Intuitive Intro to Gödel's Incompleteness TheoremIlya Kuzovkin
Intuitive introduction into meaning of Gödel's incompleteness theorem. Slides are prepared for "Philosophy of Artificial Intelligence" class at University of Tartu
Logicians sometimes talk about sentences being “true but unprovable." What does this mean? This presentation includes a fairly thorough introduction to mathematical logic.
Intuitive Intro to Gödel's Incompleteness TheoremIlya Kuzovkin
Intuitive introduction into meaning of Gödel's incompleteness theorem. Slides are prepared for "Philosophy of Artificial Intelligence" class at University of Tartu
Logicians sometimes talk about sentences being “true but unprovable." What does this mean? This presentation includes a fairly thorough introduction to mathematical logic.
Incompleteness Theorems: Logical Necessity of InconsistencyCarl Hewitt
These are slides for video of "Wittgenstein versus Gödel on the Foundations of Logic" Stanford Logic Colloquium on April 23, 2010.
Video can be viewed at:
http://wh-stream.stanford.edu/MediaX/CarlHewittEdit.mp4
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
What is a theory? What makes a good theory?
We also look at accuracy and precision, historic examples, Karl Popper's ideas of a theories, Occam's razor, the scientific model etc. for an extensive look into the concept of a theory and its place in any discipline.
[DSC Europe 22] Amazing things that happen with Human-AI synergy - Petar Veli...DataScienceConferenc1
For the past few years, I have been part of a collaboration with top-tier mathematicians on a challenging project: teaching machines to assist humans with proving difficult theorems and conjecturing new approaches to long-standing open problems. My team has demonstrated that analysing and interpreting the outputs of (graph) neural networks is a concrete solution. Our efforts independently derived novel top-tier mathematical results in areas as diverse as representation theory and knot theory. The significance of our findings was recognised by the journal Nature, where it featured on the cover page. I will share our findings from a personal perspective, with key details of our team’s modelling work.
Logic and mathematics history and overview for studentsBob Marcus
Math and logic overview for students. Covers a wide range of topics including algorithms, proofs, probability, networks, number theory, statistics, causality, WolframAlpha, and Python programs.
This short and informal article shows that, although Godel's theorem is valid using classical logic, there exists some four-valued logical system that is able to prove that arithmetic is both sound and complete. This article also describes a four-valued Prolog in some informal, brief and intuitive manner.
Trick or Treat?: Bitcoin for Non-Believers, Cryptocurrencies for CypherpunksDavid Evans
David Evans
DC Area Crypto Day
Johns Hopkins University
30 October 2015
This (non-research) talk will start with a tutorial introduction to cryptocurrencies and how bitcoin works (and doesn’t work) today. We’ll touch on some of the legal, policy, and business aspects of bitcoin and discuss some potential research opportunities in cryptocurrencies.
Incompleteness Theorems: Logical Necessity of InconsistencyCarl Hewitt
These are slides for video of "Wittgenstein versus Gödel on the Foundations of Logic" Stanford Logic Colloquium on April 23, 2010.
Video can be viewed at:
http://wh-stream.stanford.edu/MediaX/CarlHewittEdit.mp4
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
What is a theory? What makes a good theory?
We also look at accuracy and precision, historic examples, Karl Popper's ideas of a theories, Occam's razor, the scientific model etc. for an extensive look into the concept of a theory and its place in any discipline.
[DSC Europe 22] Amazing things that happen with Human-AI synergy - Petar Veli...DataScienceConferenc1
For the past few years, I have been part of a collaboration with top-tier mathematicians on a challenging project: teaching machines to assist humans with proving difficult theorems and conjecturing new approaches to long-standing open problems. My team has demonstrated that analysing and interpreting the outputs of (graph) neural networks is a concrete solution. Our efforts independently derived novel top-tier mathematical results in areas as diverse as representation theory and knot theory. The significance of our findings was recognised by the journal Nature, where it featured on the cover page. I will share our findings from a personal perspective, with key details of our team’s modelling work.
Logic and mathematics history and overview for studentsBob Marcus
Math and logic overview for students. Covers a wide range of topics including algorithms, proofs, probability, networks, number theory, statistics, causality, WolframAlpha, and Python programs.
This short and informal article shows that, although Godel's theorem is valid using classical logic, there exists some four-valued logical system that is able to prove that arithmetic is both sound and complete. This article also describes a four-valued Prolog in some informal, brief and intuitive manner.
Trick or Treat?: Bitcoin for Non-Believers, Cryptocurrencies for CypherpunksDavid Evans
David Evans
DC Area Crypto Day
Johns Hopkins University
30 October 2015
This (non-research) talk will start with a tutorial introduction to cryptocurrencies and how bitcoin works (and doesn’t work) today. We’ll touch on some of the legal, policy, and business aspects of bitcoin and discuss some potential research opportunities in cryptocurrencies.
Exploring the Mindfulness Understanding Its Benefits.pptxMartaLoveguard
Slide 1: Title: Exploring the Mindfulness: Understanding Its Benefits
Slide 2: Introduction to Mindfulness
Mindfulness, defined as the conscious, non-judgmental observation of the present moment, has deep roots in Buddhist meditation practice but has gained significant popularity in the Western world in recent years. In today's society, filled with distractions and constant stimuli, mindfulness offers a valuable tool for regaining inner peace and reconnecting with our true selves. By cultivating mindfulness, we can develop a heightened awareness of our thoughts, feelings, and surroundings, leading to a greater sense of clarity and presence in our daily lives.
Slide 3: Benefits of Mindfulness for Mental Well-being
Practicing mindfulness can help reduce stress and anxiety levels, improving overall quality of life.
Mindfulness increases awareness of our emotions and teaches us to manage them better, leading to improved mood.
Regular mindfulness practice can improve our ability to concentrate and focus our attention on the present moment.
Slide 4: Benefits of Mindfulness for Physical Health
Research has shown that practicing mindfulness can contribute to lowering blood pressure, which is beneficial for heart health.
Regular meditation and mindfulness practice can strengthen the immune system, aiding the body in fighting infections.
Mindfulness may help reduce the risk of chronic diseases such as type 2 diabetes and obesity by reducing stress and improving overall lifestyle habits.
Slide 5: Impact of Mindfulness on Relationships
Mindfulness can help us better understand others and improve communication, leading to healthier relationships.
By focusing on the present moment and being fully attentive, mindfulness helps build stronger and more authentic connections with others.
Mindfulness teaches us how to be present for others in difficult times, leading to increased compassion and understanding.
Slide 6: Mindfulness Techniques and Practices
Focusing on the breath and mindful breathing can be a simple way to enter a state of mindfulness.
Body scan meditation involves focusing on different parts of the body, paying attention to any sensations and feelings.
Practicing mindful walking and eating involves consciously focusing on each step or bite, with full attention to sensory experiences.
Slide 7: Incorporating Mindfulness into Daily Life
You can practice mindfulness in everyday activities such as washing dishes or taking a walk in the park.
Adding mindfulness practice to daily routines can help increase awareness and presence.
Mindfulness helps us become more aware of our needs and better manage our time, leading to balance and harmony in life.
Slide 8: Summary: Embracing Mindfulness for Full Living
Mindfulness can bring numerous benefits for physical and mental health.
Regular mindfulness practice can help achieve a fuller and more satisfying life.
Mindfulness has the power to change our perspective and way of perceiving the world, leading to deeper se
The Book of Joshua is the sixth book in the Hebrew Bible and the Old Testament, and is the first book of the Deuteronomistic history, the story of Israel from the conquest of Canaan to the Babylonian exile.
HANUMAN STORIES: TIMELESS TEACHINGS FOR TODAY’S WORLDLearnyoga
Hanuman Stories: Timeless Teachings for Today’s World" delves into the inspiring tales of Hanuman, highlighting lessons of devotion, strength, and selfless service that resonate in modern life. These stories illustrate how Hanuman's unwavering faith and courage can guide us through challenges and foster resilience. Through these timeless narratives, readers can find profound wisdom to apply in their daily lives.
2 Peter 3: Because some scriptures are hard to understand and some will force them to say things God never intended, Peter warns us to take care.
https://youtu.be/nV4kGHFsEHw
Lesson 9 - Resisting Temptation Along the Way.pptxCelso Napoleon
Lesson 9 - Resisting Temptation Along the Way
SBs – Sunday Bible School
Adult Bible Lessons 2nd quarter 2024 CPAD
MAGAZINE: THE CAREER THAT IS PROPOSED TO US: The Path of Salvation, Holiness and Perseverance to Reach Heaven
Commentator: Pastor Osiel Gomes
Presentation: Missionary Celso Napoleon
Renewed in Grace
What Should be the Christian View of Anime?Joe Muraguri
We will learn what Anime is and see what a Christian should consider before watching anime movies? We will also learn a little bit of Shintoism religion and hentai (the craze of internet pornography today).
Why is this So? ~ Do Seek to KNOW (English & Chinese).pptxOH TEIK BIN
A PowerPoint Presentation based on the Dhamma teaching of Kamma-Vipaka (Intentional Actions-Ripening Effects).
A Presentation for developing morality, concentration and wisdom and to spur us to practice the Dhamma diligently.
The texts are in English and Chinese.
The PBHP DYC ~ Reflections on The Dhamma (English).pptxOH TEIK BIN
A PowerPoint Presentation based on the Dhamma Reflections for the PBHP DYC for the years 1993 – 2012. To motivate and inspire DYC members to keep on practicing the Dhamma and to do the meritorious deed of Dhammaduta work.
The texts are in English.
For the Video with audio narration, comments and texts in English, please check out the Link:
https://www.youtube.com/watch?v=zF2g_43NEa0
The Good News, newsletter for June 2024 is hereNoHo FUMC
Our monthly newsletter is available to read online. We hope you will join us each Sunday in person for our worship service. Make sure to subscribe and follow us on YouTube and social media.
The Chakra System in our body - A Portal to Interdimensional Consciousness.pptxBharat Technology
each chakra is studied in greater detail, several steps have been included to
strengthen your personal intention to open each chakra more fully. These are designed
to draw forth the highest benefit for your spiritual growth.
In Jude 17-23 Jude shifts from piling up examples of false teachers from the Old Testament to a series of practical exhortations that flow from apostolic instruction. He preserves for us what may well have been part of the apostolic catechism for the first generation of Christ-followers. In these instructions Jude exhorts the believer to deal with 3 different groups of people: scoffers who are "devoid of the Spirit", believers who have come under the influence of scoffers and believers who are so entrenched in false teaching that they need rescue and pose some real spiritual risk for the rescuer. In all of this Jude emphasizes Jesus' call to rescue straying sheep, leaving the 99 safely behind and pursuing the 1.
2. Announcements
Don’t forget to submit your PS8 choice before
5pm today!
If you don’t submit anything, you are
expected to do all three options!
Friday’s Class: Rice Hall Dedication
11am: Dedication
12-3pm: Tours (Visit our lab in Rice 442)
3pm: Dean Kamen talk
5pm: Scavenger Hunt
2
3. Exam 2
Out: Monday, 21 November (one week from today)
Due: Wednesday, 30 November, 11:01am
Covers everything in the course
Classes 1-37 (through Friday’s class – yes there
may be a question about Rice Dedication)
Course book Chapters 1-12 (Entire book!)
Problem Sets 1-7 (and posted comments)
Emphasis on material since Exam 1
You will be allowed to use Scheme, Python, and
Charme interpreters
3
4. Recap Last Class
Since Aristotle, humans have been trying to
formalize reasoning.
An axiomatic system is a set of axioms and
inference rules.
Russell and Whitehead wrote Principia
Mathematica (1910-1913) to formalize all
number knowledge as an axiomatic system.
Encountered paradoxes…but claimed a
complete and consistent system.
4
6. In its absolutely barest
form, Gödel’s discovery involves
the translation of an ancient
paradox in philosophy into
mathematical terms. That
paradox is the so-called
Epimenides paradox, or liar
paradox. Epimenides was a
Cretan who made one immortal
statement: “All Cretans are
liars.” A sharper version of the
statement is simply “I am lying”;
or, “This statement is false.”
6
8. What Epimenides “Really” Said
They fashioned a tomb for
thee, O holy and high one
The Cretans, always
liars, evil beasts, idle
bellies!
But thou art not dead: thou
livest and abidest forever,
For in thee we live and
move and have our being.
— Epimenides, Cretica
(as quoted, no actual Minos speaking to Zeus
text survived)
8
9. One of Crete’s own prophets has said it:
“Cretans are always liars, evil brutes, lazy
gluttons”.
He has surely told the truth.
— First Epistle of Paul to Titus
(in the New Testament Bible)
9
10. The Real Paradox
This statement is false.
What Gödel did:
Showed the formal system in Principia
Mathematica could state a sentence
equivalent to: “This statement has no proof.”
10
11. Gödel’s Stronger Result
All consistent axiomatic formulations of
number theory include undecidable
propositions.
undecidable: cannot be proven either
true or false inside the system.
12. The Information, Chapter 6
Kurt Gödel
Born 1906 in Brno (now
Czech Republic, then
Austria-Hungary)
1931: publishes Über formal
unentscheidbare Sätze der
Principia Mathematica und
verwandter Systeme (On
Formally Undecidable
Propositions of Principia
Mathematica and Related
Systems)
13. 1939: flees Vienna
Institute for Advanced
Study, Princeton
Died in 1978 –
convinced everything
was poisoned and
refused to eat
14. Gödel’s Theorem
All logical systems of any complexity
are incomplete: there are statements
that are true that cannot be proven
within the system.
15. Proof – General Idea
Theorem: In the Principia
Mathematica system, there are
statements that cannot be
proven either true or false.
Proof: Find such a statement
16. Gödel’s Statement
G: This statement does not
have any proof in the
system of Principia
Mathematica.
G is unprovable, but true!
17. Gödel’s Proof Idea
G: This statement does not have any
proof in the system of PM.
If G is provable, PM would be inconsistent.
If G is unprovable, PM would be incomplete.
Thus, PM cannot be complete and consistent!
19. Gödel’s Statement
G: This statement does not have
any proof in the system of PM.
Possibilities:
1. G is true G has no proof
System is incomplete
2. G is false G has a proof
System is inconsistent
20. incomplete Pick one: some false
statements
Derives all true
Derives statements, and some
some, but not all true
statements, and no false false statements starting
statements starting from a from a finite number of
finite number of axioms axioms and following
and following mechanical
inference rules.
mechanical
inference rules.
Incomplete Inconsistent Axiomatic
Axiomatic System System
21. Inconsistent Axiomatic System
Derives
all true
statements, and some false
statements starting from a
finite number of axioms
and following mechanical
inference rules. some false
Once you can prove one false statement, statements
everything can be proven! false anything
22. Finishing The Proof
Turn G into a statement in the Principia
Mathematica system
Is PM powerful enough to express G:
“This statement does not have any
proof in the PM system.”
?
23. How to express “does not have any
proof in the system of PM”
What does “have a proof of S in PM” mean?
There is a sequence of steps that follow the
inference rules that starts with the initial axioms
and ends with S
What does it mean to “not have any proof of S
in PM”?
There is no sequence of steps that follow the
inference rules that starts with the initial axioms
and ends with S
24. Can PM express unprovability?
There is no sequence of steps that follows the
inference rules that starts with the initial
axioms and ends with S
Sequence of steps:
T0, T1, T2, ..., TN
T0 must be the axioms
TN must include S
Every step must follow from the previous
using an inference rule
25. Can we express
“This statement”?
Yes!
If you don’t believe me (and you
shouldn’t) read Gödel, Escher, Bach over
winter break.
We can write every statement as a
number, so we can turn “This statement
does not have any proof in the system”
into a number which can be written in PM.
26. Gödel’s Proof
G: This statement does not have any proof
in the system of PM.
If G is provable, PM would be inconsistent.
If G is unprovable, PM would be incomplete.
PM can express G.
Thus, PM cannot be complete and consistent!
27. Generalization
All logical systems of any
complexity are incomplete:
there are statements that are true
that cannot be proven within the
system.
28. “Practical” Implications
There are mathematical truths that cannot be
determined mechanically.
We can write a program that automatically
proves only true theorems about number
theory, but if it cannot prove something we do
not know whether or not it is a true theorem.
Mathematicians will never be completely replaced by computers.
29. What does it mean for an axiomatic system
to be complete and consistent?
Derives all true
statements, and no false
statements starting from a
finite number of axioms
and following mechanical
inference rules.
30. What does it mean for an axiomatic system
to be complete and consistent?
It means the axiomatic system is weak.
Indeed, it is so weak, it cannot express:
“This statement has no proof.”
31. Impossibility Results
Mathematics (Declarative Knowledge)
Gödel: Any powerful axiomatic system cannot be
both complete and consistent
If it is possible to express “This statement has no
proof.” in the system, it must be incomplete or
inconsistent.
Computer Science (Imperative Knowledge)
Are there (well-defined) problems that cannot be
solved by any algorithm?
Alan Turing (and Alonzo Church): Yes!
32. Computability
A problem is computable if there is an
algorithm that solves it.
What is an algorithm? A procedure that always finishes.
What is a procedure? A precise description of a series of steps
that can be followed mechanically*
(without any thought).
*A formal definition of computable requires a more formal definition of a procedure.
What does it mean to have an algorithm that solves
a problem?
We have a procedure that always finished, and always provides
a correct output for any problem instance.
33. Computability
Is there an algorithm that solves a problem?
Computable (decidable) problems can be solved by some
algorithm.
Make a photomosaic, sorting, drug discovery,
winning chess (it doesn’t mean we know the
algorithm, but there is one)
Noncomputable (undecidable) problems cannot be
solved by any algorithm.
There might be a procedure (but it doesn’t
finish for some inputs).
34. The (Pythonized) Halting Problem
Input: a string representing a Python
program.
Output: If evaluating the input
program would ever finish, output
true. Otherwise, output false.
35. Suppose halts solves Halting Problem
Input: a string representing a
def halts(code): Python program.
Output: If evaluating the input
... ? ...
program would ever finish, output
true. Otherwise, output false.
>>> halts('3 + 3')
True
>>> halts("""
i=0
while i < 100:
i = i * 2""")
False
36. Halting Examples
>>> halts(""" >>> halts("""
def fact(n): def fact(n):
if n = 1: return 1 if n = 1: return 1
else: return n * fact(n - 1) else: return n * fact(n - 1)
fact(7) fact(0)
""") """)
True False
halts(''''''
def fibo(n):
if n == 1 or n == 2: return 1
else: return fibo(n 1) + fibo(n 2)
fibo(60)
'''''')
37. Can we define halts?
Attempt #1: Attempt #2:
def halts(code): def halts(code):
eval(code) try:
return True with Timer(100):
eval(code)
return True
except Timer:
return False
These two approaches fail, but not a proof it cannot be done!
38. Impossibility of Halts
Recall how Gödel showed incompleteness of PM:
Find a statement that leads to a contradiction
Gödel’s statement: “This statement has no proof.”
Is there an input to halts that leads to a contradiction?
39. Charge
Wednesday’s class (and Chapter 12):
Are there any noncomputable problems?
Problem Set 8 Commitments:
Due before 5pm today
Problem Set 7: Due Wednesday (but not until
after class)
39