconsistent obviously it proves every formula so the above intuitive
notion will be provable. If however

F

is consistent it cannot prove

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

⇒

ϕ

n

for arbitrary formula, because

if it could

, for formally refutable formula

ϕ

k

≡

[

0

=

1

]

,

F

could

prove by modus tollens

¬

⁢

ϕ

k

⇒

¬

⁢

∃

x

⁢

P

⁢

(

x

,

k

ˇ

)

, and since

¬

⁢

ϕ

k

is formally provable formula, then the formal system

could

infer its own consistency, or said in ordinary language: from

¬

⁢

ϕ

k

,

F

could

have proved ”there exists at least one unprovable formula in

F

”, which is impossible according to Gödel’s second theorem.

Corollary: If

F

is consistent formal system then

F

cannot prove for arbitrary formula

ϕ

n

that from formula that asserts “exists proof of

ϕ

n

” follows

ϕ

n

.

The proof of the corollary is straigthforward either directly from Löb’s theorem, or by

independent

reasoning using the fact that consistent

F

cannot prove

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

⇒

ϕ

k

for refutable

ϕ

k

.

The relationship between the direct text of Löb’s theorem in the proof provided by Podnieks, and the corrolary is this one: if the system

F

is consistent, it might be able to prove

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

⇒

ϕ

n

for provable formulas

ϕ

n

, because the provability of

ϕ

n

taken together with the consistency of

F

ensure unprovability of

¬

⁢

ϕ

n

and hence modus tollens reversed formula of

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

⇒

ϕ

n

cannot be used for

F

to prove its own consistency. Also taking into account the second Gödel theorem it is easy to be seen that

F

should not be able to prove

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

⇒

ϕ

k

for disprovable (refutable) formulas

ϕ

k

, a result that alone is

sufficient

to prove the corollary. Still the corollary is a weaker result than Löb’s theorem, because the corrolary does not give us clue whether

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

⇒

ϕ

p

is provable for

undecidable

formulas

ϕ

p

. This was indeed the ”

open problem

” proposed by Leon . The Löb’s theorem answers this question, and shows that it is impossible for consistent

F

to prove

∃

x

⁢

P

⁢

(

x

,

n

ˇ

)

⇒

ϕ

p

for undecidable formulas

ϕ

p

. Therefore summarized, Löb’s theorem says that for refutable or undecidable formulas

ϕ

, the intuition ”if “exists proof of

ϕ

” then

ϕ

” is erroneous.

Modus tollens inversed Löb’s theorem: If

F

is consistent formal system then

F

cannot prove for any unprovable formula

ϕ

k

that from formula that asserts “exists proof of

ϕ

k

” follows

ϕ

k

.

Comparing the above theorem with the corrolary shows small but significant

difference

, which makes it stronger

proposition

(note: in consistent formal system

F

unprovable formulas are all refutable (disprovable) formulas, as well as all undecidable formulas).

References

1. Crossley JN, Brickhill C, Ash C, Stillwell J, Williams
N (1972)

What is mathematical logic?

Oxford University Press.

2. Detlovs V, Podnieks K (2006)

http://www.ltn.lv/ podnieks/mlog/ml.htm

Introduction

to
Mathematical Logic

. Hypertextbook for students in mathematical logic

3. Gödel K (1931) Über formal unentscheidbare Sätze der
Principia und verwandter Systeme I.

Monatshefte
für Mathematik und Physik

38

: 173-198.

4. Löb MH (1955) Solution of a Problem of Leon Henkin.

The Journal of Symbolic Logic

20

(2): 115-118.

5. Podnieks K (2006)

http://www.ltn.lv/ podnieks/gt.html

What is Mathematics? Gödel’s
Theorem and Around

. Hypertextbook for students in mathematical logic

Title

Löb’s theorem

Canonical name

LobsTheorem

Date of creation

2013-03-22 17:05:19

Last modified on

2013-03-22 17:05:19

Owner

dankomed (17058)

Last modified by

dankomed (17058)

Numerical id

25

Author

dankomed (17058)

Entry type

Theorem

Classification

msc 03F03

Classification

msc 03F45

Classification

msc 03F40

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