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Understanding everything is set in axiomatic set theory
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When I read a book of set theory written by Charles Pinter, in chapter 1 section 7, the author says
If $A$ is any set, there is an element $ain A$ such that $acap A=emptyset$.
The author calls it the axiom of foundation. Let $A=red,blue$, by this axiom, we know $red cap A=emptyset$, but in native set theory $red cap A$ makes no sense, I know in axiomatic set theory everything is set, but how to explain this contradiction? Are set in naive set theory and set in axiomatic set theory the same?
set-theory
asked Jul 17 at 3:32
noname1014
1,19211131
add a comment |Â
up vote
3
down vote
favorite
When I read a book of set theory written by Charles Pinter, in chapter 1 section 7, the author says
If $A$ is any set, there is an element $ain A$ such that $acap A=emptyset$.
The author calls it the axiom of foundation. Let $A=red,blue$, by this axiom, we know $red cap A=emptyset$, but in native set theory $red cap A$ makes no sense, I know in axiomatic set theory everything is set, but how to explain this contradiction? Are set in naive set theory and set in axiomatic set theory the same?
set-theory
asked Jul 17 at 3:32
noname1014
1,19211131
Does en.wikipedia.org/wiki/… answer your question?
– user76284
Jul 17 at 3:37
why $redcap A$ makes no sense?
– zaphodxvii
Jul 17 at 5:52
1
@zaphodxvii - because $text red$ is a color.
– Mauro ALLEGRANZA
Jul 17 at 6:13
1
How do you explain "everything is binary data" in computers? Surely this comment is not binary data per se. So... How do you cope with that?
– Asaf Karagila♦
Jul 17 at 9:49
It doesn't matter what $red$ is... What matters is what $xcap y=z$ means, since $cap$ is not a fundamental symbol in the Language Of Set Theory, but a "defined notion". In other words, $xcap y=z$ is an abbreviation for a "formula" in LOST. What is that formula?
– DanielWainfleet
Jul 24 at 19:11
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
When I read a book of set theory written by Charles Pinter, in chapter 1 section 7, the author says
If $A$ is any set, there is an element $ain A$ such that $acap A=emptyset$.
The author calls it the axiom of foundation. Let $A=red,blue$, by this axiom, we know $red cap A=emptyset$, but in native set theory $red cap A$ makes no sense, I know in axiomatic set theory everything is set, but how to explain this contradiction? Are set in naive set theory and set in axiomatic set theory the same?
set-theory
asked Jul 17 at 3:32
noname1014
1,19211131
When I read a book of set theory written by Charles Pinter, in chapter 1 section 7, the author says
If $A$ is any set, there is an element $ain A$ such that $acap A=emptyset$.
The author calls it the axiom of foundation. Let $A=red,blue$, by this axiom, we know $red cap A=emptyset$, but in native set theory $red cap A$ makes no sense, I know in axiomatic set theory everything is set, but how to explain this contradiction? Are set in naive set theory and set in axiomatic set theory the same?
set-theory
asked Jul 17 at 3:32
noname1014
1,19211131
asked Jul 17 at 3:32
noname1014
1,19211131
asked Jul 17 at 3:32
noname1014
1,19211131
asked Jul 17 at 3:32
noname1014
1,19211131
1,19211131
Does en.wikipedia.org/wiki/… answer your question?
– user76284
Jul 17 at 3:37
why $redcap A$ makes no sense?
– zaphodxvii
Jul 17 at 5:52
1
@zaphodxvii - because $text red$ is a color.
– Mauro ALLEGRANZA
Jul 17 at 6:13
1
How do you explain "everything is binary data" in computers? Surely this comment is not binary data per se. So... How do you cope with that?
– Asaf Karagila♦
Jul 17 at 9:49
It doesn't matter what $red$ is... What matters is what $xcap y=z$ means, since $cap$ is not a fundamental symbol in the Language Of Set Theory, but a "defined notion". In other words, $xcap y=z$ is an abbreviation for a "formula" in LOST. What is that formula?
– DanielWainfleet
Jul 24 at 19:11
add a comment |Â
Does en.wikipedia.org/wiki/… answer your question?
– user76284
Jul 17 at 3:37
why $redcap A$ makes no sense?
– zaphodxvii
Jul 17 at 5:52
1
@zaphodxvii - because $text red$ is a color.
– Mauro ALLEGRANZA
Jul 17 at 6:13
1
How do you explain "everything is binary data" in computers? Surely this comment is not binary data per se. So... How do you cope with that?
– Asaf Karagila♦
Jul 17 at 9:49
It doesn't matter what $red$ is... What matters is what $xcap y=z$ means, since $cap$ is not a fundamental symbol in the Language Of Set Theory, but a "defined notion". In other words, $xcap y=z$ is an abbreviation for a "formula" in LOST. What is that formula?
– DanielWainfleet
Jul 24 at 19:11
Does en.wikipedia.org/wiki/… answer your question?
– user76284
Jul 17 at 3:37
Does en.wikipedia.org/wiki/… answer your question?
– user76284
Jul 17 at 3:37
why $redcap A$ makes no sense?
– zaphodxvii
Jul 17 at 5:52
why $redcap A$ makes no sense?
– zaphodxvii
Jul 17 at 5:52
1
1
@zaphodxvii - because $text red$ is a color.
– Mauro ALLEGRANZA
Jul 17 at 6:13
@zaphodxvii - because $text red$ is a color.
– Mauro ALLEGRANZA
Jul 17 at 6:13
1
1
How do you explain "everything is binary data" in computers? Surely this comment is not binary data per se. So... How do you cope with that?
– Asaf Karagila♦
Jul 17 at 9:49
How do you explain "everything is binary data" in computers? Surely this comment is not binary data per se. So... How do you cope with that?
– Asaf Karagila♦
Jul 17 at 9:49
It doesn't matter what $red$ is... What matters is what $xcap y=z$ means, since $cap$ is not a fundamental symbol in the Language Of Set Theory, but a "defined notion". In other words, $xcap y=z$ is an abbreviation for a "formula" in LOST. What is that formula?
– DanielWainfleet
Jul 24 at 19:11
It doesn't matter what $red$ is... What matters is what $xcap y=z$ means, since $cap$ is not a fundamental symbol in the Language Of Set Theory, but a "defined notion". In other words, $xcap y=z$ is an abbreviation for a "formula" in LOST. What is that formula?
– DanielWainfleet
Jul 24 at 19:11
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
8
down vote
accepted
There are several tensions between naive and axiomatic set theory. The big one, of course, is "naive" versus "axiomatic" - are we looking at an informal or formal theory? This question, though, focuses on a different tension: the tension between being maximally inclusive and being meaningfully specific.
In short, the situation is this:
Not every "naive set" is a set in the sense of axiomatic set theory.
Here's a bit more detail:
Things like "red" and "blue" simply don't exist from the point of view of axiomatic set theory (at least, the usual version; not every form of axiomatic set theory includes the axiom of foundation, or even asserts that all objects are sets!). If you wish, you can think of standard axiomatic set theory as describing a portion of the mathematical universe - namely, the part which can be "built out of $emptyset$" just by the basic set operations (including powerset and transfinite recursion - so, maybe "basic" is misleading!).
Put another way, it's obvious that any theory of sets needs to be able to talk about $emptyset$. Well, $emptyset$ is also a set, so that should be included. By contrast, if all we want to do is talk about sets, there's no obvious reason why something like "red" should be in our universe. Axiomatic set theory studies the concept of set on its own in a sense. In light of this, it should be surprising that something actually interesting (let alone powerful enough to implement all of mathematics) comes out!
answered Jul 17 at 3:42
Noah Schweber
111k9140263
thank you, I still have a question, in wiki en.wikipedia.org/wiki/… it assumes A is not an empty set, but in Pinter's book, it has no such assumption, is it assumption necessary?
– noname1014
Jul 17 at 6:53
1
@noname1014: Yes, the assumption that $A$ is nonempty is definitely necessary in the Axiom of Foundation. It is clearly not true for $A=varnothing$. If your author has forgotten it, that's an error in the book.
– Henning Makholm
Jul 17 at 11:43
add a comment |Â
up vote
0
down vote
He is probably saying that $a$ is also a set. For example $$A=0,1$$let $a=1in A$ therefore $$acap A=phi$$
answered Jul 17 at 9:42
Mostafa Ayaz
8,6023630
That's really not the question here.
– Asaf Karagila♦
Jul 17 at 9:50
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
8
down vote
accepted
There are several tensions between naive and axiomatic set theory. The big one, of course, is "naive" versus "axiomatic" - are we looking at an informal or formal theory? This question, though, focuses on a different tension: the tension between being maximally inclusive and being meaningfully specific.
In short, the situation is this:
Not every "naive set" is a set in the sense of axiomatic set theory.
Here's a bit more detail:
Things like "red" and "blue" simply don't exist from the point of view of axiomatic set theory (at least, the usual version; not every form of axiomatic set theory includes the axiom of foundation, or even asserts that all objects are sets!). If you wish, you can think of standard axiomatic set theory as describing a portion of the mathematical universe - namely, the part which can be "built out of $emptyset$" just by the basic set operations (including powerset and transfinite recursion - so, maybe "basic" is misleading!).
Put another way, it's obvious that any theory of sets needs to be able to talk about $emptyset$. Well, $emptyset$ is also a set, so that should be included. By contrast, if all we want to do is talk about sets, there's no obvious reason why something like "red" should be in our universe. Axiomatic set theory studies the concept of set on its own in a sense. In light of this, it should be surprising that something actually interesting (let alone powerful enough to implement all of mathematics) comes out!
answered Jul 17 at 3:42
Noah Schweber
111k9140263
thank you, I still have a question, in wiki en.wikipedia.org/wiki/… it assumes A is not an empty set, but in Pinter's book, it has no such assumption, is it assumption necessary?
– noname1014
Jul 17 at 6:53
1
@noname1014: Yes, the assumption that $A$ is nonempty is definitely necessary in the Axiom of Foundation. It is clearly not true for $A=varnothing$. If your author has forgotten it, that's an error in the book.
– Henning Makholm
Jul 17 at 11:43
add a comment |Â
up vote
8
down vote
accepted
There are several tensions between naive and axiomatic set theory. The big one, of course, is "naive" versus "axiomatic" - are we looking at an informal or formal theory? This question, though, focuses on a different tension: the tension between being maximally inclusive and being meaningfully specific.
In short, the situation is this:
Not every "naive set" is a set in the sense of axiomatic set theory.
Here's a bit more detail:
Things like "red" and "blue" simply don't exist from the point of view of axiomatic set theory (at least, the usual version; not every form of axiomatic set theory includes the axiom of foundation, or even asserts that all objects are sets!). If you wish, you can think of standard axiomatic set theory as describing a portion of the mathematical universe - namely, the part which can be "built out of $emptyset$" just by the basic set operations (including powerset and transfinite recursion - so, maybe "basic" is misleading!).
Put another way, it's obvious that any theory of sets needs to be able to talk about $emptyset$. Well, $emptyset$ is also a set, so that should be included. By contrast, if all we want to do is talk about sets, there's no obvious reason why something like "red" should be in our universe. Axiomatic set theory studies the concept of set on its own in a sense. In light of this, it should be surprising that something actually interesting (let alone powerful enough to implement all of mathematics) comes out!
answered Jul 17 at 3:42
Noah Schweber
111k9140263
thank you, I still have a question, in wiki en.wikipedia.org/wiki/… it assumes A is not an empty set, but in Pinter's book, it has no such assumption, is it assumption necessary?
– noname1014
Jul 17 at 6:53
1
@noname1014: Yes, the assumption that $A$ is nonempty is definitely necessary in the Axiom of Foundation. It is clearly not true for $A=varnothing$. If your author has forgotten it, that's an error in the book.
– Henning Makholm
Jul 17 at 11:43
add a comment |Â
up vote
8
down vote
accepted
up vote
8
down vote
accepted
There are several tensions between naive and axiomatic set theory. The big one, of course, is "naive" versus "axiomatic" - are we looking at an informal or formal theory? This question, though, focuses on a different tension: the tension between being maximally inclusive and being meaningfully specific.
In short, the situation is this:
Not every "naive set" is a set in the sense of axiomatic set theory.
Here's a bit more detail:
Things like "red" and "blue" simply don't exist from the point of view of axiomatic set theory (at least, the usual version; not every form of axiomatic set theory includes the axiom of foundation, or even asserts that all objects are sets!). If you wish, you can think of standard axiomatic set theory as describing a portion of the mathematical universe - namely, the part which can be "built out of $emptyset$" just by the basic set operations (including powerset and transfinite recursion - so, maybe "basic" is misleading!).
Put another way, it's obvious that any theory of sets needs to be able to talk about $emptyset$. Well, $emptyset$ is also a set, so that should be included. By contrast, if all we want to do is talk about sets, there's no obvious reason why something like "red" should be in our universe. Axiomatic set theory studies the concept of set on its own in a sense. In light of this, it should be surprising that something actually interesting (let alone powerful enough to implement all of mathematics) comes out!
answered Jul 17 at 3:42
Noah Schweber
111k9140263
There are several tensions between naive and axiomatic set theory. The big one, of course, is "naive" versus "axiomatic" - are we looking at an informal or formal theory? This question, though, focuses on a different tension: the tension between being maximally inclusive and being meaningfully specific.
In short, the situation is this:
Not every "naive set" is a set in the sense of axiomatic set theory.
Here's a bit more detail:
Things like "red" and "blue" simply don't exist from the point of view of axiomatic set theory (at least, the usual version; not every form of axiomatic set theory includes the axiom of foundation, or even asserts that all objects are sets!). If you wish, you can think of standard axiomatic set theory as describing a portion of the mathematical universe - namely, the part which can be "built out of $emptyset$" just by the basic set operations (including powerset and transfinite recursion - so, maybe "basic" is misleading!).
Put another way, it's obvious that any theory of sets needs to be able to talk about $emptyset$. Well, $emptyset$ is also a set, so that should be included. By contrast, if all we want to do is talk about sets, there's no obvious reason why something like "red" should be in our universe. Axiomatic set theory studies the concept of set on its own in a sense. In light of this, it should be surprising that something actually interesting (let alone powerful enough to implement all of mathematics) comes out!
answered Jul 17 at 3:42
Noah Schweber
111k9140263
answered Jul 17 at 3:42
Noah Schweber
111k9140263
answered Jul 17 at 3:42
Noah Schweber
111k9140263
answered Jul 17 at 3:42
Noah Schweber
111k9140263
111k9140263
thank you, I still have a question, in wiki en.wikipedia.org/wiki/… it assumes A is not an empty set, but in Pinter's book, it has no such assumption, is it assumption necessary?
– noname1014
Jul 17 at 6:53
1
@noname1014: Yes, the assumption that $A$ is nonempty is definitely necessary in the Axiom of Foundation. It is clearly not true for $A=varnothing$. If your author has forgotten it, that's an error in the book.
– Henning Makholm
Jul 17 at 11:43
add a comment |Â
thank you, I still have a question, in wiki en.wikipedia.org/wiki/… it assumes A is not an empty set, but in Pinter's book, it has no such assumption, is it assumption necessary?
– noname1014
Jul 17 at 6:53
1
@noname1014: Yes, the assumption that $A$ is nonempty is definitely necessary in the Axiom of Foundation. It is clearly not true for $A=varnothing$. If your author has forgotten it, that's an error in the book.
– Henning Makholm
Jul 17 at 11:43
thank you, I still have a question, in wiki en.wikipedia.org/wiki/… it assumes A is not an empty set, but in Pinter's book, it has no such assumption, is it assumption necessary?
– noname1014
Jul 17 at 6:53
thank you, I still have a question, in wiki en.wikipedia.org/wiki/… it assumes A is not an empty set, but in Pinter's book, it has no such assumption, is it assumption necessary?
– noname1014
Jul 17 at 6:53
1
1
@noname1014: Yes, the assumption that $A$ is nonempty is definitely necessary in the Axiom of Foundation. It is clearly not true for $A=varnothing$. If your author has forgotten it, that's an error in the book.
– Henning Makholm
Jul 17 at 11:43
@noname1014: Yes, the assumption that $A$ is nonempty is definitely necessary in the Axiom of Foundation. It is clearly not true for $A=varnothing$. If your author has forgotten it, that's an error in the book.
– Henning Makholm
Jul 17 at 11:43
add a comment |Â
up vote
0
down vote
He is probably saying that $a$ is also a set. For example $$A=0,1$$let $a=1in A$ therefore $$acap A=phi$$
answered Jul 17 at 9:42
Mostafa Ayaz
8,6023630
That's really not the question here.
– Asaf Karagila♦
Jul 17 at 9:50
add a comment |Â
up vote
0
down vote
He is probably saying that $a$ is also a set. For example $$A=0,1$$let $a=1in A$ therefore $$acap A=phi$$
answered Jul 17 at 9:42
Mostafa Ayaz
8,6023630
That's really not the question here.
– Asaf Karagila♦
Jul 17 at 9:50
add a comment |Â
up vote
0
down vote
up vote
0
down vote
He is probably saying that $a$ is also a set. For example $$A=0,1$$let $a=1in A$ therefore $$acap A=phi$$
answered Jul 17 at 9:42
Mostafa Ayaz
8,6023630
He is probably saying that $a$ is also a set. For example $$A=0,1$$let $a=1in A$ therefore $$acap A=phi$$
answered Jul 17 at 9:42
Mostafa Ayaz
8,6023630
answered Jul 17 at 9:42
Mostafa Ayaz
8,6023630
answered Jul 17 at 9:42
Mostafa Ayaz
8,6023630
answered Jul 17 at 9:42
Mostafa Ayaz
8,6023630
8,6023630
That's really not the question here.
– Asaf Karagila♦
Jul 17 at 9:50
add a comment |Â
That's really not the question here.
– Asaf Karagila♦
Jul 17 at 9:50
That's really not the question here.
– Asaf Karagila♦
Jul 17 at 9:50
That's really not the question here.
– Asaf Karagila♦
Jul 17 at 9:50
add a comment |Â
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Does en.wikipedia.org/wiki/… answer your question?
– user76284
Jul 17 at 3:37
why $redcap A$ makes no sense?
– zaphodxvii
Jul 17 at 5:52
1
@zaphodxvii - because $text red$ is a color.
– Mauro ALLEGRANZA
Jul 17 at 6:13
1
How do you explain "everything is binary data" in computers? Surely this comment is not binary data per se. So... How do you cope with that?
– Asaf Karagila♦
Jul 17 at 9:49
It doesn't matter what $red$ is... What matters is what $xcap y=z$ means, since $cap$ is not a fundamental symbol in the Language Of Set Theory, but a "defined notion". In other words, $xcap y=z$ is an abbreviation for a "formula" in LOST. What is that formula?
– DanielWainfleet
Jul 24 at 19:11