gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · tarski: undefinability of truth gödel:...
TRANSCRIPT
![Page 1: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/1.jpg)
Gödel in class
Achim FeldmeierBrno - Oct 2010
![Page 2: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/2.jpg)
![Page 3: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/3.jpg)
![Page 4: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/4.jpg)
right life (happyness, morals)
science and technology
social issues
Christianity
Natural Sciences
Politics
Philosophy lost key competence
to specialized disciplines:
Scientists not interested in administrational issueswant to do sciencehow can we help with science in education?
![Page 5: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/5.jpg)
- do research
- write programs
- analyze data
- argue and defend
- write
} thinkstraight
core abilities
![Page 6: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/6.jpg)
Describing Nature
Physical Law and Free Will
Why is Math the Language of Physics?
Modernism in Physics
Physics of Consciousness
Natural Philosophy
My recent experience from lectures on
summer 2010, with M. Wilkens
summer 2009, with M. Wilkens
summer 2008, with M. Wilkens
summer 2007, winter 2006/07
winter 2009/10
winter 2010/11
after reading texts by
backto logics
jour-
nalists
neur
o-sc
ien-
tists
lawyers
histori-
ans
priests
![Page 7: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/7.jpg)
after all: its for the fun of it
![Page 8: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/8.jpg)
and is there place for the empirical sciences?
1. Mathematical vs Philosophical Logic
Frege: invents formal language
Russell: paradox; type theory Ramsey: Ramsey sentences (ontological reduction)
Tarski: undefinability of truth
Gödel: incompleteness theorem
Carnap: diagonal argument
Quine: new Gödel numbering
Cohen: forcing
Putnam: Löwenheim-Skolem as epistemic principle
Kripke: modal logics. Gödel without self-reference
Hintikka: Gödel & Wittgenstein
![Page 9: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/9.jpg)
![Page 10: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/10.jpg)
![Page 11: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/11.jpg)
2. What is first order logic?(old name: predicate logic)
variables x, y, z, . . . constants π, e, . . .
functions +,×, . . . relations =, <,∈, . . .
a)
b) plus propositional logic (Aristotle) ¬,∧,→
c) plus quantifiers (over variables only) ∀x ∃y
![Page 12: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/12.jpg)
∀f ∃x f(x) = . . .
quantification
over functions
and predicates
2nd order logic:
![Page 13: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/13.jpg)
3. Löwenheim-Skolem
first, and deepest result in model theoryLöwenheim 1915 Skolem 1920, 1922
If statements have interpretation in a set, then they have interpretation in a countable set
x2 = 2
x2 = −1
x2 = 1 has interpret. in...
...
N,R,CR,CC
![Page 14: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/14.jpg)
![Page 15: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/15.jpg)
4. Objects of Logic & Science
no REALS in first-orderlogic
von Neumann:All from empty set.
Only sets
0 = ∅1 = {∅}2 = {∅, {∅}}3 = {∅, {∅}, {∅, {∅}}}
2 < 3 2 ∈ 3iff
What areobjects ofphysics?
Elements ofsets
![Page 16: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/16.jpg)
5. Completeness Theorem
in first-order logic and are synonymsTRUE PROVABLE
iffΦ � φ Φ � φ
Gödel 1930, PhD thesis
![Page 17: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/17.jpg)
6. Incompleteness
diagonalization self-re
feren
ce
There are true statements that cannot be proved within(!) a consistent (!) system
![Page 18: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/18.jpg)
Fact: all mathematicsis set theory
Gödel II: cannotprove consistencyof set theorywithin set theory
cannot know, whether there is a Gödel sen-tence
Consistencyis required inGödel I
![Page 19: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/19.jpg)
1. Gödel number: symbols numbers (Leibniz!)n = G[P (x)]
2. diagonalization: for each predicate there is :
y = G[P (y)]
P y
3. ‘provable’ is a predicate:
is true iff is provablePr[G(P )] P
4. apply (2) to predicate
g = G[¬Pr(g)]
¬Pr(x)
is true & unprovable
![Page 20: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/20.jpg)
adapted from Smullyan, “Gödel’s incompleteness theorems”
![Page 21: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/21.jpg)
Philosophy behind Gödel’s proof
1. Gödel: liar paradox2. (postmoderns): self-reference3. Hintikka: acting; theater play
4. Putnam, Kripke: no self-reference needed
![Page 22: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/22.jpg)
7. The continuumCantor: there is no isomorphism f : N → P(N)Proof: assume there is such f
consider the set L = {n|n /∈ f(n)}claim: there is no l such that L = f(l)
proof: assume, there is such lcase 1: l ∈ L → l /∈ f(l) = Lcase I1: l /∈ L → ¬(l /∈ f(l)) → l ∈ f(l) = L
both cases lead to contradictionthus assumption is wrongthus L has no number l
thus there is no f
![Page 23: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/23.jpg)
is there a cardinality between and ?
Since 1963 (P. Cohen)we cannot know
Continuum Problem (Cantor)
N P(N)
invented method of forcing(which Badiou calls “revolutionary” for philosophy)
![Page 24: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/24.jpg)
8. What is a proof?high-school students: “these observations prove Einstein’s theory”
Kant
Math issynthetical
a priori
Viennacircle
Math isanalyticala priori
Zer-melo
well-orderingtheorem(1904)
![Page 25: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/25.jpg)
![Page 26: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/26.jpg)
9. Logic of empirical sciences
![Page 27: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/27.jpg)
Entangle-ment
Uncertaintyprinciple
9.a Quantum mechanics
and Relativity
HorizonsUniverse Hawking
radiation
Role ofobserver
Schrödinger’scat
![Page 28: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/28.jpg)
9.b Quotation marks(e.g. in social sciences)
Quotation marks cannot help to define “truth”.Tarski, “The concept of truth in formalized languages” 1936
![Page 29: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/29.jpg)
9.c Category error
Aristotle
false
error oflogic
mean-inglesserror ofcategory
“the brain
thinks”
![Page 30: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/30.jpg)
9.d I TuringPut-nam
Touringmachines
what isa computer?
semantic argument:Chinese room
syntactic argument:window as computer
calc-ulating vs registermachine
Searle
INTENTI
ONALITY
about-ness
Husserl
![Page 31: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/31.jpg)
9.e Free Will
is freedom of will compatible
with causation?
![Page 32: Gödel in class - uni-potsdam.deafeld/goedelinclass.pdf · Tarski: undefinability of truth Gödel: incompleteness theorem Carnap: diagonal argument Quine: new Gödel numbering Cohen:](https://reader033.vdocument.in/reader033/viewer/2022042216/5ebef8b0607646403a7dad06/html5/thumbnails/32.jpg)
contribution of empirical
sciences to logic
are wecurrentlylosing ourexpertise?
like philo-sophy did
before (andregained it)
Math: too hardPhilo: too strange
Physics: pragmatic
SUMMARY