Download The theorem, it`s meaning and the central concepts

En ufuldstændig gennemgang af Gödels ufuldstændigheds-teorem
The theorem, it’s meaning and the central concepts
Mathematical systems concist of axioms and theorems. The axioms are propositions that you don’t
proove to be true, but are assumed true to “start off the system”. You need some simple, basic
assumptions to be able to proove things. The theorems are propositions that you can proove via the
axioms and some simple rules of logical deduction. The ideal is to assume the fewest possible
number of axiom and proove all possible propositions or their negations.
But Gödel’s incompleteness theorem shows (it’s proven) that every mathematical system strong
enough to contain simple arithmetics will contain at least one true sentence that can’t be proven.
Gödels første ufuldstændigheds-teorem er en sætning indenfor matematisk logik, der kan
formuleres således:
Gödel’s first incompleteness theorem is a proposition in mathematical logic that can be formulated:
”Under the assumption that N is ω-consistent, neither the wellformed formula U or its negation ~U
is a theorem in N. Therefore: If N is ω-consistent, then N is not complete.” 1
Which can be roughly ”translated” to:
”If the logical system N, that describes simple arithmetics, is ω-consistent, there is a sentence U that
can be formulated in N, but where neither U or its negation can be proven in N.
”Consistency” is a mathematical concept that means, that you can’t both prove a sentence and its
negation in a system. If you could do this, you would be able to prove every sentence (even a
contradiction) via the two true sentences and the rules of deduction – which mean that every
sentence would be true in the system. Which is not particularly smart
ω-consistency is a slightly stronger version of consistency.
The incompleteness theorem says, that if a system contains simple arithmetics and is ω-consistent
(and thereby interesting from a mathematical point of view), there will be a sentence in the system
that is true but can’t be proven (either U is true or its negation ~U is true, but neither can be proven
in the system). This is very problematic, since the way you find out if a sentence in mathematics is
true is exactly to prove it. Thus, the mathematical systems will be incomplete.
“Completeness” means, that you will either be able to prove a sentence U or its negation ~U in a
mathematical system. In the mathematical environment in the 1920s people were very determined
to try to show, that mathematics was secure knowledge and that mathematics as a field was
complete. It was therefore a hard blow to the formalists (Hilbert) and the logicists (Russel and
Whitehead) when Gödel published his proof of the incompleteness theorem.
N is chosen to denote “the system of first order arithmetics”. This is a logical system that describes
the set of natural numbers (1,2,3,...) and 0 and the rules for addition and multiplication of the
numbers.
N consists of a logical language LN that contain:
Wikipedia formulerer det således: “Any effectively generated theory capable of expressing elementary arithmetic
cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves
certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory.”
1
Variables: x1, x2,... (for the natural numbers)
The individual constant: a1 (for the number 0)
Function letters: f11 (successor function) , f21 (sum) og f22 (product)
The predicate symbol “=”
Grammar: “(“, “)”, and “,”
Logical symbols: The connectives (actually you only need one – the NAND- or the
NOR-operator. But usually you use ~ (negation) and -> (implication)) and the
quantifier
In addition to this, some axioms are added: Axioms about first order language, axioms about
equality (to ensure that the predicate symbol “=” behave as we want it to) and axioms about
arithmetics (to ensure that the functions behave as we want them to)
Proof sketch
The proof can be broken down to the following points:
1) The invention of Gödel numbers and the proof that every symbol, term, wellformed formula
(“wf”), and sequence of wfs in the system N can be represented by Gödel numbers
2) The construction of certain relations that express that a wf can be proven in the system and
proof that these relations are recursive
3) The construction of a wf U that together with the relation W can be interpreted as saying:
“This sentence cannot be proven”
4) Analysis of the formula U that shows, that neither U or ~U can be proven in the system N
The idea in the proof is to construct a sentence U that is similar to “The liar’s paradox”: “This
sentence is a lie”. But unlike the paradox, the sentence U is NOT a true paradox. Therefore Gödel
construct the concept of Gödel numbering, so that wfs in a logical system can be represented by
numbers. You can then let wfs in the system refer to these numbers to create a kind of self-reference
on N. And because of the reference back and forth between the natural numbers (the Gödel
numbers) and the logical system N, one avoids a paradox.
The Gödel numbers are numbers assigned to every logical system and every wf or string of wfs in
the system.
We define the function g on the set of symbols in L thus::
g(() = 3
g()) = 5
g(,) = 7
g(~) = 9
g(->) = 11
g(quantifier) = 13
g(xk) = 7 + 8k for k = 1,2,…,
g(ak) = 9 + 8k for k = 1,2,…,
g(fnk) = 11 + 8*(2n * 3k) for n = 1,2,...; k = 1,2,…,
g(Ank) = 13 + 8*(2n * 3k) for n = 1,2,…; k = 1,2,…
The point is, that we can encode every wf or every string of wfs in a language, and that we can
“decode” which wf or string of wfs each Gödel number represents.
We have therefore created a connection between LN and N.
Gödel then continues to construct a lot of relations, which he proves are recursive. These are for
example P(x,y): “x is the Gödel number of a proof of the wf with Gödel number y”
He then construct the wf U that together with a relation W says something like “This wf can’t be
proven” when it’s analysed.
By the ω-consistency of the system N, we finally get that neither U or its negation ~U is a theorem
in the system N. Which means that there is at least one true wf in N that can’t be proven in N.
If you try to “save” the sytem N by adding either U og ~U as an axiom, you just face another Gödel
sentence which arises in the new system “N plus either U or ~U”.