- published: 22 Jan 2016
- views: 1110
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.
Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Set_theory
Θ (set theory)
In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection from the reals onto α.
If the axiom of choice (AC) holds (or even if the reals can be wellordered) then Θ is simply 
, the cardinal successor of the cardinality of the continuum. However, Θ is often studied in contexts where the axiom of choice fails, such as models of the axiom of determinacy.
Θ is also the supremum of the lengths of all prewellorderings of the reals.
Proof of existence
It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto which there is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinals are wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the set of all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals into the set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the powerset axiom). Now the axiom of replacement shows that the class of all ordinals is in fact a set. But that is impossible, by the Burali-Forti paradox.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Θ_(set_theory)
Set theory (music)
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson (1960) in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte (1973), drawing on the work in twelve-tone theory of Milton Babbitt. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally tempered tuning system, and to some extent more generally than that.
One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm as well.
Mathematical set theory versus musical set theory
Although musical set theory is often thought to involve the application of mathematical set theory to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection. Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of ordered sets, and although these can be seen to include the musical kind in some sense, they are far more involved).
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Set_theory_(music)
John von Neumann
John von Neumann (/vɒn ˈnɔɪmən/; Hungarian: Neumann János (Hungarian pronunciation: [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian-American pure and applied mathematician, physicist, inventor, and polymath. He made major contributions to a number of fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics, fluid dynamics and quantum statistical mechanics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics.
He was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor and the digital computer. He published 150 papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. His last work, an unfinished manuscript written while in the hospital, was later published in book form as The Computer and the Brain.
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Von Neumann (disambiguation)
John von Neumann (1903–1957) was a Hungarian American mathematician.
Von Neumann may also refer to:
See also
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Von_Neumann_(disambiguation)
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical Zermelo–Fraenkel set theory (ZFC). A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes proper classes, objects having members but that cannot be members of other entities. NBG's principle of class comprehension is predicative; quantified variables in the defining formula can range only over sets. Allowing impredicative comprehension turns NBG into Morse-Kelley set theory (MK). NBG, unlike ZFC and MK, can be finitely axiomatized.
Ontology
The defining aspect of NBG is the distinction between proper class and set. Let a and s be two individuals. Then the atomic sentence 
is defined if a is a set and s is a class. In other words, is defined unless a is a proper class. A proper class is very large; NBG even admits of "the class of all sets", the universal class called V. However, NBG does not admit "the class of all classes" (which fails because proper classes are not "objects" that can be put into classes in NBG) or "the set of all sets" (whose existence cannot be justified with NBG axioms).
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Von_Neumann–Bernays–Gödel_set_theory
Podcasts:
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Von Neumann–Bernays–Gödel set theory
If you find our videos helpful you can support us by buying something from amazon. https://www.amazon.com/?tag=wiki-audio-20 Von Neumann–Bernays–Gödel set theory In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical Zermelo–Fraenkel set theory (ZFC).A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. -Video is targeted to blind users Attribution: Article text available under CC-BY-SA image source in video https://www.youtube.com/watch?v=2enHBkdT6Ec
published: 22 Jan 2016 -
Von Neumann–Bernays–Gödel set theory | Wikipedia audio article
This is an audio version of the Wikipedia Article: https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory 00:01:50 1 Classes in set theory 00:02:00 1.1 The uses of classes 00:05:21 1.2 Axiom schema versus class existence theorem 00:07:49 2 Axiomatization of NBG 00:07:59 2.1 Classes and sets 00:12:01 2.2 Definitions and axioms of extensionality and pairing 00:16:24 2.3 Class existence axioms and axiom of regularity 00:34:56 2.4 Class existence theorem 00:55:06 2.5 Extending the class existence theorem 01:03:16 2.6 Set axioms 01:15:50 2.7 Axiom of global choice 01:18:44 3 History 01:18:53 3.1 Von Neumann's 1925 axiom system 01:23:59 3.2 Von Neumann's 1929 axiom system 01:27:07 3.3 Bernays' axiom system 01:28:12 3.4 Gödel's axiom system (NBG) 01:30:43 4 ...
published: 25 Dec 2018 -
A Lesser-Known Mathematical Framework: The Godel Bernays Set Theory
Embark on a captivating journey through the lesser-known corridors of mathematical philosophy with "The Gödel-Bernays Set Theory: Unveiling the Universe of Mathematics." This video takes you deep into the heart of a mathematical framework that challenges our understanding of the universe. From its historical origins to its profound philosophical implications, discover how the Gödel-Bernays set theory offers a unique lens through which we can explore the infinite and the unknown. Through a blend of art, history, and science, this narrative delves into the minds of Kurt Gödel, Paul Bernays, and the pioneering thinkers who dared to question the foundations of mathematical reality. As we navigate the complex interplay between sets and classes, and the philosophical quandaries they pose, we inv...
published: 25 Mar 2024 -
What is a Basic Universe? (Axiomatic Set Theory)
An explanation of a Basic Universe for Neumann Berneays Gödel Set/Class Theory. This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets. Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Ri...
published: 19 Oct 2018 -
Classes vs Sets (Axiomatic Set Theory)
The difference between Classes and Sets as defined by Neumann Berneays Gödel (NBG) set theory, members vs subclasses of the Universal Class. This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets. Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssi...
published: 09 Oct 2018 -
NBG Theory: A Theory of Classes
Final presentation for Math 682 at University of Michigan. Topic is Von Neumann–Bernays–Gödel set theory: comparison to ZFC set theory, proof of its finite axiomatizability, and the limitation of size axiom. 00:00 Historical introduction 07:30 Axioms 19:20 Finite axiomatizability 42:10 Limitation of size axiom 59:00 Q&A
published: 29 May 2023 -
Roger Penrose explains Godel's incompleteness theorem in 3 minutes
good explanation from his interview with joe rogan https://www.youtube.com/watch?v=GEw0ePZUMHA
published: 01 Jun 2020 -
Is {∅} the Universal Class? (Axiomatic Set Theory)
This video shows that with only Axioms 1, 2, and 3, of Neumann Berneays Gödel {∅} could be the universal class. This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets. Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthi...
published: 25 Oct 2018 -
What is a Supercomplete Set? (Axiomatic Set Theory)
An explanation of what a supercomplete set or class is, a formal definition, and several examples and non examples. This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets. Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Ga...
published: 18 Oct 2018 -
What is a Swelled Set? (Axiomatic Set Theory)
A explanation of a swelled class or set including examples and a formal definition. This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets. Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, ...
published: 17 Oct 2018
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21:42
Von Neumann–Bernays–Gödel set theory
- Order: Reorder
- Duration: 21:42
- Uploaded Date: 22 Jan 2016
- views: 1110
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Von Neumann–Bernays–Gödel set theo...
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Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical Zermelo–Fraenkel set theory (ZFC).A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC.
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
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https://wn.com/Von_Neumann–Bernays–Gödel_Set_Theory
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Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical Zermelo–Fraenkel set theory (ZFC).A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC.
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
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1:35:41
Von Neumann–Bernays–Gödel set theory | Wikipedia audio article
- Order: Reorder
- Duration: 1:35:41
- Uploaded Date: 25 Dec 2018
- views: 295
This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
00:01:50 1 Clas...
This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
00:01:50 1 Classes in set theory
00:02:00 1.1 The uses of classes
00:05:21 1.2 Axiom schema versus class existence theorem
00:07:49 2 Axiomatization of NBG
00:07:59 2.1 Classes and sets
00:12:01 2.2 Definitions and axioms of extensionality and pairing
00:16:24 2.3 Class existence axioms and axiom of regularity
00:34:56 2.4 Class existence theorem
00:55:06 2.5 Extending the class existence theorem
01:03:16 2.6 Set axioms
01:15:50 2.7 Axiom of global choice
01:18:44 3 History
01:18:53 3.1 Von Neumann's 1925 axiom system
01:23:59 3.2 Von Neumann's 1929 axiom system
01:27:07 3.3 Bernays' axiom system
01:28:12 3.4 Gödel's axiom system (NBG)
01:30:43 4 NBG, ZFC, and MK
01:33:13 4.1 Models
01:34:12 5 Category theory
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"There is only one good, knowledge, and one evil, ignorance."
- Socrates
SUMMARY
=======
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.
A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas (membership and equality) and finitely many logical symbols, only finitely many axioms are needed to build the classes satisfying them. This is why NBG is finitely axiomatizable. Classes are also used for other constructions, for handling the set-theoretic paradoxes, and for stating the axiom of global choice, which is stronger than ZFC's axiom of choice.
John von Neumann introduced classes into set theory in 1925. The primitive notions of his theory were function and argument. Using these notions, he defined class and set. Paul Bernays reformulated von Neumann's theory by taking class and set as primitive notions. Kurt Gödel simplified Bernays' theory for his relative consistency proof of the axiom of choice and the generalized continuum hypothesis.
https://wn.com/Von_Neumann–Bernays–Gödel_Set_Theory_|_Wikipedia_Audio_Article
This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
00:01:50 1 Classes in set theory
00:02:00 1.1 The uses of classes
00:05:21 1.2 Axiom schema versus class existence theorem
00:07:49 2 Axiomatization of NBG
00:07:59 2.1 Classes and sets
00:12:01 2.2 Definitions and axioms of extensionality and pairing
00:16:24 2.3 Class existence axioms and axiom of regularity
00:34:56 2.4 Class existence theorem
00:55:06 2.5 Extending the class existence theorem
01:03:16 2.6 Set axioms
01:15:50 2.7 Axiom of global choice
01:18:44 3 History
01:18:53 3.1 Von Neumann's 1925 axiom system
01:23:59 3.2 Von Neumann's 1929 axiom system
01:27:07 3.3 Bernays' axiom system
01:28:12 3.4 Gödel's axiom system (NBG)
01:30:43 4 NBG, ZFC, and MK
01:33:13 4.1 Models
01:34:12 5 Category theory
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
"There is only one good, knowledge, and one evil, ignorance."
- Socrates
SUMMARY
=======
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.
A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas (membership and equality) and finitely many logical symbols, only finitely many axioms are needed to build the classes satisfying them. This is why NBG is finitely axiomatizable. Classes are also used for other constructions, for handling the set-theoretic paradoxes, and for stating the axiom of global choice, which is stronger than ZFC's axiom of choice.
John von Neumann introduced classes into set theory in 1925. The primitive notions of his theory were function and argument. Using these notions, he defined class and set. Paul Bernays reformulated von Neumann's theory by taking class and set as primitive notions. Kurt Gödel simplified Bernays' theory for his relative consistency proof of the axiom of choice and the generalized continuum hypothesis.
- published: 25 Dec 2018
- views: 295
11:14
A Lesser-Known Mathematical Framework: The Godel Bernays Set Theory
- Order: Reorder
- Duration: 11:14
- Uploaded Date: 25 Mar 2024
- views: 202
Embark on a captivating journey through the lesser-known corridors of mathematical philosophy with "The Gödel-Bernays Set Theory: Unveiling the Universe of Math...
Embark on a captivating journey through the lesser-known corridors of mathematical philosophy with "The Gödel-Bernays Set Theory: Unveiling the Universe of Mathematics." This video takes you deep into the heart of a mathematical framework that challenges our understanding of the universe. From its historical origins to its profound philosophical implications, discover how the Gödel-Bernays set theory offers a unique lens through which we can explore the infinite and the unknown. Through a blend of art, history, and science, this narrative delves into the minds of Kurt Gödel, Paul Bernays, and the pioneering thinkers who dared to question the foundations of mathematical reality. As we navigate the complex interplay between sets and classes, and the philosophical quandaries they pose, we invite you to reflect on the nature of existence, knowledge, and the endless pursuit of truth in the mathematical cosmos. Join us on this enlightening journey, and let the mysteries of Gödel-Bernays set theory illuminate your understanding of the world. Thank you for your curiosity and your willingness to explore the depths of mathematical philosophy with us.
#Mathematics #Philosophy #GödelBernays #SetTheory #MathematicalLogic
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https://wn.com/A_Lesser_Known_Mathematical_Framework_The_Godel_Bernays_Set_Theory
Embark on a captivating journey through the lesser-known corridors of mathematical philosophy with "The Gödel-Bernays Set Theory: Unveiling the Universe of Mathematics." This video takes you deep into the heart of a mathematical framework that challenges our understanding of the universe. From its historical origins to its profound philosophical implications, discover how the Gödel-Bernays set theory offers a unique lens through which we can explore the infinite and the unknown. Through a blend of art, history, and science, this narrative delves into the minds of Kurt Gödel, Paul Bernays, and the pioneering thinkers who dared to question the foundations of mathematical reality. As we navigate the complex interplay between sets and classes, and the philosophical quandaries they pose, we invite you to reflect on the nature of existence, knowledge, and the endless pursuit of truth in the mathematical cosmos. Join us on this enlightening journey, and let the mysteries of Gödel-Bernays set theory illuminate your understanding of the world. Thank you for your curiosity and your willingness to explore the depths of mathematical philosophy with us.
#Mathematics #Philosophy #GödelBernays #SetTheory #MathematicalLogic
Become a member of this channel to enjoy benefits:
https://www.youtube.com/channel/UCh6gJYYFDueIzAk_Jn6SmZw/join
- published: 25 Mar 2024
- views: 202
3:31
What is a Basic Universe? (Axiomatic Set Theory)
- Order: Reorder
- Duration: 3:31
- Uploaded Date: 19 Oct 2018
- views: 2017
An explanation of a Basic Universe for Neumann Berneays Gödel Set/Class Theory.
This series covers the basics of set theory and higher order logic. In this ...
An explanation of a Basic Universe for Neumann Berneays Gödel Set/Class Theory.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
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Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
https://wn.com/What_Is_A_Basic_Universe_(Axiomatic_Set_Theory)
An explanation of a Basic Universe for Neumann Berneays Gödel Set/Class Theory.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
Buy stuff with Zazzle: http://www.zazzle.com/carneades
Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
- published: 19 Oct 2018
- views: 2017
3:25
Classes vs Sets (Axiomatic Set Theory)
- Order: Reorder
- Duration: 3:25
- Uploaded Date: 09 Oct 2018
- views: 10671
The difference between Classes and Sets as defined by Neumann Berneays Gödel (NBG) set theory, members vs subclasses of the Universal Class.
This series covers...
The difference between Classes and Sets as defined by Neumann Berneays Gödel (NBG) set theory, members vs subclasses of the Universal Class.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
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Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
https://wn.com/Classes_Vs_Sets_(Axiomatic_Set_Theory)
The difference between Classes and Sets as defined by Neumann Berneays Gödel (NBG) set theory, members vs subclasses of the Universal Class.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
Buy stuff with Zazzle: http://www.zazzle.com/carneades
Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
- published: 09 Oct 2018
- views: 10671
1:03:54
NBG Theory: A Theory of Classes
- Order: Reorder
- Duration: 1:03:54
- Uploaded Date: 29 May 2023
- views: 319
Final presentation for Math 682 at University of Michigan. Topic is Von Neumann–Bernays–Gödel set theory: comparison to ZFC set theory, proof of its finite axio...
Final presentation for Math 682 at University of Michigan. Topic is Von Neumann–Bernays–Gödel set theory: comparison to ZFC set theory, proof of its finite axiomatizability, and the limitation of size axiom.
00:00 Historical introduction
07:30 Axioms
19:20 Finite axiomatizability
42:10 Limitation of size axiom
59:00 Q&A
https://wn.com/Nbg_Theory_A_Theory_Of_Classes
Final presentation for Math 682 at University of Michigan. Topic is Von Neumann–Bernays–Gödel set theory: comparison to ZFC set theory, proof of its finite axiomatizability, and the limitation of size axiom.
00:00 Historical introduction
07:30 Axioms
19:20 Finite axiomatizability
42:10 Limitation of size axiom
59:00 Q&A
- published: 29 May 2023
- views: 319
3:39
Roger Penrose explains Godel's incompleteness theorem in 3 minutes
- Order: Reorder
- Duration: 3:39
- Uploaded Date: 01 Jun 2020
- views: 1303750
good explanation
from his interview with joe rogan
https://www.youtube.com/watch?v=GEw0ePZUMHA
good explanation
from his interview with joe rogan
https://www.youtube.com/watch?v=GEw0ePZUMHA
https://wn.com/Roger_Penrose_Explains_Godel's_Incompleteness_Theorem_In_3_Minutes
good explanation
from his interview with joe rogan
https://www.youtube.com/watch?v=GEw0ePZUMHA
- published: 01 Jun 2020
- views: 1303750
10:52
Is {∅} the Universal Class? (Axiomatic Set Theory)
- Order: Reorder
- Duration: 10:52
- Uploaded Date: 25 Oct 2018
- views: 876
This video shows that with only Axioms 1, 2, and 3, of Neumann Berneays Gödel {∅} could be the universal class.
This series covers the basics of set theory and...
This video shows that with only Axioms 1, 2, and 3, of Neumann Berneays Gödel {∅} could be the universal class.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
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Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
https://wn.com/Is_∅_The_Universal_Class_(Axiomatic_Set_Theory)
This video shows that with only Axioms 1, 2, and 3, of Neumann Berneays Gödel {∅} could be the universal class.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
Buy stuff with Zazzle: http://www.zazzle.com/carneades
Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
- published: 25 Oct 2018
- views: 876
6:47
What is a Supercomplete Set? (Axiomatic Set Theory)
- Order: Reorder
- Duration: 6:47
- Uploaded Date: 18 Oct 2018
- views: 725
An explanation of what a supercomplete set or class is, a formal definition, and several examples and non examples.
This series covers the basics of set theory...
An explanation of what a supercomplete set or class is, a formal definition, and several examples and non examples.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
Buy stuff with Zazzle: http://www.zazzle.com/carneades
Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
https://wn.com/What_Is_A_Supercomplete_Set_(Axiomatic_Set_Theory)
An explanation of what a supercomplete set or class is, a formal definition, and several examples and non examples.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
Buy stuff with Zazzle: http://www.zazzle.com/carneades
Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
- published: 18 Oct 2018
- views: 725
13:10
What is a Swelled Set? (Axiomatic Set Theory)
- Order: Reorder
- Duration: 13:10
- Uploaded Date: 17 Oct 2018
- views: 1068
A explanation of a swelled class or set including examples and a formal definition.
This series covers the basics of set theory and higher order logic. In thi...
A explanation of a swelled class or set including examples and a formal definition.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
Buy stuff with Zazzle: http://www.zazzle.com/carneades
Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
https://wn.com/What_Is_A_Swelled_Set_(Axiomatic_Set_Theory)
A explanation of a swelled class or set including examples and a formal definition.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
Buy stuff with Zazzle: http://www.zazzle.com/carneades
Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
- published: 17 Oct 2018
- views: 1068
21:42
Von Neumann–Bernays–Gödel set theory
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published: 22 Jan 2016
Von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory
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Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical Zermelo–Fraenkel set theory (ZFC).A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC.
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1:35:41
Von Neumann–Bernays–Gödel set theory | Wikipedia audio article
This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Von_Neum...
published: 25 Dec 2018
Von Neumann–Bernays–Gödel set theory | Wikipedia audio article
Von Neumann–Bernays–Gödel set theory | Wikipedia audio article
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This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
00:01:50 1 Classes in set theory
00:02:00 1.1 The uses of classes
00:05:21 1.2 Axiom schema versus class existence theorem
00:07:49 2 Axiomatization of NBG
00:07:59 2.1 Classes and sets
00:12:01 2.2 Definitions and axioms of extensionality and pairing
00:16:24 2.3 Class existence axioms and axiom of regularity
00:34:56 2.4 Class existence theorem
00:55:06 2.5 Extending the class existence theorem
01:03:16 2.6 Set axioms
01:15:50 2.7 Axiom of global choice
01:18:44 3 History
01:18:53 3.1 Von Neumann's 1925 axiom system
01:23:59 3.2 Von Neumann's 1929 axiom system
01:27:07 3.3 Bernays' axiom system
01:28:12 3.4 Gödel's axiom system (NBG)
01:30:43 4 NBG, ZFC, and MK
01:33:13 4.1 Models
01:34:12 5 Category theory
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"There is only one good, knowledge, and one evil, ignorance."
- Socrates
SUMMARY
=======
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.
A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas (membership and equality) and finitely many logical symbols, only finitely many axioms are needed to build the classes satisfying them. This is why NBG is finitely axiomatizable. Classes are also used for other constructions, for handling the set-theoretic paradoxes, and for stating the axiom of global choice, which is stronger than ZFC's axiom of choice.
John von Neumann introduced classes into set theory in 1925. The primitive notions of his theory were function and argument. Using these notions, he defined class and set. Paul Bernays reformulated von Neumann's theory by taking class and set as primitive notions. Kurt Gödel simplified Bernays' theory for his relative consistency proof of the axiom of choice and the generalized continuum hypothesis.
11:14
A Lesser-Known Mathematical Framework: The Godel Bernays Set Theory
Embark on a captivating journey through the lesser-known corridors of mathematical philoso...
published: 25 Mar 2024
A Lesser-Known Mathematical Framework: The Godel Bernays Set Theory
A Lesser-Known Mathematical Framework: The Godel Bernays Set Theory
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- published: 25 Mar 2024
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Embark on a captivating journey through the lesser-known corridors of mathematical philosophy with "The Gödel-Bernays Set Theory: Unveiling the Universe of Mathematics." This video takes you deep into the heart of a mathematical framework that challenges our understanding of the universe. From its historical origins to its profound philosophical implications, discover how the Gödel-Bernays set theory offers a unique lens through which we can explore the infinite and the unknown. Through a blend of art, history, and science, this narrative delves into the minds of Kurt Gödel, Paul Bernays, and the pioneering thinkers who dared to question the foundations of mathematical reality. As we navigate the complex interplay between sets and classes, and the philosophical quandaries they pose, we invite you to reflect on the nature of existence, knowledge, and the endless pursuit of truth in the mathematical cosmos. Join us on this enlightening journey, and let the mysteries of Gödel-Bernays set theory illuminate your understanding of the world. Thank you for your curiosity and your willingness to explore the depths of mathematical philosophy with us.
#Mathematics #Philosophy #GödelBernays #SetTheory #MathematicalLogic
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3:31
What is a Basic Universe? (Axiomatic Set Theory)
An explanation of a Basic Universe for Neumann Berneays Gödel Set/Class Theory.
This se...
published: 19 Oct 2018
What is a Basic Universe? (Axiomatic Set Theory)
What is a Basic Universe? (Axiomatic Set Theory)
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- published: 19 Oct 2018
- views: 2017
An explanation of a Basic Universe for Neumann Berneays Gödel Set/Class Theory.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
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Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
3:25
Classes vs Sets (Axiomatic Set Theory)
The difference between Classes and Sets as defined by Neumann Berneays Gödel (NBG) set the...
published: 09 Oct 2018
Classes vs Sets (Axiomatic Set Theory)
Classes vs Sets (Axiomatic Set Theory)
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- published: 09 Oct 2018
- views: 10671
The difference between Classes and Sets as defined by Neumann Berneays Gödel (NBG) set theory, members vs subclasses of the Universal Class.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
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Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
1:03:54
NBG Theory: A Theory of Classes
Final presentation for Math 682 at University of Michigan. Topic is Von Neumann–Bernays–Gö...
published: 29 May 2023
NBG Theory: A Theory of Classes
NBG Theory: A Theory of Classes
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Final presentation for Math 682 at University of Michigan. Topic is Von Neumann–Bernays–Gödel set theory: comparison to ZFC set theory, proof of its finite axiomatizability, and the limitation of size axiom.
00:00 Historical introduction
07:30 Axioms
19:20 Finite axiomatizability
42:10 Limitation of size axiom
59:00 Q&A
3:39
Roger Penrose explains Godel's incompleteness theorem in 3 minutes
good explanation
from his interview with joe rogan
https://www.youtube.com/watch?v=GEw0eP...
published: 01 Jun 2020
Roger Penrose explains Godel's incompleteness theorem in 3 minutes
Roger Penrose explains Godel's incompleteness theorem in 3 minutes
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- published: 01 Jun 2020
- views: 1303750
good explanation
from his interview with joe rogan
https://www.youtube.com/watch?v=GEw0ePZUMHA
10:52
Is {∅} the Universal Class? (Axiomatic Set Theory)
This video shows that with only Axioms 1, 2, and 3, of Neumann Berneays Gödel {∅} could be...
published: 25 Oct 2018
Is {∅} the Universal Class? (Axiomatic Set Theory)
Is {∅} the Universal Class? (Axiomatic Set Theory)
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- published: 25 Oct 2018
- views: 876
This video shows that with only Axioms 1, 2, and 3, of Neumann Berneays Gödel {∅} could be the universal class.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
Buy stuff with Zazzle: http://www.zazzle.com/carneades
Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
6:47
What is a Supercomplete Set? (Axiomatic Set Theory)
An explanation of what a supercomplete set or class is, a formal definition, and several e...
published: 18 Oct 2018
What is a Supercomplete Set? (Axiomatic Set Theory)
What is a Supercomplete Set? (Axiomatic Set Theory)
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- published: 18 Oct 2018
- views: 725
An explanation of what a supercomplete set or class is, a formal definition, and several examples and non examples.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
Sponsors: João Costa Neto, Dakota Jones, Thorin Isaiah Malmgren, Prince Otchere, Mike Samuel, Daniel Helland, Mohammad Azmi Banibaker, Dennis Sexton, kdkdk, Yu Saburi, Mauricino Andrade, Diéssica, Will Roberts, Greg Gauthier, Christian Bay, Joao Sa, Richard Seaton, Edward Jacobson, isenshi, and √2. Thanks for your support!
Donate on Patreon: https://www.patreon.com/Carneades
Buy stuff with Zazzle: http://www.zazzle.com/carneades
Follow us on Twitter: @CarneadesCyrene https://twitter.com/CarneadesCyrene
Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
13:10
What is a Swelled Set? (Axiomatic Set Theory)
A explanation of a swelled class or set including examples and a formal definition.
This ...
published: 17 Oct 2018
What is a Swelled Set? (Axiomatic Set Theory)
What is a Swelled Set? (Axiomatic Set Theory)
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- published: 17 Oct 2018
- views: 1068
A explanation of a swelled class or set including examples and a formal definition.
This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory. In the next month we will look at relationships between sets.
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Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy, Set Theory and the Continuum Problem by Smullyan and Fitting, Set Theory The Structure of Arithmetic by Hamilton and Landin, and more! (#SetTheory)
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.
Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Set_theory
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