Kurt Godel
(28 April 1906
Brno
, then
Austria-Hungary
, now
Czech Republic
? 14 January 1978
Princeton, New Jersey
) was a
logician
,
mathematician
, and
philosopher
.
Some people believe Godel was one of the most significant logicians of all time. Godel's work has had a big impact on scientific and philosophical thinking in the
20th century
. Many people, such as
Bertrand Russell
,
A. N. Whitehead
, and
David Hilbert
, tried to use
logic
and
set theory
at that time. They wanted to understand the foundations of
mathematics
.
Godel is best known for his two
incompleteness theorems
. The theorems were published in 1931. He was 25 years of age, and had just finished his doctorate at the
University of Vienna
one year earlier.
The more famous of the two theorems says that if there are consistent
axiomatic
systems that are powerful enough to describe themselves, there will be things that are true in those systems that can not be proved within the system itself.
To prove this theorem, Godel developed a technique now known as
Godel numbering
, which codes formal expressions as natural numbers.
He also showed that the
continuum hypothesis
cannot be disproved from the accepted
axioms of set theory
, if those axioms are consistent. He made important contributions to
proof theory
. He did this by clarifying the connections between
classical logic
,
intuitionistic logic
, and
modal logic
.
Later in his life, Godel probably suffered from
paranoia
. He thought that some people would come and poison his food. So when his wife Adele was no longer able to prepare his food, he stopped eating. He died of
starvation
.
To put his theories in simple terms: What he discovered was that some
theorems
in mathematics may be true, but they cannot be proved to be true.
[
source?
]
This sentence is false
.
- If the sentence were false, the statement would be true, which it cannot be (because we thought it was false)
- If the sentence were true, then the statement would be false. But we said it was true.
Therefore, we found a
contradiction
.
This sentence is false
is a proposition that is called
undecidable
. We cannot say if it is true or false.
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