 2nd: Alain Badiou – Logic Course (1981 – 82) [Here is the second translation that I promised. It is a little more interesting than the first, but the two do flow together quite well. Enjoy!]

Alain Badiou – Logic Course (1981 – 1982)

Lacan: “the real is the impasse of formalization” as well as the symbolic. Its nature pierces
the symbolic.
Theorems of limitation: theorems of the formal real
Gödel: impossibility of nomology, of completeness, of the decidable. Every formal
system is pierced. There is an excess of the true in the calculable.
Tarski: impossibility to formalize the notion of truth (of a statement): The true is
unrepresentable in arithmetic.
Beth: impossibility to demonstrate that such a notion is definable in a given system.
Lowenheim-Skolem: superiority of the infinite in relation to the numerable. There is an
excess of the calculable in the true.
However, this theorem = theorems also of excess! There is something of the truth that
exceeds the formal.

General Crises

Greeks: Crisis of irrationals. The idea that being is given in numbers, that all are counted,
is called into question by a “being” that is no longer a “rational” number.
17th – 18th century: differential and integral calculus. They can have a mathematics of
infinity and no longer only of finitude. They can therefore have a mathematics of movement.
Beginning of the 19th century: Non-Euclidian geometry. Doubt over the absolutivity of
Euclidian space taken by Kant as a structural category of experience.
End of 19th century: Paradoxes of set-theory. Attained through the reliability of the
intuition of the multiple.

Complete result of signs → correct formulas → true formulas → demonstrable formulas
The indecidable that limits calculation is wandering and cannot be assigned.

Theorem of Lowenheim-Skolem (TLS or LS):
if a formal system has an infinite model, it necessarily has a demonstrable (pathological)
model. All formal theories necessarily have an infinite demonstrable model. The infinite non-demonstrable does not have a specific formalization! All formalization is in excess in relation to possible interpretations. There is an over-powering logic in regards to that which one might formalize.
Gödel: true (and rational) > calculable; or: the semantic exceeds the syntax.
LS: the contrary: There is a point that flees formalization.
Torsion = this double excess of something at an indecidable and non-conforming
(pathological) time.

Deviations:
• true → calculable → the only indecidable → reversal in the irrational!
• purely productive insight by overlooking the point fleeing formalization → reversal by ruining the rigor of calculus at the same time

Metatheorem of deduction:
1. ((S+A)→B)→(S→(A→B)) ≡ deduce B from S+A again by deducing (A→B) from S ≡ joining A to S and in deducing B again from by deducing B from A in S alone.
2. This demonstrates a connection between the rational notion of the hypothesis (cf. S+A) and this deduction of implication! It’s a connection between the rational and the calculable.
3. This demonstrates this by recurrence along the length of deductions.
4. The calculable → the finite law where the infinite is on this side of reality. Theology posits the contrary: the world is finite but the law of God is infinte.
5. Grand mathematical thought is not secondarily calculating: it is conceptual. Hegel despised mathematics because he believed it was reducible to calculation. He criticized mathematics because the negative was not operative (because the negation of negation = affirmation).
6. Calculation is not the absence of the subject but its withdrawal. In calculation the subject does not lack but comes to be missing.
7. Formalism is mechanisable. The machine, if it deals with the real, has no relation to the real. Only the subject has a relationship.
8. Axiomatic schema: in a deduction, one can replace any variable by any concrete expression.
9. The logic of predicates of the 1st order: one only quantifies the individuals and not the predicates or properties. There is not, therefore, properties of properties.
10. Most of the algebraic theories are axiomatizable in this logic. On the contrary, topological structure necessitate, in general, a superior order logic.
11. Existence defined by double negation. But ∃ and ¬¬; are they equivalent? Existence is then posed as a limit to the total power of negation. And yet the universal does not have the ontological range that the existential does. Example: All gods are infinite → There exists an infinite God. This last statement commits me, but not the first where the verb has no existential value. The equivalence of ∃ and∀ is purely logical and would not know to infer on the ontological.

3 transformations for substituting ∃ and ∀:
1. replace (A → B) with ¬(B→A)
2.  substitute ∃ and ∀
3. put a negation (¬) in front

An erasure is always a marker, but on its borders.
TLS shows that the homogeneity of the inscription does not arrive to homogenize the
semantic.
Univocity does not exist, therefore, more for artificial languages (langues) than it does for
natural languages. It does not exist as an absolutely univocal language (cf. the failed dream of Leibniz). All languages are poorly made. The subject is effaceable from all language.
TLS is a massive return of the repressed.

Formal theory:
0. underlying logic
1. list of individual constants
2. list of individual predicates

Theory of identity:
1. no individual constants
2. predicative constants at 2 places: I
3. Three axioms:
∀xI(x,x) =  reflexivity
(∀x)(∀y)I(x,y) → I(y,x) = commutativity or symmetry
(∀x)(∀y)(∀z)I(x,y) and I(y,z) → I(x,z) = transivity
This theory gives 1st-order predicate calculus with equality. One can sometimes consider it not even being mathematical and remaining purely logical.

Theory of the strict partial order:
1. not individual constants
2. predicative constants at 2 places <(x,y) or x<y
3. three axioms: irreflexivity, anti-symmetry and transivity

Theory of the strict total order:
Supplementary axiom: ∀x∀y x<y or y>x

Theory of successor:
1. no individual constants
2. binary relation S(x,y) = y is the successor of x
3. four axioms:
¬∃x S(x,0) ≡ 0 is not succeeded
S(x,y) → [S(x,z) → y = z] ≡ there is a single successor.
S(x,y) → [S(z,y) → x = z] ≡ there is a single predecessor.
All numbers have a successor.

~ by stellarcartographies on July 6, 2008.

3 Responses to “2nd: Alain Badiou – Logic Course (1981 – 82)”

1. […] two different selections of course notes from Badiou’s lectures circa 1980-82 here and here. This translation is short, but extremely concise, so there’s a lot of material to absorb. In […]

2. 3 transformations for substituting ∃ and ∀:
1. replace (A → B) with ¬(B→A)
2. substitute ∃ and ∀
3. put a negation (¬) in front