[Here is the final part of Badiou’s logic course from 1982 – 83. I find the brief discussion of the infinite and infinitesimal interesting. Enjoy.]

Alain Badiou – Logic Course (1982 – 1983)

Kant: What can I know? → the phenomena and the thing-in-itself. What must I do? → conform my action to the categorical imperative and ensure that my maxim is universalizable without contradiction. What must I hope for? → The immortality of the soul.

Gödel: the not-all of the calculable in relation to the rational.

Lowenhein-Skolem: the not-one of the rational in relation to the calculable.

Tarski: the impossibility of restoring to the calculable the crucial predicate of the rational that is said of this truth. The rational concept of truth is not representable in calculation: what is not representable are true statements that should be true.

Metatheorem of equivalence

Let C be a theorem of L. Let A be a part of a correct formula C. Suppose that A/B. Let C′ be where A is replaced by B. Then C′ is also a theorem of L.

Language LP of propositional calculus

1) alphabet =

• propositional variables: A, A′, B, B′ …

• logical operators: negation (¬), implication (→) [ The operators ‘or’, ‘and’, ‘equivalence’, are logically deduced from these two operators]

• punctuation signs: ‘(‘ ‘,’ ‘[‘ …

2) grammar

3) three axioms

• p → (q→p)

• [p → (q → r)] → [(p → q) → (p → r)]

• (¬p → ¬ q) → (q → p) = transposition

4) rule of deduction

• Modus ponens (MP): (p→q / p // q)

5) Theorem of deduction

• Theorem of Kalmer (1922): theorem of the completeness of propositional calculus that is said of all tautologies/contradictions: syntacto(decidable)-semantic(being a tautology). Where the language LP is coherent, complete, and decidable.

Language L of the predicate calculus of the 1st order

In addition to LP:

1) alphabet =

• individual constants or proper names: a, a′, b, b′ …

• individual variables: x, y, x′, y′ …

• unary predicates: P, P′, Q, Q′ …

• binary (R, R’ …) and tertiary (T, T′) relations

• Logical connectors:

• Universal quantifier (∀) [ Cf. ∃ is logically deduced from ∀]

2) grammar

3) two axioms

• [∀a (A→B)] → [A→(∀a) B] if ‘a’ is not free in A

• (∀x)A → Subst(y/x)A: one can replace in A x with y if it does not have a particular role in A

4) rule of deduction

• rule of generalization: If A then (∀a)A (one can universalize a statement) [The rule of existentialization is logically deduced from the precedent as ∃ is logically deduced from ∀]

→ (Meta)theorem of equivalence

→ Lowenheim-Skolem’s Theorem (1st)

→ Lowenheim-Skolem’s Theorem (2nd)

Gödel’s Theorem (1930): says the completeness of 1st order logic which is introduced in universally valid statements. Where the language L is coherent, complete but decidable. Cf.:

• Church’s Theorem: 1st order logic is not decidable

• Zermelo’s Axiom of Choice is formulable.

• Herbrand, then Henkin, have also worked on the completeness of 1st order logic for universally valid statements.

Logic of arithmetic by Peano = theory of identity + theory of successor + axiomatic schema for demonstration by repetition: {A[0] and (∀x)(∀y)[S(x,y) → A[x] → A[y]} → (∀x)A[x]

Properties of a Theory

a) consistency and coherence: one cannot deduce all at once A and ¬A; one cannot, therefore, deduce, only introduce. Cf. if I am → A and →¬A then:

• ¬A → (A → B): cf. “ex facto”

• ¬A and ¬A → (A → B) brings about (A → B) by Modus Ponens

• then A and (A → B) brings about (by MP) B!

• An inconsistent theory is a theory where at least one statement is not deducible.

• Lowenheim-Skolem’s Theorem (1st): If (closed formula) A is not deducible in (consistent theory) T the T + ¬A is consistent.

• Lowenheim-Skolem’s Theorem (2nd): If T is coherent, the theory obtaining an additional list of individual constants (= proper names), eventually becoming infinite, is equally coherent.

b) completeness (syntax): all statements are deducible. [semantic completeness is where all true statements are demonstrable. This forbids Gödel’s Theorem.]

• more precisely: a theory is syntactical complete if all of the closed universal formulas are decided. A complete theory is maximal: one cannot add them without rendering them incoherent.

• Cf. Closing the universe of open formulas: one adds to a formula a ∀ for every free variable. An open statement can be true or false depending on this or that interpretation. A closed statement, however, is constrained by a single interpretation. Cf. the open statement: (∃x)P(x) → (∀x)P(x) is indecidable. The open statement P(x) → P(x) is transformed into universally closed statement: (∀x)[P(x) → P(x)]

• A universally closed statement:

ß does not have a free variable

ß is closed by the universal quantifier (≠ existential closure)

• Gödel’s Theorem → incomplete in this sense

• Lindenbaum’s Theorem → ∃ extension completes all theory

• One does not have a complete theory of completeness. One cannot know why a theory is complete or not.

• Cf. Complete theories:

ß Additive arithmetic

ß Boole’s non-atomistic algebra

ß Naïve set-theory

• Most of the grand theories are incomplete.

• Lindenbaum’s Theorem: every theory admits a syntaxically complete extension

• Remarks: this extension is monstrous in three ways:

ß This extension is entirely ineffective: one cannot know whether the given statement is or is not.

ß This extension is not given by axioms but in a single package. It is very difficult to axiomatize. Cf. there exists a tension between completeness and axiomatic presentation.

ß One cannot know at all models of extension taken into consideration by the initial model

ß Extension is also ineffective, axiomatically inaccessible and pathological in terms of the model.

c) decidability = There exists a mechanical procedure that permits the verification for all formulas if it is a theorem.

• Cf. idea of a mechanical procedure that is a decidability machine. Cf. concept of recursive function, Turing machine, Markov algorithm…for these verifiable procedures (and not demonstrable)

• It is a calculable on the calculable.

• Example of decidable theory: the formal theory of commutative groups.

• Why a theory is decidable? One cannot know. Its not a question of simplicity.

• One shows that there is no decidability as decidability!

• One shows that in common decidable theories (cf. naïve arithmetic is additive arithmetic) there always exists formulas where numbers approach the gigantic.

Concept of the Model

Semantic: that yields 5 things, minimally:

• a domain of objects

• proper names distinguishing these objects

• there can exist properties of these objects

• there can exist relations between these objects

ß Intensive relations: given in their definition

ß Extensive relations: given in a list of these n-uplets

• rule of the true and the false

The interpretation of a theory T according to a model M makes:

• variable of T → object of M

• individual constant T → fixed element of M

• predicate in place of T → property of M

• relation of T → relation of M

One translated in this way all elementary formulas of T and one cuts on true/false.

non p p and q v f p or q v f p → q v f p ↔ q v f

v f v v f v v v v v f v v f

f v f f f f v f f v v f f v

The logical operators are not translated but one has directly the rules of evaluation. The logical operators are impossible to translate. Towards them, one has a disposition of evaluation, not translation.

Critique of modal logic (with possible, necessity, etc.): one can formalize the theories of probability with classical logic.

N.B. Formalization is the experimental moment of mathematics. It is not true to say, from this point of view, that mathematics is a formal science.

Mathematics exemplifies, in the simplest way, the fact that reality is an impasse. And all experience of the reality is a tense experience, extreme,

History

Boole (1850): 1st version of propositional calculus (with ¬, ø; but without quantification)

Frege, Peirce (1870): idea of Frege that everything returns ideography, that is to say, a writing where the concept would be transparent.

Whitehead and Russell: cf. criticize an inconsistency in Frege’s system holding to the possibility of applying the predicates to the themselves, holding quantifications over properties. Where stratification of syntax and the theory of types: what is to the left and to the right of the sign ∈ is not the same type. Cumbersome and useless (One needed an axiom of reducibility to destratify.

Hilbert: project of the auto-foundation of mathematics over calculus; to seek that everything true be demonstrable = reinstate without rest the truth of demonstrability. Auto-founding the demonstrable.

1915: Lowenheim’s 1st version

1920: Skolem’s recaptures this

1930: Gödel’s theorem

Universally valid formula = valid formula in all interpretive domains.

In logical language, all theorems are universally valid.

Kalmer’s theorem (1920) = theorem of completeness of propositional calculus is said of all tautologies/antilogies. Where language LP is coherent, complete, decidable.

Gödel’s Theorem (1930): says of the completeness of 1st order logic that introduced into universally valid statements. Where the language L is coherent, complete, but not decidable. Cf.:

Church’s Theorem (1937): 1st order logic is not decidable.

Cf. in logic, one cannot know what one says. Cf. mathematics ≠ logic. In mathematics, contrary to Russell, one knows of what they speak.

Logicist School (cf. Russell): logic and mathematics indistinguishable, under the order of a single syntax and argument that if two things are axiomatizable, they are the same.

Therefore:

• Propositional calculus → tautologies → language LP with “¬” and “→” → three axioms and rules of deduction.

• 1st order logic → universally valid formulas → Language L with another ∀, an axiom and rule of deduction.

All coherent theories have at least one model (theorem of Léon Henkins) = anti-idealist position: coherence allows the reference. There is not uninterpretable coherence. All coherence makes sense. Cf. Hegel: “all rationality is real”

All theory that have a model are coherent.

Lowenheim-Skolem’s (1st): all coherent theory has a numerable model

Lowenheim-Skolem’s (2nd): all coherent theory has a model of poor infinity

Categoricity of a theory:

The models of a type infinity given are isomorphs? If yes for a given infinity α the theory is said α-categoricity. It is not of an absolute categoricity since two different models of infinity are not isomorphs.

Examples:

• theory categorical (and therefore univocal) in numeration but not categorical for the superior types of infinity.

• theories α-catergorical for α > numerable but not for the numerable; ex. group theory communitive to unique divisor.

• theories categorical for all infinites separately; ex. theory of vectorial spaces of the entire modulo 2.

• theories never categorical; ex. group theory

Conjecture of Los: it would not have to have these 4 cases.

Morley’s Theorem: has demonstrated this conjecture! It is the longest and most difficult theory of all of logic.

In this way, categoricity does not distinguish the two types of infinity: the numerable, all that is beyond and which makes for categoricity. It is close to the numerability of writing.

Infinites, transcendence

Indefinite ≠ infinite, as potential infinite ≠ actual infinite

The infinite is not connected to the one in mathematics for Cantor. Before this connection was theological. (Uncreated) God was infinite and the (created) world was finite. Their one takes precedence over their infinity: It is the united transcendence of God that brings about their infinity.

One has therefore, before the 17th-century:

• a theological infinite (God)

• a physical finite (world)

• a purely potential logico-mathematical infinite.

Shaking the 17th-century: introduce the actual infinite in the physical world (cf. dialectical dramatization of this point by Pascal) but still not the concept of the actual infinite in mathematics.

The mathematical shaking comes instead from infinitesimal quantities. One progressively eliminates (between Newton = end of 17th and Cauchy = beginning of the 19th) these quantities by the introduction of a calculus of limits: “tightening toward” restoring the potential infinite. (Cauchy ≡ Russell more than Brouwer in the crises of sets)

Cantor introduces the radical revolution in secularizing the infinite by calculating the actual infinite. The introduction of the Hebrew alphabet!

The mathematical concept of infinitesimals is not far behind. Cf. Abraham Robinson (1960) and non-standards analysis. Nevertheless this has little repercussions: one redeems these known results for a long tome to come!

In this way, the mathematical revolution of the infinite has more than a century whereas the infinitesimal only has 25 years.

Pascal makes these two infinites symmetrical. Cantor does not believe in the infinitesimal.

Large Cardinals

• inaccessible

• ineffables

• of Ramsey

• low compacts [faiblement compacts]

• compacts

• super compacts

• enormous

• measurable

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