Thanks for sharing this video! Its a fun and clear introduction. Here's an excellent summary of current technical results on the continuum hypothesis and the mathematics of large cardinals by an authoritative researcher in the field.
Corina Marinescu Ha! I think there are some videos of Woodin or Dana Scott talking about this stuff, but I have to look. The book version would probably be the wonderful and authoritative "The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings" by Akihiro Kanamori: http://books.google.com/books/about/The_Higher_Infinite.html?id=FH_n84ocuSMC
Sometimes, it is better not to answer any unanswered question. It can be funny, but the mathematicians of 20th century can never match with the reality of the infinite set of numbers. If we try to compare the vast space of universe with the ever-extending ocean of number systems, they really mess up with so much complexity that no one can track the actual pattern. Similar had happened with them.
Hi.. So if there is somebody from you interested to learn more about cantor his diagonal Argument and its failure.. Than let me know..:-) i am really interested to Show you a List wich contains cantors non listable members of the Unit Intervall.. there are Lot of Strange Things i would like to Show you.. So here is something to get you interested..:-)
This is a difficult question to give a short answer to.
The concept of the "Universe of Sets" is to me that of a genuine nonphysical realm. The relationship of the physical universe to the universe of sets is absolutely unclear to me, though our understanding of the universe of sets does make physical predictions. I certainly would not claim that the universe of sets contains the physical universe. I also believe that the understanding of the universe of sets really has nothing to do with our physical universe except of course we happen to inhabit the physical universe and so our search for truth is subject to the physical constraints of our home-universe (i.e. we cannot yet do infinite searches, or design a physical experiment to determine if the Continuum Hypothesis is true). But to me this makes the progress we have achieved all the more remarkable and contributes to the beauty of our conception of the universe of sets (to use a decidedly human adjective).
All of the above also holds of every applied set theory, such as Quine's set theories WITH "INDIVIDUALS" [= all objects other than the null set which have no elements]. That is, if you add individuals to sets as the basic objects of your logical theory, and if you assume that individuals are material objects with all their usual attributes, then such an applied set theory simultaneously includes the entire physical universe by default, which is INCLUDED INSIDE THE SET UNIVERSE. (The only things "missing" are natural laws; however, these could be included after all among the individuals' attributes, if we allow such higher-level theoretical attributes.) This, by the way, completely accords with Frege's view that logic (just like mathematics) holds for EVERYTHING (not just Plato's ideal objects).
Most set theories have no individuals, so then it appears that set theory and natural sciences have two different domains and two different kinds of laws. That's typical for Platonic Dualism, the position held by most "working mathematicians". But the problem is that, e.g. theoretical physics already contains all kinds of powerful sets --- for example, Quantum Mechanics uses Hilbert spaces, i.e. fields with infinitely many dimensions. [Some logicians think that theoretical physics uses at least seven levels of sets, or seven orders in type theory. But that's probably much too few.] So you can't "separate" the domains of objects, as Plato (and unfortunately even Gödel) tried to do. In other words, there's something correct about Aristotle's view that ideas "inhere" in matter and are inseparable.
What IS SEPARATE and distinguishable, however, is the MODALITY of the propositions stating laws of logic and laws of nature. Logical laws of sets and numbers, etc. are "analytic", or "logically necessary", whereas natural laws (and all observations) are "contingent", or "synthetic". Quine has famously claimed that the analytic/synthetic dichotomy which Carnap supported collapses. But that's because Quine accepts only sensory observation as a support for knowledge. If, in addition, we accept logical intuition as our evidence for logic (and set theory, and number theory, etc.), then we can tell the difference between two classes of propositions. (But NOT between two classes of objects !!!) This is the point I make in my article "Gödels Platonismus", which you should go back and read again.
Summarizing: 1. It is useless to try separating the set universe from the physical universe, i.e. the domain of sets from the domain of physical entities. Set theoreticians of course don't deal with physical entities if no individuals are included; but all of set theory applies "automatically" to all natural sciences, hence set theoreticians IMPLICITLY always do deal with the physical universe. IN THAT SENSE, the universe of sets "automatically" INCLUDES the physical universe --- because all work on set theory is intended for (or at least "welcomes") applications. 2. What IS separate are two categories of propositions, depending on the kind of support they enjoy. One category is supported by sensory observation --- and what may be inferred deductively or inductively from observation. The other category is supported by logical/mathematical intuition. And intuition may be clearly distinguished from sensory observation. Gödel didn't know how to do this, but I do.
I have written a long manuscript on intuition, where I show how and why it differs from sensory observation, which I will soon send off to a publisher, where most of these things are discussed in much more detail.
Yours, Eckehart Köhler
PS: You may forward this to Woodin, if you like. He wrote "the relation . . . is absolutely unclear to me", so maybe what I write here becomes less unclear.
PPS: Your problems with the "Absolute" are probably easily solvable, if you just define your concepts clearly enough, and maybe read e.g. Weingartner's book on Omniscience.
Thomas A. Anderson wrote: ---------- Forwarded message ---------- From: W. Hugh Woodin Date: Mon, Oct 6, 2008 at 9:41 PM Subject: Re: is there a concept: natural sets + pure sets = To: "Thomas A. Anderson"
This is a difficult question to give a short answer to.
The concept of the "Universe of Sets" is to me that of a genuine nonphysical realm. The relationship of the physical universe to the universe of sets is absolutely unclear to me, though our understanding of the universe of sets does make physical predictions. I certainly would not claim that the universe of sets contains the physical universe. I also believe that the understanding of the universe of sets really has nothing to do with our physical universe except of course we happen to inhabit the physical universe and so our search for truth is subject to the physical constraints of our home-universe (i.e. we cannot yet do infinite searches, or design a physical experiment to determine if the Continuum Hypothesis is true). But to me this makes the progress we have achieved all the more remarkable and contributes to the beauty of our conception of the universe of sets (to use a decidedly human adjective).
All of the above also holds of every applied set theory, such as Quine's set theories WITH "INDIVIDUALS" [= all objects other than the null set which have no elements]. That is, if you add individuals to sets as the basic objects of your logical theory, and if you assume that individuals are material objects with all their usual attributes, then such an applied set theory simultaneously includes the entire physical universe by default, which is INCLUDED INSIDE THE SET UNIVERSE. (The only things "missing" are natural laws; however, these could be included after all among the individuals' attributes, if we allow such higher-level theoretical attributes.) This, by the way, completely accords with Frege's view that logic (just like mathematics) holds for EVERYTHING (not just Plato's ideal objects).
Most set theories have no individuals, so then it appears that set theory and natural sciences have two different domains and two different kinds of laws. That's typical for Platonic Dualism, the position held by most "working mathematicians". But the problem is that, e.g. theoretical physics already contains all kinds of powerful sets --- for example, Quantum Mechanics uses Hilbert spaces, i.e. fields with infinitely many dimensions. [Some logicians think that theoretical physics uses at least seven levels of sets, or seven orders in type theory. But that's probably much too few.] So you can't "separate" the domains of objects, as Plato (and unfortunately even Gödel) tried to do. In other words, there's something correct about Aristotle's view that ideas "inhere" in matter and are inseparable.
What IS SEPARATE and distinguishable, however, is the MODALITY of the propositions stating laws of logic and laws of nature. Logical laws of sets and numbers, etc. are "analytic", or "logically necessary", whereas natural laws (and all observations) are "contingent", or "synthetic". Quine has famously claimed that the analytic/synthetic dichotomy which Carnap supported collapses. But that's because Quine accepts only sensory observation as a support for knowledge. If, in addition, we accept logical intuition as our evidence for logic (and set theory, and number theory, etc.), then we can tell the difference between two classes of propositions. (But NOT between two classes of objects !!!) This is the point I make in my article "Gödels Platonismus", which you should go back and read again.
Summarizing: 1. It is useless to try separating the set universe from the physical universe, i.e. the domain of sets from the domain of physical entities. Set theoreticians of course don't deal with physical entities if no individuals are included; but all of set theory applies "automatically" to all natural sciences, hence set theoreticians IMPLICITLY always do deal with the physical universe. IN THAT SENSE, the universe of sets "automatically" INCLUDES the physical universe --- because all work on set theory is intended for (or at least "welcomes") applications. 2. What IS separate are two categories of propositions, depending on the kind of support they enjoy. One category is supported by sensory observation --- and what may be inferred deductively or inductively from observation. The other category is supported by logical/mathematical intuition. And intuition may be clearly distinguished from sensory observation. Gödel didn't know how to do this, but I do.
I have written a long manuscript on intuition, where I show how and why it differs from sensory observation, which I will soon send off to a publisher, where most of these things are discussed in much more detail.
Yours, Eckehart Köhler
PS: You may forward this to Woodin, if you like. He wrote "the relation . . . is absolutely unclear to me", so maybe what I write here becomes less unclear.
PPS: Your problems with the "Absolute" are probably easily solvable, if you just define your concepts clearly enough, and maybe read e.g. Weingartner's book on Omniscience.
Thomas A. Anderson wrote: ---------- Forwarded message ---------- From: W. Hugh Woodin Date: Mon, Oct 6, 2008 at 9:41 PM Subject: Re: is there a concept: natural sets + pure sets = To: "Thomas A. Anderson"
This is a difficult question to give a short answer to.
The concept of the "Universe of Sets" is to me that of a genuine nonphysical realm. The relationship of the physical universe to the universe of sets is absolutely unclear to me, though our understanding of the universe of sets does make physical predictions. I certainly would not claim that the universe of sets contains the physical universe. I also believe that the understanding of the universe of sets really has nothing to do with our physical universe except of course we happen to inhabit the physical universe and so our search for truth is subject to the physical constraints of our home-universe (i.e. we cannot yet do infinite searches, or design a physical experiment to determine if the Continuum Hypothesis is true). But to me this makes the progress we have achieved all the more remarkable and contributes to the beauty of our conception of the universe of sets (to use a decidedly human adjective).
Infinitely big, of course!
ReplyDeleteBut really, infinity has different sizes.
It is a concept not a number
ReplyDeleteThanks for sharing this video! Its a fun and clear introduction. Here's an excellent summary of current technical results on the continuum hypothesis and the mathematics of large cardinals by an authoritative researcher in the field.
ReplyDeletehttp://www.math.unicaen.fr/~dehornoy/Surveys/DgtUS.pdf
Walter Salgado thanks, can I have the audio version of that? If not..I'll settle even with a book :D
ReplyDeleteDanke
Corina Marinescu Ha! I think there are some videos of Woodin or Dana Scott talking about this stuff, but I have to look. The book version would probably be the wonderful and authoritative "The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings" by Akihiro Kanamori: http://books.google.com/books/about/The_Higher_Infinite.html?id=FH_n84ocuSMC
ReplyDeleteIt's a beautiful, but advanced, book.
Merci beaucoup Walter Salgado
ReplyDeletein this way should be taken lessons at school!!!!
ReplyDeleteThis guy has an awesome 4th grade teacher.
ReplyDeleteSometimes, it is better not to answer any unanswered question. It can be funny, but the mathematicians of 20th century can never match with the reality of the infinite set of numbers. If we try to compare the vast space of universe with the ever-extending ocean of number systems, they really mess up with so much complexity that no one can track the actual pattern. Similar had happened with them.
ReplyDeleteHi.. So if there is somebody from you interested to learn more about cantor his diagonal Argument and its failure.. Than let me know..:-) i am really interested to Show you a List wich contains cantors non listable members of the Unit Intervall.. there are Lot of Strange Things i would like to Show you.. So here is something to get you interested..:-)
ReplyDeleteThis is a difficult question to give a short answer to.
The concept of the "Universe of Sets" is to me that of a
genuine nonphysical realm. The relationship of the physical
universe to the universe of sets is absolutely unclear to me,
though our understanding of the universe of sets does make
physical predictions. I certainly would not claim that
the universe of sets contains the physical universe. I also
believe that the understanding of the universe of sets really
has nothing to do with our physical universe except of course
we happen to inhabit the physical universe and so our search
for truth is subject to the physical constraints of our
home-universe (i.e. we cannot yet do infinite searches, or
design a physical experiment to determine if the Continuum
Hypothesis is true). But to me this makes the progress we
have achieved all the more remarkable and contributes to
the beauty of our conception of the universe of sets (to
use a decidedly human adjective).
--Hugh
All of the above also holds of every applied set theory, such as Quine's set theories WITH "INDIVIDUALS" [= all objects other than the null set which have no elements]. That is, if you add individuals to sets as the basic objects of your logical theory, and if you assume that individuals are material objects with all their usual attributes, then such an applied set theory simultaneously includes the entire physical universe by default, which is INCLUDED INSIDE THE SET UNIVERSE. (The only things "missing" are natural laws; however, these could be included after all among the individuals' attributes, if we allow such higher-level theoretical attributes.) This, by the way, completely accords with Frege's view that logic (just like mathematics) holds for EVERYTHING (not just Plato's ideal objects).
ReplyDeleteMost set theories have no individuals, so then it appears that set theory and natural sciences have two different domains and two different kinds of laws. That's typical for Platonic Dualism, the position held by most "working mathematicians". But the problem is that, e.g. theoretical physics already contains all kinds of powerful sets --- for example, Quantum Mechanics uses Hilbert spaces, i.e. fields with infinitely many dimensions. [Some logicians think that theoretical physics uses at least seven levels of sets, or seven orders in type theory. But that's probably much too few.] So you can't "separate" the domains of objects, as Plato (and unfortunately even Gödel) tried to do. In other words, there's something correct about Aristotle's view that ideas "inhere" in matter and are inseparable.
What IS SEPARATE and distinguishable, however, is the MODALITY of the propositions stating laws of logic and laws of nature. Logical laws of sets and numbers, etc. are "analytic", or "logically necessary", whereas natural laws (and all observations) are "contingent", or "synthetic". Quine has famously claimed that the analytic/synthetic dichotomy which Carnap supported collapses. But that's because Quine accepts only sensory observation as a support for knowledge. If, in addition, we accept logical intuition as our evidence for logic (and set theory, and number theory, etc.), then we can tell the difference between two classes of propositions. (But NOT between two classes of objects !!!) This is the point I make in my article "Gödels Platonismus", which you should go back and read again.
Summarizing:
1. It is useless to try separating the set universe from the physical universe, i.e. the domain of sets from the domain of physical entities. Set theoreticians of course don't deal with physical entities if no individuals are included; but all of set theory applies "automatically" to all natural sciences, hence set theoreticians IMPLICITLY always do deal with the physical universe. IN THAT SENSE, the universe of sets "automatically" INCLUDES the physical universe --- because all work on set theory is intended for (or at least "welcomes") applications.
2. What IS separate are two categories of propositions, depending on the kind of support they enjoy. One category is supported by sensory observation --- and what may be inferred deductively or inductively from observation. The other category is supported by logical/mathematical intuition. And intuition may be clearly distinguished from sensory observation. Gödel didn't know how to do this, but I do.
I have written a long manuscript on intuition, where I show how and why it differs from sensory observation, which I will soon send off to a publisher, where most of these things are discussed in much more detail.
Yours,
Eckehart Köhler
PS: You may forward this to Woodin, if you like. He wrote "the relation . . . is absolutely unclear to me", so maybe what I write here becomes less unclear.
PPS: Your problems with the "Absolute" are probably easily solvable, if you just define your concepts clearly enough, and maybe read e.g. Weingartner's book on Omniscience.
ReplyDeleteThomas A. Anderson wrote:
---------- Forwarded message ----------
From: W. Hugh Woodin
Date: Mon, Oct 6, 2008 at 9:41 PM
Subject: Re: is there a concept: natural sets + pure sets =
To: "Thomas A. Anderson"
This is a difficult question to give a short answer to.
The concept of the "Universe of Sets" is to me that of a
genuine nonphysical realm. The relationship of the physical
universe to the universe of sets is absolutely unclear to me,
though our understanding of the universe of sets does make
physical predictions. I certainly would not claim that
the universe of sets contains the physical universe. I also
believe that the understanding of the universe of sets really
has nothing to do with our physical universe except of course
we happen to inhabit the physical universe and so our search
for truth is subject to the physical constraints of our
home-universe (i.e. we cannot yet do infinite searches, or
design a physical experiment to determine if the Continuum
Hypothesis is true). But to me this makes the progress we
have achieved all the more remarkable and contributes to
the beauty of our conception of the universe of sets (to
use a decidedly human adjective).
--Hugh
All of the above also holds of every applied set theory, such as Quine's set theories WITH "INDIVIDUALS" [= all objects other than the null set which have no elements]. That is, if you add individuals to sets as the basic objects of your logical theory, and if you assume that individuals are material objects with all their usual attributes, then such an applied set theory simultaneously includes the entire physical universe by default, which is INCLUDED INSIDE THE SET UNIVERSE. (The only things "missing" are natural laws; however, these could be included after all among the individuals' attributes, if we allow such higher-level theoretical attributes.) This, by the way, completely accords with Frege's view that logic (just like mathematics) holds for EVERYTHING (not just Plato's ideal objects).
ReplyDeleteMost set theories have no individuals, so then it appears that set theory and natural sciences have two different domains and two different kinds of laws. That's typical for Platonic Dualism, the position held by most "working mathematicians". But the problem is that, e.g. theoretical physics already contains all kinds of powerful sets --- for example, Quantum Mechanics uses Hilbert spaces, i.e. fields with infinitely many dimensions. [Some logicians think that theoretical physics uses at least seven levels of sets, or seven orders in type theory. But that's probably much too few.] So you can't "separate" the domains of objects, as Plato (and unfortunately even Gödel) tried to do. In other words, there's something correct about Aristotle's view that ideas "inhere" in matter and are inseparable.
What IS SEPARATE and distinguishable, however, is the MODALITY of the propositions stating laws of logic and laws of nature. Logical laws of sets and numbers, etc. are "analytic", or "logically necessary", whereas natural laws (and all observations) are "contingent", or "synthetic". Quine has famously claimed that the analytic/synthetic dichotomy which Carnap supported collapses. But that's because Quine accepts only sensory observation as a support for knowledge. If, in addition, we accept logical intuition as our evidence for logic (and set theory, and number theory, etc.), then we can tell the difference between two classes of propositions. (But NOT between two classes of objects !!!) This is the point I make in my article "Gödels Platonismus", which you should go back and read again.
Summarizing:
1. It is useless to try separating the set universe from the physical universe, i.e. the domain of sets from the domain of physical entities. Set theoreticians of course don't deal with physical entities if no individuals are included; but all of set theory applies "automatically" to all natural sciences, hence set theoreticians IMPLICITLY always do deal with the physical universe. IN THAT SENSE, the universe of sets "automatically" INCLUDES the physical universe --- because all work on set theory is intended for (or at least "welcomes") applications.
2. What IS separate are two categories of propositions, depending on the kind of support they enjoy. One category is supported by sensory observation --- and what may be inferred deductively or inductively from observation. The other category is supported by logical/mathematical intuition. And intuition may be clearly distinguished from sensory observation. Gödel didn't know how to do this, but I do.
I have written a long manuscript on intuition, where I show how and why it differs from sensory observation, which I will soon send off to a publisher, where most of these things are discussed in much more detail.
Yours,
Eckehart Köhler
PS: You may forward this to Woodin, if you like. He wrote "the relation . . . is absolutely unclear to me", so maybe what I write here becomes less unclear.
PPS: Your problems with the "Absolute" are probably easily solvable, if you just define your concepts clearly enough, and maybe read e.g. Weingartner's book on Omniscience.
ReplyDeleteThomas A. Anderson wrote:
---------- Forwarded message ----------
From: W. Hugh Woodin
Date: Mon, Oct 6, 2008 at 9:41 PM
Subject: Re: is there a concept: natural sets + pure sets =
To: "Thomas A. Anderson"
This is a difficult question to give a short answer to.
The concept of the "Universe of Sets" is to me that of a
genuine nonphysical realm. The relationship of the physical
universe to the universe of sets is absolutely unclear to me,
though our understanding of the universe of sets does make
physical predictions. I certainly would not claim that
the universe of sets contains the physical universe. I also
believe that the understanding of the universe of sets really
has nothing to do with our physical universe except of course
we happen to inhabit the physical universe and so our search
for truth is subject to the physical constraints of our
home-universe (i.e. we cannot yet do infinite searches, or
design a physical experiment to determine if the Continuum
Hypothesis is true). But to me this makes the progress we
have achieved all the more remarkable and contributes to
the beauty of our conception of the universe of sets (to
use a decidedly human adjective).
--Hugh
Sorry for the Double Feature it was a Problem on the google+ side to handle the Long copy and Paste Procedere.. So it crashed and i Sent again.. Sorry
ReplyDelete