#settheory — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #settheory, aggregated by home.social.
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I'm looking for an intro into set theory for a very non mathematical person. Everything I find uses numbers, sets of sets of sets, and such things.
Did the mathematicians loose their marbles? :-)
I mean, explaining sets using marbles should be an easy way to introduce the concepts. Only if those are understood, it may be sensible to go to Hilberts hotel, Russells paradox, and maybe, just maybe to Cantors diagonal argument.
Suggestions welcome.
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I'm looking for an intro into set theory for a very non mathematical person. Everything I find uses numbers, sets of sets of sets, and such things.
Did the mathematicians loose their marbles? :-)
I mean, explaining sets using marbles should be an easy way to introduce the concepts. Only if those are understood, it may be sensible to go to Hilberts hotel, Russells paradox, and maybe, just maybe to Cantors diagonal argument.
Suggestions welcome.
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I'm looking for an intro into set theory for a very non mathematical person. Everything I find uses numbers, sets of sets of sets, and such things.
Did the mathematicians loose their marbles? :-)
I mean, explaining sets using marbles should be an easy way to introduce the concepts. Only if those are understood, it may be sensible to go to Hilberts hotel, Russells paradox, and maybe, just maybe to Cantors diagonal argument.
Suggestions welcome.
-
I'm looking for an intro into set theory for a very non mathematical person. Everything I find uses numbers, sets of sets of sets, and such things.
Did the mathematicians loose their marbles? :-)
I mean, explaining sets using marbles should be an easy way to introduce the concepts. Only if those are understood, it may be sensible to go to Hilberts hotel, Russells paradox, and maybe, just maybe to Cantors diagonal argument.
Suggestions welcome.
-
I'm looking for an intro into set theory for a very non mathematical person. Everything I find uses numbers, sets of sets of sets, and such things.
Did the mathematicians loose their marbles? :-)
I mean, explaining sets using marbles should be an easy way to introduce the concepts. Only if those are understood, it may be sensible to go to Hilberts hotel, Russells paradox, and maybe, just maybe to Cantors diagonal argument.
Suggestions welcome.
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Wasn't expecting a set theory round on university challenge - a gift to all three set theorists who watched the episode!
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RE: https://friendica.world/display/84b6ef2b-1569-fb6a-c9a4-fc9241315409
This topic is too close to my heart for me to find it funny 😅
#math #mathematics #settheory #infinity -
RE: https://mastodon.social/@sflorg/116517535576002455
The human behavioral range, filters overlapping data segments by learned, hard encoded, and individual bias...
#BehavioralScience #Scalar #Differential #Equations #DoingTheMath #BehavioralRange #ValueTheory #SetTheory
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RE: https://mastodon.social/@sflorg/116517535576002455
The human behavioral range, filters overlapping data segments by learned, hard encoded, and individual bias...
#BehavioralScience #Scalar #Differential #Equations #DoingTheMath #BehavioralRange #ValueTheory #SetTheory
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RE: https://mastodon.social/@sflorg/116517535576002455
The human behavioral range, filters overlapping data segments by learned, hard encoded, and individual bias...
#BehavioralScience #Scalar #Differential #Equations #DoingTheMath #BehavioralRange #ValueTheory #SetTheory
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Late to the party, but I heart “Rethinking Set Theory”, Tom Leinster’s presentation of ETCS (https://arxiv.org/abs/1212.6543). My natal foundation is higher-order logic, and this is the first time set theory has made any sense to me, other than as a technical device.
Bonus lecture notes: https://webhomes.maths.ed.ac.uk/~tl/ast/ast.pdf
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Late to the party, but I heart “Rethinking Set Theory”, Tom Leinster’s presentation of ETCS (https://arxiv.org/abs/1212.6543). My natal foundation is higher-order logic, and this is the first time set theory has made any sense to me, other than as a technical device.
Bonus lecture notes: https://webhomes.maths.ed.ac.uk/~tl/ast/ast.pdf
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Late to the party, but I heart “Rethinking Set Theory”, Tom Leinster’s presentation of ETCS (https://arxiv.org/abs/1212.6543). My natal foundation is higher-order logic, and this is the first time set theory has made any sense to me, other than as a technical device.
Bonus lecture notes: https://webhomes.maths.ed.ac.uk/~tl/ast/ast.pdf
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Late to the party, but I heart “Rethinking Set Theory”, Tom Leinster’s presentation of ETCS (https://arxiv.org/abs/1212.6543). My natal foundation is higher-order logic, and this is the first time set theory has made any sense to me, other than as a technical device.
Bonus lecture notes: https://webhomes.maths.ed.ac.uk/~tl/ast/ast.pdf
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Begriffsschrift (1879), is one of the first manuscripts on #SymbolicLogic. As such, it literally invents a new language to describe the subjects the author, #GottlobFrege, wants to introduce. And this notation is very unlike what we see in math before or after this.
So I will list some theorems adapted (by me, circa 2020) from #Frege with proper set-theoretical bounds.
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Alright, future engineers!
The **Union** of sets A & B (A U B) is all elements in A *or* B.
Ex: A={1,2}, B={2,3}. A U B = {1,2,3}.
Pro-Tip: Think 'OR' for Union. It includes everything from either set!
#SetTheory #DiscreteMath #STEM #StudyNotes -
Alright, future engineers!
The **Union** of sets A & B (A U B) is all elements in A *or* B.
Ex: A={1,2}, B={2,3}. A U B = {1,2,3}.
Pro-Tip: Think 'OR' for Union. It includes everything from either set!
#SetTheory #DiscreteMath #STEM #StudyNotes -
Take a set, 𝑥. Is it finite or infinite? Well what definition do you use?
① ∃𝑦 ∈ ω₀ 𝑥 ≈ 𝑦 ; There is a natural number, 𝑦, such that you may may map 𝑦 one-to-one onto the set 𝑥, enumerating each of its members. So 𝑥 is finite for the same reason { 1, 2, 3 } is finite.
② ¬ ∃𝑧 ∈ (On ∖ ω₀) 𝑥 ≈ 𝑧 ; There is no such infinite ordinal, 𝑧, such that you may map 𝑧 one-to-one onto the set 𝑥. So 𝑥 is finite because ω₀, the smallest infinite ordinal, cannot be mapped 1-to-1 into it.
Do ① and ② say the same thing?
Obviously, ① implies ② in all cases for if it didn't there would be at least one natural number, 𝑦, which may be used to enumerate at least one infinite ordinal, 𝑧.
But does ② imply ①? If ② doesn't imply ① then there must be sets which can't be be placed side-by-side with any infinite ordinal and yet can't be placed side-by-side with any finite ordinal. In short, there must be sets which can't be well-ordered. But the axiom of choice says all sets may be well-ordered, even if it doesn't provide a recipe.
Not only does the axiom of choice say all sets can be well-ordered and thus ② implies ①, but assuming ② implies ① is equivalent to the axiom of choice. It was invented by Zermelo for this purpose and so that ①, ②, and 6 alternative definitions of "finite set" all mean the same thing.
The axiom of choice is basically saying the border between finite and infinite has nothing trapped in it.
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Take a set, 𝑥. Is it finite or infinite? Well what definition do you use?
① ∃𝑦 ∈ ω₀ 𝑥 ≈ 𝑦 ; There is a natural number, 𝑦, such that you may may map 𝑦 one-to-one onto the set 𝑥, enumerating each of its members. So 𝑥 is finite for the same reason { 1, 2, 3 } is finite.
② ¬ ∃𝑧 ∈ (On ∖ ω₀) 𝑥 ≈ 𝑧 ; There is no such infinite ordinal, 𝑧, such that you may map 𝑧 one-to-one onto the set 𝑥. So 𝑥 is finite because ω₀, the smallest infinite ordinal, cannot be mapped 1-to-1 into it.
Do ① and ② say the same thing?
Obviously, ① implies ② in all cases for if it didn't there would be at least one natural number, 𝑦, which may be used to enumerate at least one infinite ordinal, 𝑧.
But does ② imply ①? If ② doesn't imply ① then there must be sets which can't be be placed side-by-side with any infinite ordinal and yet can't be placed side-by-side with any finite ordinal. In short, there must be sets which can't be well-ordered. But the axiom of choice says all sets may be well-ordered, even if it doesn't provide a recipe.
Not only does the axiom of choice say all sets can be well-ordered and thus ② implies ①, but assuming ② implies ① is equivalent to the axiom of choice. It was invented by Zermelo for this purpose and so that ①, ②, and 6 alternative definitions of "finite set" all mean the same thing.
The axiom of choice is basically saying the border between finite and infinite has nothing trapped in it.
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Take a set, 𝑥. Is it finite or infinite? Well what definition do you use?
① ∃𝑦 ∈ ω₀ 𝑥 ≈ 𝑦 ; There is a natural number, 𝑦, such that you may may map 𝑦 one-to-one onto the set 𝑥, enumerating each of its members. So 𝑥 is finite for the same reason { 1, 2, 3 } is finite.
② ¬ ∃𝑧 ∈ (On ∖ ω₀) 𝑥 ≈ 𝑧 ; There is no such infinite ordinal, 𝑧, such that you may map 𝑧 one-to-one onto the set 𝑥. So 𝑥 is finite because ω₀, the smallest infinite ordinal, cannot be mapped 1-to-1 into it.
Do ① and ② say the same thing?
Obviously, ① implies ② in all cases for if it didn't there would be at least one natural number, 𝑦, which may be used to enumerate at least one infinite ordinal, 𝑧.
But does ② imply ①? If ② doesn't imply ① then there must be sets which can't be be placed side-by-side with any infinite ordinal and yet can't be placed side-by-side with any finite ordinal. In short, there must be sets which can't be well-ordered. But the axiom of choice says all sets may be well-ordered, even if it doesn't provide a recipe.
Not only does the axiom of choice say all sets can be well-ordered and thus ② implies ①, but assuming ② implies ① is equivalent to the axiom of choice. It was invented by Zermelo for this purpose and so that ①, ②, and 6 alternative definitions of "finite set" all mean the same thing.
The axiom of choice is basically saying the border between finite and infinite has nothing trapped in it.
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Take a set, 𝑥. Is it finite or infinite? Well what definition do you use?
① ∃𝑦 ∈ ω₀ 𝑥 ≈ 𝑦 ; There is a natural number, 𝑦, such that you may may map 𝑦 one-to-one onto the set 𝑥, enumerating each of its members. So 𝑥 is finite for the same reason { 1, 2, 3 } is finite.
② ¬ ∃𝑧 ∈ (On ∖ ω₀) 𝑥 ≈ 𝑧 ; There is no such infinite ordinal, 𝑧, such that you may map 𝑧 one-to-one onto the set 𝑥. So 𝑥 is finite because ω₀, the smallest infinite ordinal, cannot be mapped 1-to-1 into it.
Do ① and ② say the same thing?
Obviously, ① implies ② in all cases for if it didn't there would be at least one natural number, 𝑦, which may be used to enumerate at least one infinite ordinal, 𝑧.
But does ② imply ①? If ② doesn't imply ① then there must be sets which can't be be placed side-by-side with any infinite ordinal and yet can't be placed side-by-side with any finite ordinal. In short, there must be sets which can't be well-ordered. But the axiom of choice says all sets may be well-ordered, even if it doesn't provide a recipe.
Not only does the axiom of choice say all sets can be well-ordered and thus ② implies ①, but assuming ② implies ① is equivalent to the axiom of choice. It was invented by Zermelo for this purpose and so that ①, ②, and 6 alternative definitions of "finite set" all mean the same thing.
The axiom of choice is basically saying the border between finite and infinite has nothing trapped in it.
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Take a set, 𝑥. Is it finite or infinite? Well what definition do you use?
① ∃𝑦 ∈ ω₀ 𝑥 ≈ 𝑦 ; There is a natural number, 𝑦, such that you may may map 𝑦 one-to-one onto the set 𝑥, enumerating each of its members. So 𝑥 is finite for the same reason { 1, 2, 3 } is finite.
② ¬ ∃𝑧 ∈ (On ∖ ω₀) 𝑥 ≈ 𝑧 ; There is no such infinite ordinal, 𝑧, such that you may map 𝑧 one-to-one onto the set 𝑥. So 𝑥 is finite because ω₀, the smallest infinite ordinal, cannot be mapped 1-to-1 into it.
Do ① and ② say the same thing?
Obviously, ① implies ② in all cases for if it didn't there would be at least one natural number, 𝑦, which may be used to enumerate at least one infinite ordinal, 𝑧.
But does ② imply ①? If ② doesn't imply ① then there must be sets which can't be be placed side-by-side with any infinite ordinal and yet can't be placed side-by-side with any finite ordinal. In short, there must be sets which can't be well-ordered. But the axiom of choice says all sets may be well-ordered, even if it doesn't provide a recipe.
Not only does the axiom of choice say all sets can be well-ordered and thus ② implies ①, but assuming ② implies ① is equivalent to the axiom of choice. It was invented by Zermelo for this purpose and so that ①, ②, and 6 alternative definitions of "finite set" all mean the same thing.
The axiom of choice is basically saying the border between finite and infinite has nothing trapped in it.
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Alright, future engineers!
A **Cartesian Product (A x B)** is the set of all ordered pairs (a,b) where a∈A & b∈B.
Ex: A={1,2}, B={x,y} => A x B = {(1,x),(1,y),(2,x),(2,y)}.
Pro-Tip: It's how you model all possible pairs across two sets! -
Alright, future engineers!
A **Cartesian Product (A x B)** is the set of all ordered pairs (a,b) where a∈A & b∈B.
Ex: A={1,2}, B={x,y} => A x B = {(1,x),(1,y),(2,x),(2,y)}.
Pro-Tip: It's how you model all possible pairs across two sets! -
Alright, future engineers!
A **Cartesian Product (A x B)** is the set of all ordered pairs (a,b) where a∈A & b∈B.
Ex: A={1,2}, B={x,y} => A x B = {(1,x),(1,y),(2,x),(2,y)}.
Pro-Tip: It's how you model all possible pairs across two sets! -
Alright, future engineers!
A **Cartesian Product (A x B)** is the set of all ordered pairs (a,b) where a∈A & b∈B.
Ex: A={1,2}, B={x,y} => A x B = {(1,x),(1,y),(2,x),(2,y)}.
Pro-Tip: It's how you model all possible pairs across two sets! -
Alright, future engineers!
A **Cartesian Product (A x B)** is the set of all ordered pairs (a,b) where a∈A & b∈B.
Ex: A={1,2}, B={x,y} => A x B = {(1,x),(1,y),(2,x),(2,y)}.
Pro-Tip: It's how you model all possible pairs across two sets! -
Alright, future engineers!
A **Power Set** P(S) is the set of *all* possible subsets of set S. Ex: If `S = {1, 2}`, then `P(S) = {{}, {1}, {2}, {1, 2}}`. Pro-Tip: Its size (cardinality) is always `2^|S|`!
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Alright, future engineers!
A **Power Set** P(S) is the set of *all* possible subsets of set S. Ex: If `S = {1, 2}`, then `P(S) = {{}, {1}, {2}, {1, 2}}`. Pro-Tip: Its size (cardinality) is always `2^|S|`!
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Alright, future engineers!
A **Power Set** P(S) is the set of *all* possible subsets of set S. Ex: If `S = {1, 2}`, then `P(S) = {{}, {1}, {2}, {1, 2}}`. Pro-Tip: Its size (cardinality) is always `2^|S|`!
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Alright, future engineers!
A **Power Set** P(S) is the set of *all* possible subsets of set S. Ex: If `S = {1, 2}`, then `P(S) = {{}, {1}, {2}, {1, 2}}`. Pro-Tip: Its size (cardinality) is always `2^|S|`!
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Alright, future engineers!
A **Power Set** P(S) is the set of *all* possible subsets of set S. Ex: If `S = {1, 2}`, then `P(S) = {{}, {1}, {2}, {1, 2}}`. Pro-Tip: Its size (cardinality) is always `2^|S|`!
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Alright, future engineers! **Cartesian Product (A x B)** creates all ordered pairs (a,b) where 'a' is from set A & 'b' from set B.
Ex: A={1,2}, B={x,y}. A x B = {(1,x), (1,y), (2,x), (2,y)}.
Pro-Tip: Essential for understanding relations and functions – think of it as building a grid of possibilities!
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Alright, future engineers!
The **Power Set** of a set A is the set of *all* possible subsets of A. Ex: For A={1,2}, P(A) = {{}, {1}, {2}, {1,2}}.
Pro-Tip: If a set has 'n' elements, its power set will always have 2^n elements. Handy for combinatorics! -
Set Intersection (A ∩ B) finds elements common to *both* sets A & B. Ex: A={1,2,3}, B={2,3,4} -> A ∩ B = {2,3}. Pro-Tip: Think of it as AND – an element must be in A AND in B to be in the intersection!
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I have the greatest admiration for the theorems and proofs of transfinite set theory, what we've called Cantor's transfinite set theory.
I taught it for years, wrote restatements for my students, and wrote a piece viewing it in the perspective of historical thinking about the infinite.
http://nrs.harvard.edu/urn-3:HUL.InstRepos:3715468Now I've learned that #Cantor deliberately suppressed the role of #Dedekind in some of his work, particularly the proof that the set of real numbers is larger than the sets of natural and rational numbers. This was the first glimpse of the infinite hierarchy of infinite cardinalities.
https://www.quantamagazine.org/the-man-who-stole-infinity-20260225/Cantor had the first germ of the proof, so props for that. But Dedekind helped him clarify it and Cantor published the clarified version without credit to Dedekind. I respect the math as much as ever but am now dealing with a serious case of flawed-hero syndrome.
First, make amends. Kudos to Dedekind. Second, give thanks. Kudos to the sleuths who turned up the empirical evidence of Cantor's #plagiarism -- Emmy Noether, Ivor Grattan-Guinness, José Ferreirós, and (decisively) Demian Goos.
Also thanks to Joseph Howlett for the Quanta article summarizing the evidence -- and in passing for calling Leopold #Kronecker an ideologue. Exactly!
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I have the greatest admiration for the theorems and proofs of transfinite set theory, what we've called Cantor's transfinite set theory.
I taught it for years, wrote restatements for my students, and wrote a piece viewing it in the perspective of historical thinking about the infinite.
http://nrs.harvard.edu/urn-3:HUL.InstRepos:3715468Now I've learned that #Cantor deliberately suppressed the role of #Dedekind in some of his work, particularly the proof that the set of real numbers is larger than the sets of natural and rational numbers. This was the first glimpse of the infinite hierarchy of infinite cardinalities.
https://www.quantamagazine.org/the-man-who-stole-infinity-20260225/Cantor had the first germ of the proof, so props for that. But Dedekind helped him clarify it and Cantor published the clarified version without credit to Dedekind. I respect the math as much as ever but am now dealing with a serious case of flawed-hero syndrome.
First, make amends. Kudos to Dedekind. Second, give thanks. Kudos to the sleuths who turned up the empirical evidence of Cantor's #plagiarism -- Emmy Noether, Ivor Grattan-Guinness, José Ferreirós, and (decisively) Demian Goos.
Also thanks to Joseph Howlett for the Quanta article summarizing the evidence -- and in passing for calling Leopold #Kronecker an ideologue. Exactly!
-
I have the greatest admiration for the theorems and proofs of transfinite set theory, what we've called Cantor's transfinite set theory.
I taught it for years, wrote restatements for my students, and wrote a piece viewing it in the perspective of historical thinking about the infinite.
http://nrs.harvard.edu/urn-3:HUL.InstRepos:3715468Now I've learned that #Cantor deliberately suppressed the role of #Dedekind in some of his work, particularly the proof that the set of real numbers is larger than the sets of natural and rational numbers. This was the first glimpse of the infinite hierarchy of infinite cardinalities.
https://www.quantamagazine.org/the-man-who-stole-infinity-20260225/Cantor had the first germ of the proof, so props for that. But Dedekind helped him clarify it and Cantor published the clarified version without credit to Dedekind. I respect the math as much as ever but am now dealing with a serious case of flawed-hero syndrome.
First, make amends. Kudos to Dedekind. Second, give thanks. Kudos to the sleuths who turned up the empirical evidence of Cantor's #plagiarism -- Emmy Noether, Ivor Grattan-Guinness, José Ferreirós, and (decisively) Demian Goos.
Also thanks to Joseph Howlett for the Quanta article summarizing the evidence -- and in passing for calling Leopold #Kronecker an ideologue. Exactly!
-
I have the greatest admiration for the theorems and proofs of transfinite set theory, what we've called Cantor's transfinite set theory.
I taught it for years, wrote restatements for my students, and wrote a piece viewing it in the perspective of historical thinking about the infinite.
http://nrs.harvard.edu/urn-3:HUL.InstRepos:3715468Now I've learned that #Cantor deliberately suppressed the role of #Dedekind in some of his work, particularly the proof that the set of real numbers is larger than the sets of natural and rational numbers. This was the first glimpse of the infinite hierarchy of infinite cardinalities.
https://www.quantamagazine.org/the-man-who-stole-infinity-20260225/Cantor had the first germ of the proof, so props for that. But Dedekind helped him clarify it and Cantor published the clarified version without credit to Dedekind. I respect the math as much as ever but am now dealing with a serious case of flawed-hero syndrome.
First, make amends. Kudos to Dedekind. Second, give thanks. Kudos to the sleuths who turned up the empirical evidence of Cantor's #plagiarism -- Emmy Noether, Ivor Grattan-Guinness, José Ferreirós, and (decisively) Demian Goos.
Also thanks to Joseph Howlett for the Quanta article summarizing the evidence -- and in passing for calling Leopold #Kronecker an ideologue. Exactly!
-
I have the greatest admiration for the theorems and proofs of transfinite set theory, what we've called Cantor's transfinite set theory.
I taught it for years, wrote restatements for my students, and wrote a piece viewing it in the perspective of historical thinking about the infinite.
http://nrs.harvard.edu/urn-3:HUL.InstRepos:3715468Now I've learned that #Cantor deliberately suppressed the role of #Dedekind in some of his work, particularly the proof that the set of real numbers is larger than the sets of natural and rational numbers. This was the first glimpse of the infinite hierarchy of infinite cardinalities.
https://www.quantamagazine.org/the-man-who-stole-infinity-20260225/Cantor had the first germ of the proof, so props for that. But Dedekind helped him clarify it and Cantor published the clarified version without credit to Dedekind. I respect the math as much as ever but am now dealing with a serious case of flawed-hero syndrome.
First, make amends. Kudos to Dedekind. Second, give thanks. Kudos to the sleuths who turned up the empirical evidence of Cantor's #plagiarism -- Emmy Noether, Ivor Grattan-Guinness, José Ferreirós, and (decisively) Demian Goos.
Also thanks to Joseph Howlett for the Quanta article summarizing the evidence -- and in passing for calling Leopold #Kronecker an ideologue. Exactly!
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@unnick @CenTdemeern1 Another angle to see this would be set theroy; like a normal d6 would conform to the set
{1, 2, 3, 4, 5, 6}while your special die with only one face showing seven would be the set{7}. A true zero-sided set would therefor be the empty set ∅ /{}.Now we define the action of rolling to picking a random entry of the set... however I'm unsure what this means in the case of ∅:
We could argue that randomly picking is reducing the set to just a set by throwing our all other numbers, this would mean that a d6 would be reduced to
{4}(if you roll a 4); in the same manner, rolling ∅ would mean the result still is ∅ since we cant reduce further.On the other hand we can go the normal way and say it's not reducing the set but producing a number, but numbers themselv can be represented using sets by converting the number (i.e. 3) into a series of nested sets build by their previous components:
0 = ∅
1 = {0} = {∅}
2 = {0,1} = {∅, {∅}}
3 = {0,1,2} = {∅, {∅}, {∅, {∅}}}(Called Zermelo-Fraenkel set theory)
... so ultemately I think the right answer would be ∅? (I'm not a mathematician, I just like nerdy stuff and happen to know about set theory bc. of uni, but I'm happy to be proven wrong)
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Every well-ordered set is isomorphic to an ordinal. Common notion, for example see Introduction to https://arxiv.org/abs/2409.07352
⊢((𝐴∈V∧𝑅We𝐴)↔∃𝑓(dom𝑓∈On∧𝑓IsomE,𝑅(dom𝑓,𝐴)))
\[ \vdash ( ( A \in \mathrm{V} \wedge R \mathrm{We} A ) \leftrightarrow \exists f ( \mathrm{dom} f \in \mathrm{On} \wedge f \mathrm{Isom} \mathrm{E} , R ( \mathrm{dom} f , A ) ) ) \]
Every well-ordered set is isomorphic to
a unique ordinal.⊢((𝐴∈V∧𝑅We𝐴)↔∃!𝑜∈On∃𝑓∈(𝐴↑ₘ𝑜)𝑓IsomE,𝑅(𝑜,𝐴))
\[ \vdash ( ( A \in \mathrm{V} \wedge R \mathrm{We} A ) \leftrightarrow \exists{!} o \in \mathrm{On} \exists f \in ( A \uparrow_\mathrm{m} o ) f \mathrm{Isom} \mathrm{E} , R ( o , A ) ) \]
We can phrase the Axiom of Choice as "Every set injects into an ordinal."
⊢(CHOICE↔∀𝑥∃𝑜∈On𝑥≼𝑜)
\[ \vdash ( \mathrm{CHOICE} \leftrightarrow \forall x \exists o \in \mathrm{On} x \preccurlyeq o ) \]
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Every well-ordered set is isomorphic to an ordinal. Common notion, for example see Introduction to https://arxiv.org/abs/2409.07352
⊢((𝐴∈V∧𝑅We𝐴)↔∃𝑓(dom𝑓∈On∧𝑓IsomE,𝑅(dom𝑓,𝐴)))
\[ \vdash ( ( A \in \mathrm{V} \wedge R \mathrm{We} A ) \leftrightarrow \exists f ( \mathrm{dom} f \in \mathrm{On} \wedge f \mathrm{Isom} \mathrm{E} , R ( \mathrm{dom} f , A ) ) ) \]
Every well-ordered set is isomorphic to
a unique ordinal.⊢((𝐴∈V∧𝑅We𝐴)↔∃!𝑜∈On∃𝑓∈(𝐴↑ₘ𝑜)𝑓IsomE,𝑅(𝑜,𝐴))
\[ \vdash ( ( A \in \mathrm{V} \wedge R \mathrm{We} A ) \leftrightarrow \exists{!} o \in \mathrm{On} \exists f \in ( A \uparrow_\mathrm{m} o ) f \mathrm{Isom} \mathrm{E} , R ( o , A ) ) \]
We can phrase the Axiom of Choice as "Every set injects into an ordinal."
⊢(CHOICE↔∀𝑥∃𝑜∈On𝑥≼𝑜)
\[ \vdash ( \mathrm{CHOICE} \leftrightarrow \forall x \exists o \in \mathrm{On} x \preccurlyeq o ) \]
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Every well-ordered set is isomorphic to an ordinal. Common notion, for example see Introduction to https://arxiv.org/abs/2409.07352
⊢((𝐴∈V∧𝑅We𝐴)↔∃𝑓(dom𝑓∈On∧𝑓IsomE,𝑅(dom𝑓,𝐴)))
\[ \vdash ( ( A \in \mathrm{V} \wedge R \mathrm{We} A ) \leftrightarrow \exists f ( \mathrm{dom} f \in \mathrm{On} \wedge f \mathrm{Isom} \mathrm{E} , R ( \mathrm{dom} f , A ) ) ) \]
Every well-ordered set is isomorphic to
a unique ordinal.⊢((𝐴∈V∧𝑅We𝐴)↔∃!𝑜∈On∃𝑓∈(𝐴↑ₘ𝑜)𝑓IsomE,𝑅(𝑜,𝐴))
\[ \vdash ( ( A \in \mathrm{V} \wedge R \mathrm{We} A ) \leftrightarrow \exists{!} o \in \mathrm{On} \exists f \in ( A \uparrow_\mathrm{m} o ) f \mathrm{Isom} \mathrm{E} , R ( o , A ) ) \]
We can phrase the Axiom of Choice as "Every set injects into an ordinal."
⊢(CHOICE↔∀𝑥∃𝑜∈On𝑥≼𝑜)
\[ \vdash ( \mathrm{CHOICE} \leftrightarrow \forall x \exists o \in \mathrm{On} x \preccurlyeq o ) \]
-
Every well-ordered set is isomorphic to an ordinal. Common notion, for example see Introduction to https://arxiv.org/abs/2409.07352
⊢((𝐴∈V∧𝑅We𝐴)↔∃𝑓(dom𝑓∈On∧𝑓IsomE,𝑅(dom𝑓,𝐴)))
\[ \vdash ( ( A \in \mathrm{V} \wedge R \mathrm{We} A ) \leftrightarrow \exists f ( \mathrm{dom} f \in \mathrm{On} \wedge f \mathrm{Isom} \mathrm{E} , R ( \mathrm{dom} f , A ) ) ) \]
Every well-ordered set is isomorphic to
a unique ordinal.⊢((𝐴∈V∧𝑅We𝐴)↔∃!𝑜∈On∃𝑓∈(𝐴↑ₘ𝑜)𝑓IsomE,𝑅(𝑜,𝐴))
\[ \vdash ( ( A \in \mathrm{V} \wedge R \mathrm{We} A ) \leftrightarrow \exists{!} o \in \mathrm{On} \exists f \in ( A \uparrow_\mathrm{m} o ) f \mathrm{Isom} \mathrm{E} , R ( o , A ) ) \]
We can phrase the Axiom of Choice as "Every set injects into an ordinal."
⊢(CHOICE↔∀𝑥∃𝑜∈On𝑥≼𝑜)
\[ \vdash ( \mathrm{CHOICE} \leftrightarrow \forall x \exists o \in \mathrm{On} x \preccurlyeq o ) \]
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@FishFace
Here's the paper from 1958:http://matwbn.icm.edu.pl/ksiazki/fm/fm46/fm4611.pdf
Where reference [7] is
https://eudml.org/doc/213059 where 𝔖 is defined to have ur-ElementsPrior work:
(𝔖 or ZF) + AC ⊢ Fin₁ = Fin₁ₐ = Fin₂ = Fin₃ = Fin₄ = Fin₅ = Fin₆ = Fin₇EScbO = the axiom "Every set can be (linearly, strictly) ordered"
Theorem 1 is
(𝔖 or ZF) ⊢ Fin₁ ⊆ Fin₁ₐ ⊆ Fin₂ ⊆ Fin₃ ⊆ Fin₄ ⊆ Fin₅ ⊆ Fin₆ ⊆ Fin₇Theorem 2 is
(𝔖 or ZF) ⊢ ( R Orders A ∧ A ∈ Fin₂ ) → A ∈ Fin₁Theorem 3 is
(𝔖 or ZF) + EScbO ⊢ Fin₁ = Fin₁ₐ = Fin₂Theorem 4 is
𝔖 + EScbO ⊢ Fin₁ ≠ Fin₃Theorem 5 is
𝔖 + EScbO ⊢ Fin₃ ≠ Fin₄Theorem 6 is
𝔖 + EScbO ⊢ Fin₄ ≠ Fin₅Theorem 7 is
𝔖 + EScbO ⊢ Fin₅ ≠ Fin₆Theorem 8 is
𝔖 + EScbO ⊢ Fin₆ ≠ Fin₇Theorem 9 is
𝔖 ⊢ Fin₁ ≠ Fin₁ₐTheorem 10 is
(𝔖 or ZF) ⊢ ( Fin₁ₐ = Fin₂ → Fin₁ = Fin₁ₐ )Theorem 11 is
𝔖 ⊢ Fin₁ₐ ≠ Fin₂As it so happens, I know of a proof:
ZF + CC ⊢ Fin₁ = Fin₄Where the axiom of Countable Choice, CC, can be expressed as ⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
See https://us.metamath.org/mpeuni/fin41.html where CC is used only for the line ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
So good guess!
And JDH on Substack isn't requiring payments which disadvantages him with respect to SEO, so please share the links. #AxiomOfCountableChoice #AxiomOfChoice #SetTheory #CountableChoice
-
@FishFace
Here's the paper from 1958:http://matwbn.icm.edu.pl/ksiazki/fm/fm46/fm4611.pdf
Where reference [7] is
https://eudml.org/doc/213059 where 𝔖 is defined to have ur-ElementsPrior work:
(𝔖 or ZF) + AC ⊢ Fin₁ = Fin₁ₐ = Fin₂ = Fin₃ = Fin₄ = Fin₅ = Fin₆ = Fin₇EScbO = the axiom "Every set can be (linearly, strictly) ordered"
Theorem 1 is
(𝔖 or ZF) ⊢ Fin₁ ⊆ Fin₁ₐ ⊆ Fin₂ ⊆ Fin₃ ⊆ Fin₄ ⊆ Fin₅ ⊆ Fin₆ ⊆ Fin₇Theorem 2 is
(𝔖 or ZF) ⊢ ( R Orders A ∧ A ∈ Fin₂ ) → A ∈ Fin₁Theorem 3 is
(𝔖 or ZF) + EScbO ⊢ Fin₁ = Fin₁ₐ = Fin₂Theorem 4 is
𝔖 + EScbO ⊢ Fin₁ ≠ Fin₃Theorem 5 is
𝔖 + EScbO ⊢ Fin₃ ≠ Fin₄Theorem 6 is
𝔖 + EScbO ⊢ Fin₄ ≠ Fin₅Theorem 7 is
𝔖 + EScbO ⊢ Fin₅ ≠ Fin₆Theorem 8 is
𝔖 + EScbO ⊢ Fin₆ ≠ Fin₇Theorem 9 is
𝔖 ⊢ Fin₁ ≠ Fin₁ₐTheorem 10 is
(𝔖 or ZF) ⊢ ( Fin₁ₐ = Fin₂ → Fin₁ = Fin₁ₐ )Theorem 11 is
𝔖 ⊢ Fin₁ₐ ≠ Fin₂As it so happens, I know of a proof:
ZF + CC ⊢ Fin₁ = Fin₄Where the axiom of Countable Choice, CC, can be expressed as ⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
See https://us.metamath.org/mpeuni/fin41.html where CC is used only for the line ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
So good guess!
And JDH on Substack isn't requiring payments which disadvantages him with respect to SEO, so please share the links. #AxiomOfCountableChoice #AxiomOfChoice #SetTheory #CountableChoice
-
@FishFace
Here's the paper from 1958:http://matwbn.icm.edu.pl/ksiazki/fm/fm46/fm4611.pdf
Where reference [7] is
https://eudml.org/doc/213059 where 𝔖 is defined to have ur-ElementsPrior work:
(𝔖 or ZF) + AC ⊢ Fin₁ = Fin₁ₐ = Fin₂ = Fin₃ = Fin₄ = Fin₅ = Fin₆ = Fin₇EScbO = the axiom "Every set can be (linearly, strictly) ordered"
Theorem 1 is
(𝔖 or ZF) ⊢ Fin₁ ⊆ Fin₁ₐ ⊆ Fin₂ ⊆ Fin₃ ⊆ Fin₄ ⊆ Fin₅ ⊆ Fin₆ ⊆ Fin₇Theorem 2 is
(𝔖 or ZF) ⊢ ( R Orders A ∧ A ∈ Fin₂ ) → A ∈ Fin₁Theorem 3 is
(𝔖 or ZF) + EScbO ⊢ Fin₁ = Fin₁ₐ = Fin₂Theorem 4 is
𝔖 + EScbO ⊢ Fin₁ ≠ Fin₃Theorem 5 is
𝔖 + EScbO ⊢ Fin₃ ≠ Fin₄Theorem 6 is
𝔖 + EScbO ⊢ Fin₄ ≠ Fin₅Theorem 7 is
𝔖 + EScbO ⊢ Fin₅ ≠ Fin₆Theorem 8 is
𝔖 + EScbO ⊢ Fin₆ ≠ Fin₇Theorem 9 is
𝔖 ⊢ Fin₁ ≠ Fin₁ₐTheorem 10 is
(𝔖 or ZF) ⊢ ( Fin₁ₐ = Fin₂ → Fin₁ = Fin₁ₐ )Theorem 11 is
𝔖 ⊢ Fin₁ₐ ≠ Fin₂As it so happens, I know of a proof:
ZF + CC ⊢ Fin₁ = Fin₄Where the axiom of Countable Choice, CC, can be expressed as ⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
See https://us.metamath.org/mpeuni/fin41.html where CC is used only for the line ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
So good guess!
And JDH on Substack isn't requiring payments which disadvantages him with respect to SEO, so please share the links. #AxiomOfCountableChoice #AxiomOfChoice #SetTheory #CountableChoice
-
@FishFace
Here's the paper from 1958:http://matwbn.icm.edu.pl/ksiazki/fm/fm46/fm4611.pdf
Where reference [7] is
https://eudml.org/doc/213059 where 𝔖 is defined to have ur-ElementsPrior work:
(𝔖 or ZF) + AC ⊢ Fin₁ = Fin₁ₐ = Fin₂ = Fin₃ = Fin₄ = Fin₅ = Fin₆ = Fin₇EScbO = the axiom "Every set can be (linearly, strictly) ordered"
Theorem 1 is
(𝔖 or ZF) ⊢ Fin₁ ⊆ Fin₁ₐ ⊆ Fin₂ ⊆ Fin₃ ⊆ Fin₄ ⊆ Fin₅ ⊆ Fin₆ ⊆ Fin₇Theorem 2 is
(𝔖 or ZF) ⊢ ( R Orders A ∧ A ∈ Fin₂ ) → A ∈ Fin₁Theorem 3 is
(𝔖 or ZF) + EScbO ⊢ Fin₁ = Fin₁ₐ = Fin₂Theorem 4 is
𝔖 + EScbO ⊢ Fin₁ ≠ Fin₃Theorem 5 is
𝔖 + EScbO ⊢ Fin₃ ≠ Fin₄Theorem 6 is
𝔖 + EScbO ⊢ Fin₄ ≠ Fin₅Theorem 7 is
𝔖 + EScbO ⊢ Fin₅ ≠ Fin₆Theorem 8 is
𝔖 + EScbO ⊢ Fin₆ ≠ Fin₇Theorem 9 is
𝔖 ⊢ Fin₁ ≠ Fin₁ₐTheorem 10 is
(𝔖 or ZF) ⊢ ( Fin₁ₐ = Fin₂ → Fin₁ = Fin₁ₐ )Theorem 11 is
𝔖 ⊢ Fin₁ₐ ≠ Fin₂As it so happens, I know of a proof:
ZF + CC ⊢ Fin₁ = Fin₄Where the axiom of Countable Choice, CC, can be expressed as ⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
See https://us.metamath.org/mpeuni/fin41.html where CC is used only for the line ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)
So good guess!
And JDH on Substack isn't requiring payments which disadvantages him with respect to SEO, so please share the links. #AxiomOfCountableChoice #AxiomOfChoice #SetTheory #CountableChoice
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So we have { 1, 2 } ∈ Fin₁ ⊆ Fin₁ₐ ⊆ Fin₂ ⊆ Fin₃ ⊆ Fin₄ ⊆ Fin₅ ⊆ Fin₆ ⊆ Fin₇
If a set is in Fin₁ then it is considered finite by all the other definitions.
But since the axiom of choice is equivalent to saying every set can be well-ordered, if we accept it VII-finite sets are equinumerous with a finite ordinal and so Fin₇ ⊆ Fin₁ and so the differences between these definitions collapse and ZF becomes ZFC, which is a widely accepted basis for Set Theory.
My consultation with math resources was inspired by a blog post:
https://www.infinitelymore.xyz/p/what-is-the-infinite
#Metamath #ZFC #SetTheory #AxiomOfChoice #FiniteSet #Infinity