Faith in mathematics

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Crisis Truth and provability True but unprovable Incompleteness Outlook

r Faith in mathematics r

34th Chaos Communication Congress

Questions are very much welcome! Please interrupt me mid-sentence.

Ingo Blechschmidt (University of Augsburg)

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1 The foundational crisis in mathematics

2 Truth and provability

3 True but unprovable statements

4 Fundamental incompleteness

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Part I

The foundational crisis in mathematics

“LetU be the set of all those sets which don’t contain themselves.” Naive mathematics isinconsistent, rendering itunreliable./

Thus theaxiomatic methodwas born.

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Part I

The foundational crisis in mathematics

“LetU be the set of all those sets which don’t contain themselves.”

Naive mathematics isinconsistent, rendering itunreliable./ Thus theaxiomatic methodwas born.

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Part I

The foundational crisis in mathematics

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Part II

Truth and provability

A syntactic quality

A statement isprovableif and only if it has aformal proofusing only thePeano axioms.

Example.1+1=S(0)+S(0)=S(S(0)+0)=S(S(0))=2.

A semantic quality

A statement istrueif and only if it holds in thestandard model.

Provable statements are true. True statements arenot necessarily provable.

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Part II

Truth and provability

A syntactic quality

A statement isprovableif and only if it has aformal proofusing only thePeano axioms

Mathematical induction and:

S(n),0 S(n)=S(m)⇒n=m n+0=n n+S(m)=S(n+m)

n·0=0 n·S(m)=n·m+n

.

Example.1+1=S(0)+S(0)=S(S(0)+0)=S(S(0))=2.

A semantic quality

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Part II

Truth and provability

A syntactic quality

n·0=0 n·S(m)=n·m+n

.

Example.1+1=S(0)+S(0)=S(S(0)+0)=S(S(0))=2.

A semantic quality

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Part II

Truth and provability

A syntactic quality

n·0=0 n·S(m)=n·m+n

.

Example.1+1=S(0)+S(0)=S(S(0)+0)=S(S(0))=2.

A semantic quality

A statement istrueif and only if it holds in thestandard model

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. Provable statements are true.

True statements arenot necessarily provable.

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Part III

True but unprovable statements

For statements aboutspecific numbers, there is no difference between provability and truth. But all of the following are unprovable:

“This statement is not provable.”

Buttake care: Consider “This statement is not true”.

“Hercula can kill any hydra.”

“BB(9000)=x.” (for the correct valuex)

“There is no proof of 1=0.” (“Peano arithmetic is

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Part IV

Fundamental incompleteness

Gödel discovered:

Any consistent and recur- sively axiomatizable formal system isincomplete.

Going deeper:

Peano arithmetic can not prove “Peano arithmetic is consistent”.

Proof idea: Get “this statement is not provable” to work. Express provability using numbers (think ASCII). Rewrite self-referentiality like this:

“»yields an unprovable statement when preceded by its quotation« yields an unprovable statement when preceded by its quotation.”

If the system is consistent, then that statement is true, but neither it nor its negation are provable.

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Part IV

Fundamental incompleteness

Gödel discovered:

Going deeper:

Proof idea: Get “this statement is not provable” to work. Express provability using numbers (think ASCII). Rewrite self-referentiality like this:

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Part IV

Fundamental incompleteness

Gödel discovered:

Going deeper:

Proof idea: Get “this statement is not provable” to work.

Express provability using numbers (think ASCII).

Rewrite self-referentiality like this:

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Outlook

We use the axiomatic method to make mathsreliable.

But any axiomatization isincomplete.

ZFC+U ZFC+CH ZFC+¬CH ZFC

HA+ε²=0 HA+CT PA HA