#metamath — Public Fediverse posts

Still working on ways to do and illustrate #CondensedDetachment but I have taken a detour in performance enhancing my Rust port of the #Metamath proof assistance and theorem verification tool.

Still working on ways to do and illustrate #CondensedDetachment but I have taken a detour in performance enhancing my Rust port of the #Metamath proof assistance and theorem verification tool.

Still working on ways to do and illustrate #CondensedDetachment but I have taken a detour in performance enhancing my Rust port of the #Metamath proof assistance and theorem verification tool.

Still working on ways to do and illustrate #CondensedDetachment but I have taken a detour in performance enhancing my Rust port of the #Metamath proof assistance and theorem verification tool.

Still working on ways to do and illustrate #CondensedDetachment but I have taken a detour in performance enhancing my Rust port of the #Metamath proof assistance and theorem verification tool.

Since 2026/05/11 I have been massaging the output of #c2rust run on the #metamath source code into increasingly idiomatic #Rust. Right now verification of large databases of theorems runs in about 80% of the time of the C binary but I'm haven't yet converted all the IO away from stdio routines which want NUL-terminated CStrings (and the super unsafe *const u8). I've stripped out the idiosyncratic private memory management in favor of pre-allocating vectors and string buffers which is an improvement in performance and maintaining my sanity.

Since 2026/05/11 I have been massaging the output of #c2rust run on the #metamath source code into increasingly idiomatic #Rust. Right now verification of large databases of theorems runs in about 80% of the time of the C binary but I'm haven't yet converted all the IO away from stdio routines which want NUL-terminated CStrings (and the super unsafe *const u8). I've stripped out the idiosyncratic private memory management in favor of pre-allocating vectors and string buffers which is an improvement in performance and maintaining my sanity.

Since 2026/05/11 I have been massaging the output of #c2rust run on the #metamath source code into increasingly idiomatic #Rust. Right now verification of large databases of theorems runs in about 80% of the time of the C binary but I'm haven't yet converted all the IO away from stdio routines which want NUL-terminated CStrings (and the super unsafe *const u8). I've stripped out the idiosyncratic private memory management in favor of pre-allocating vectors and string buffers which is an improvement in performance and maintaining my sanity.

Since 2026/05/11 I have been massaging the output of #c2rust run on the #metamath source code into increasingly idiomatic #Rust. Right now verification of large databases of theorems runs in about 80% of the time of the C binary but I'm haven't yet converted all the IO away from stdio routines which want NUL-terminated CStrings (and the super unsafe *const u8). I've stripped out the idiosyncratic private memory management in favor of pre-allocating vectors and string buffers which is an improvement in performance and maintaining my sanity.

Since 2026/05/11 I have been massaging the output of #c2rust run on the #metamath source code into increasingly idiomatic #Rust. Right now verification of large databases of theorems runs in about 80% of the time of the C binary but I'm haven't yet converted all the IO away from stdio routines which want NUL-terminated CStrings (and the super unsafe *const u8). I've stripped out the idiosyncratic private memory management in favor of pre-allocating vectors and string buffers which is an improvement in performance and maintaining my sanity.

Notation guide (adapted from Metamath):

• 𝜑, a metavariable standing for any logical formula, abbreviates the conjunction of all hypotheses listed for a given proposition; writing each line as ⊢ (𝜑 → …) puts the theorem in "deduction form," which can be easier to apply in #Metamath.

• ⊢ 𝜑 asserts that 𝜑 is true; the turnstile is descended from #Frege's own Urteilsstrich (judgment stroke).

• 𝐴𝑅𝐵 means the ordered pair ⟨𝐴, 𝐵⟩ is an element of 𝑅, or we could say 𝐵 immediately follows 𝐴

• 𝐵 = (𝑅‘𝐴) means 𝐵 is the unique set such that 𝐴𝑅𝐵 is true (when such a 𝐵 exists) which means 𝑅 is function-like when restricted to operating on the singleton {𝐴}

• (𝑅”𝐴) is the image of 𝐴

• dom 𝑅 is the domain of 𝑅, the class of all sets 𝑥 such that there is a set 𝑦 that would make 𝑥𝑅𝑦 true.

• Fun 𝑅 is true when 𝑅 is function-like for all sets in its domain.

• ◡𝑅 is the converse of 𝑅 so 𝐴◡𝑅𝐵 iff 𝐵𝑅𝐴

• (tc‘𝑅) is the #TransitiveClosure of 𝑅 (Metamath uses (t+‘𝑅) which can be awkward.) Whitehead and Russell use the term ancestral to describe how 𝐴(tc‘𝑅)𝐵 means 𝐴 is some “ancestor” of 𝐵. Alternately, we can say 𝐵 eventually follows 𝐴.

• V is the universal class, every set is a member, and only sets may be members of any class. After Frege’s later work ran into Russell’s Paradox, it was discovered that not every class {𝑥 | 𝜑} makes sense as a set and so we need the hypothesis ⊢ (𝜑 → 𝐴 ∈ V) before we can talk about the function value of 𝐴 or the ordered pair ⟨𝐴, 𝐵⟩ being an element of 𝑅. V is not italic because it is a constant symbol, like tc, dom, and Fun.

Notation guide (adapted from Metamath):

• 𝜑, a metavariable standing for any logical formula, abbreviates the conjunction of all hypotheses listed for a given proposition; writing each line as ⊢ (𝜑 → …) puts the theorem in "deduction form," which can be easier to apply in #Metamath.

• ⊢ 𝜑 asserts that 𝜑 is true; the turnstile is descended from #Frege's own Urteilsstrich (judgment stroke).

• 𝐴𝑅𝐵 means the ordered pair ⟨𝐴, 𝐵⟩ is an element of 𝑅, or we could say 𝐵 immediately follows 𝐴

• 𝐵 = (𝑅‘𝐴) means 𝐵 is the unique set such that 𝐴𝑅𝐵 is true (when such a 𝐵 exists) which means 𝑅 is function-like when restricted to operating on the singleton {𝐴}

• (𝑅”𝐴) is the image of 𝐴

• dom 𝑅 is the domain of 𝑅, the class of all sets 𝑥 such that there is a set 𝑦 that would make 𝑥𝑅𝑦 true.

• Fun 𝑅 is true when 𝑅 is function-like for all sets in its domain.

• ◡𝑅 is the converse of 𝑅 so 𝐴◡𝑅𝐵 iff 𝐵𝑅𝐴

• (tc‘𝑅) is the #TransitiveClosure of 𝑅 (Metamath uses (t+‘𝑅) which can be awkward.) Whitehead and Russell use the term ancestral to describe how 𝐴(tc‘𝑅)𝐵 means 𝐴 is some “ancestor” of 𝐵. Alternately, we can say 𝐵 eventually follows 𝐴.

• V is the universal class, every set is a member, and only sets may be members of any class. After Frege’s later work ran into Russell’s Paradox, it was discovered that not every class {𝑥 | 𝜑} makes sense as a set and so we need the hypothesis ⊢ (𝜑 → 𝐴 ∈ V) before we can talk about the function value of 𝐴 or the ordered pair ⟨𝐴, 𝐵⟩ being an element of 𝑅. V is not italic because it is a constant symbol, like tc, dom, and Fun.

Notation guide (adapted from Metamath):

• 𝜑, a metavariable standing for any logical formula, abbreviates the conjunction of all hypotheses listed for a given proposition; writing each line as ⊢ (𝜑 → …) puts the theorem in "deduction form," which can be easier to apply in #Metamath.

• ⊢ 𝜑 asserts that 𝜑 is true; the turnstile is descended from #Frege's own Urteilsstrich (judgment stroke).

• 𝐴𝑅𝐵 means the ordered pair ⟨𝐴, 𝐵⟩ is an element of 𝑅, or we could say 𝐵 immediately follows 𝐴

• 𝐵 = (𝑅‘𝐴) means 𝐵 is the unique set such that 𝐴𝑅𝐵 is true (when such a 𝐵 exists) which means 𝑅 is function-like when restricted to operating on the singleton {𝐴}

• (𝑅”𝐴) is the image of 𝐴

• dom 𝑅 is the domain of 𝑅, the class of all sets 𝑥 such that there is a set 𝑦 that would make 𝑥𝑅𝑦 true.

• Fun 𝑅 is true when 𝑅 is function-like for all sets in its domain.

• ◡𝑅 is the converse of 𝑅 so 𝐴◡𝑅𝐵 iff 𝐵𝑅𝐴

• (tc‘𝑅) is the #TransitiveClosure of 𝑅 (Metamath uses (t+‘𝑅) which can be awkward.) Whitehead and Russell use the term ancestral to describe how 𝐴(tc‘𝑅)𝐵 means 𝐴 is some “ancestor” of 𝐵. Alternately, we can say 𝐵 eventually follows 𝐴.

• V is the universal class, every set is a member, and only sets may be members of any class. After Frege’s later work ran into Russell’s Paradox, it was discovered that not every class {𝑥 | 𝜑} makes sense as a set and so we need the hypothesis ⊢ (𝜑 → 𝐴 ∈ V) before we can talk about the function value of 𝐴 or the ordered pair ⟨𝐴, 𝐵⟩ being an element of 𝑅. V is not italic because it is a constant symbol, like tc, dom, and Fun.

Notation guide (adapted from Metamath):

• 𝜑, a metavariable standing for any logical formula, abbreviates the conjunction of all hypotheses listed for a given proposition; writing each line as ⊢ (𝜑 → …) puts the theorem in "deduction form," which can be easier to apply in #Metamath.

• ⊢ 𝜑 asserts that 𝜑 is true; the turnstile is descended from #Frege's own Urteilsstrich (judgment stroke).

• 𝐴𝑅𝐵 means the ordered pair ⟨𝐴, 𝐵⟩ is an element of 𝑅, or we could say 𝐵 immediately follows 𝐴

• 𝐵 = (𝑅‘𝐴) means 𝐵 is the unique set such that 𝐴𝑅𝐵 is true (when such a 𝐵 exists) which means 𝑅 is function-like when restricted to operating on the singleton {𝐴}

• (𝑅”𝐴) is the image of 𝐴

• dom 𝑅 is the domain of 𝑅, the class of all sets 𝑥 such that there is a set 𝑦 that would make 𝑥𝑅𝑦 true.

• Fun 𝑅 is true when 𝑅 is function-like for all sets in its domain.

• ◡𝑅 is the converse of 𝑅 so 𝐴◡𝑅𝐵 iff 𝐵𝑅𝐴

• (tc‘𝑅) is the #TransitiveClosure of 𝑅 (Metamath uses (t+‘𝑅) which can be awkward.) Whitehead and Russell use the term ancestral to describe how 𝐴(tc‘𝑅)𝐵 means 𝐴 is some “ancestor” of 𝐵. Alternately, we can say 𝐵 eventually follows 𝐴.

• V is the universal class, every set is a member, and only sets may be members of any class. After Frege’s later work ran into Russell’s Paradox, it was discovered that not every class {𝑥 | 𝜑} makes sense as a set and so we need the hypothesis ⊢ (𝜑 → 𝐴 ∈ V) before we can talk about the function value of 𝐴 or the ordered pair ⟨𝐴, 𝐵⟩ being an element of 𝑅. V is not italic because it is a constant symbol, like tc, dom, and Fun.

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.

#SetTheory #Logic #Metamath

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.

#SetTheory #Logic #Metamath

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.

#SetTheory #Logic #Metamath

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.

#SetTheory #Logic #Metamath

So, just double-checking how #Lean4 and #Mathlib work:

* Lean takes 3GiB of RAM and a minute to open Mathlib
* Lean requires about 10min to build itself in CI, only verifying required theorems
* Verifying all of Mathlib is measured in hours
* Lean's kernel is untrustworthy due to junk theorems

And yet I'm a clown for using #Metamath? At some point we ought to reconsider the type-theory fetish.

So, just double-checking how #Lean4 and #Mathlib work:

* Lean takes 3GiB of RAM and a minute to open Mathlib
* Lean requires about 10min to build itself in CI, only verifying required theorems
* Verifying all of Mathlib is measured in hours
* Lean's kernel is untrustworthy due to junk theorems

And yet I'm a clown for using #Metamath? At some point we ought to reconsider the type-theory fetish.

So, just double-checking how #Lean4 and #Mathlib work:

* Lean takes 3GiB of RAM and a minute to open Mathlib
* Lean requires about 10min to build itself in CI, only verifying required theorems
* Verifying all of Mathlib is measured in hours
* Lean's kernel is untrustworthy due to junk theorems

And yet I'm a clown for using #Metamath? At some point we ought to reconsider the type-theory fetish.

So, just double-checking how #Lean4 and #Mathlib work:

* Lean takes 3GiB of RAM and a minute to open Mathlib
* Lean requires about 10min to build itself in CI, only verifying required theorems
* Verifying all of Mathlib is measured in hours
* Lean's kernel is untrustworthy due to junk theorems

And yet I'm a clown for using #Metamath? At some point we ought to reconsider the type-theory fetish.

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)

\[ \vdash ( \mathrm{CHOICE} \leftrightarrow \forall x \exists o \in \mathrm{On} x \preccurlyeq o ) \]

#math #metamath #SetTheory #WellOrdering #OrdinalNumbers

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)

\[ \vdash ( \mathrm{CHOICE} \leftrightarrow \forall x \exists o \in \mathrm{On} x \preccurlyeq o ) \]

#math #metamath #SetTheory #WellOrdering #OrdinalNumbers

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)

\[ \vdash ( \mathrm{CHOICE} \leftrightarrow \forall x \exists o \in \mathrm{On} x \preccurlyeq o ) \]

#math #metamath #SetTheory #WellOrdering #OrdinalNumbers

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)

\[ \vdash ( \mathrm{CHOICE} \leftrightarrow \forall x \exists o \in \mathrm{On} x \preccurlyeq o ) \]

#math #metamath #SetTheory #WellOrdering #OrdinalNumbers

@FishFace Update:

What I refer to as EScbO is usually called #LO the #LinearOrdering principle.

⊢ ∃y ∀a ∈ x ∀b ∈ x ∀c ∈ x (¬ aya ∧ ((ayb ∧ byc) → ayc) ∧ (ayb ∨ a = b ∨ bya))
or using abbreviations in #Metamath
⊢ ∃y y Or x

I think AC implies LO, but LO is independent of DC and CC.

@FishFace Update:

What I refer to as EScbO is usually called #LO the #LinearOrdering principle.

⊢ ∃y ∀a ∈ x ∀b ∈ x ∀c ∈ x (¬ aya ∧ ((ayb ∧ byc) → ayc) ∧ (ayb ∨ a = b ∨ bya))
or using abbreviations in #Metamath
⊢ ∃y y Or x

I think AC implies LO, but LO is independent of DC and CC.

@FishFace Update:

What I refer to as EScbO is usually called #LO the #LinearOrdering principle.

⊢ ∃y ∀a ∈ x ∀b ∈ x ∀c ∈ x (¬ aya ∧ ((ayb ∧ byc) → ayc) ∧ (ayb ∨ a = b ∨ bya))
or using abbreviations in #Metamath
⊢ ∃y y Or x

I think AC implies LO, but LO is independent of DC and CC.

@FishFace Update:

What I refer to as EScbO is usually called #LO the #LinearOrdering principle.

⊢ ∃y ∀a ∈ x ∀b ∈ x ∀c ∈ x (¬ aya ∧ ((ayb ∧ byc) → ayc) ∧ (ayb ∨ a = b ∨ bya))
or using abbreviations in #Metamath
⊢ ∃y y Or x

I think AC implies LO, but LO is independent of DC and CC.

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

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

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

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

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

Also from #Metamath I learned #infinity is hard to think about.

A . Lévy in "The independence of various definitions of finiteness" Fundamenta Mathematicae, 46:1-13 (1958) established 8 distinct set-theoretic definitions of a #finiteSet which in ZF cannot be equated without the #AxiomOfChoice

I-finite -- equinumerous with a finite ordinal. // i.e. admits a finite well-order (Numerically Finite) ⟺ the powerset of its powerset is Dedekind finite ⟺ every collection of its subsets has a maximum element ⟺ every collection of subsets has a minimal element

Ia-finite -- not the union of two sets which are not I-finite

II-finite -- every possible way of finding within the set a chain of nested subsets always contains a maximum element (Tarski finite) ⟺ equivalently every such chain contains its intersection ⟺ (Linearly Finite) ⟺ (Stäckel Finite)

|||-finite -- It's powerset is |V-finite finite, (weakly Dedekind finite) ⟺ cannot be mapped onto ordinal ω ⟺ doesn't contain a chain of subsets which can be placed in order with ω

|V-finite -- doesn't have a proper subset which is equinumerous to itself. (Dedekind finite) ⟺ there is no 1-1 map from ordinal ω to it ⟺ it is strictly dominated by the disjoint sum of it and a singleton (acts finite under successor)

V-finite -- it is either empty or strictly dominated by the disjoint sum of it with itself (acts finite under addition)

VI-finite -- it is either empty, a singleton or strictly dominated by the Cartesian product of it with itself (acts finite under multiplication)

VII-finite -- it cannot be infinitely well-ordered (not equiinumerous with the ordinal ω or any larger ordinal)

Also from #Metamath I learned #infinity is hard to think about.

A . Lévy in "The independence of various definitions of finiteness" Fundamenta Mathematicae, 46:1-13 (1958) established 8 distinct set-theoretic definitions of a #finiteSet which in ZF cannot be equated without the #AxiomOfChoice

I-finite -- equinumerous with a finite ordinal. // i.e. admits a finite well-order (Numerically Finite) ⟺ the powerset of its powerset is Dedekind finite ⟺ every collection of its subsets has a maximum element ⟺ every collection of subsets has a minimal element

Ia-finite -- not the union of two sets which are not I-finite

II-finite -- every possible way of finding within the set a chain of nested subsets always contains a maximum element (Tarski finite) ⟺ equivalently every such chain contains its intersection ⟺ (Linearly Finite) ⟺ (Stäckel Finite)

|||-finite -- It's powerset is |V-finite finite, (weakly Dedekind finite) ⟺ cannot be mapped onto ordinal ω ⟺ doesn't contain a chain of subsets which can be placed in order with ω

|V-finite -- doesn't have a proper subset which is equinumerous to itself. (Dedekind finite) ⟺ there is no 1-1 map from ordinal ω to it ⟺ it is strictly dominated by the disjoint sum of it and a singleton (acts finite under successor)

V-finite -- it is either empty or strictly dominated by the disjoint sum of it with itself (acts finite under addition)

VI-finite -- it is either empty, a singleton or strictly dominated by the Cartesian product of it with itself (acts finite under multiplication)

VII-finite -- it cannot be infinitely well-ordered (not equiinumerous with the ordinal ω or any larger ordinal)

Also from #Metamath I learned #infinity is hard to think about.

A . Lévy in "The independence of various definitions of finiteness" Fundamenta Mathematicae, 46:1-13 (1958) established 8 distinct set-theoretic definitions of a #finiteSet which in ZF cannot be equated without the #AxiomOfChoice

I-finite -- equinumerous with a finite ordinal. // i.e. admits a finite well-order (Numerically Finite) ⟺ the powerset of its powerset is Dedekind finite ⟺ every collection of its subsets has a maximum element ⟺ every collection of subsets has a minimal element

Ia-finite -- not the union of two sets which are not I-finite

II-finite -- every possible way of finding within the set a chain of nested subsets always contains a maximum element (Tarski finite) ⟺ equivalently every such chain contains its intersection ⟺ (Linearly Finite) ⟺ (Stäckel Finite)

|||-finite -- It's powerset is |V-finite finite, (weakly Dedekind finite) ⟺ cannot be mapped onto ordinal ω ⟺ doesn't contain a chain of subsets which can be placed in order with ω

|V-finite -- doesn't have a proper subset which is equinumerous to itself. (Dedekind finite) ⟺ there is no 1-1 map from ordinal ω to it ⟺ it is strictly dominated by the disjoint sum of it and a singleton (acts finite under successor)

V-finite -- it is either empty or strictly dominated by the disjoint sum of it with itself (acts finite under addition)

VI-finite -- it is either empty, a singleton or strictly dominated by the Cartesian product of it with itself (acts finite under multiplication)

VII-finite -- it cannot be infinitely well-ordered (not equiinumerous with the ordinal ω or any larger ordinal)

Also from #Metamath I learned #infinity is hard to think about.

A . Lévy in "The independence of various definitions of finiteness" Fundamenta Mathematicae, 46:1-13 (1958) established 8 distinct set-theoretic definitions of a #finiteSet which in ZF cannot be equated without the #AxiomOfChoice

I-finite -- equinumerous with a finite ordinal. // i.e. admits a finite well-order (Numerically Finite) ⟺ the powerset of its powerset is Dedekind finite ⟺ every collection of its subsets has a maximum element ⟺ every collection of subsets has a minimal element

Ia-finite -- not the union of two sets which are not I-finite

II-finite -- every possible way of finding within the set a chain of nested subsets always contains a maximum element (Tarski finite) ⟺ equivalently every such chain contains its intersection ⟺ (Linearly Finite) ⟺ (Stäckel Finite)

|||-finite -- It's powerset is |V-finite finite, (weakly Dedekind finite) ⟺ cannot be mapped onto ordinal ω ⟺ doesn't contain a chain of subsets which can be placed in order with ω

|V-finite -- doesn't have a proper subset which is equinumerous to itself. (Dedekind finite) ⟺ there is no 1-1 map from ordinal ω to it ⟺ it is strictly dominated by the disjoint sum of it and a singleton (acts finite under successor)

V-finite -- it is either empty or strictly dominated by the disjoint sum of it with itself (acts finite under addition)

VI-finite -- it is either empty, a singleton or strictly dominated by the Cartesian product of it with itself (acts finite under multiplication)

VII-finite -- it cannot be infinitely well-ordered (not equiinumerous with the ordinal ω or any larger ordinal)

Also from #Metamath I learned #infinity is hard to think about.

A . Lévy in "The independence of various definitions of finiteness" Fundamenta Mathematicae, 46:1-13 (1958) established 8 distinct set-theoretic definitions of a #finiteSet which in ZF cannot be equated without the #AxiomOfChoice

I-finite -- equinumerous with a finite ordinal. // i.e. admits a finite well-order (Numerically Finite) ⟺ the powerset of its powerset is Dedekind finite ⟺ every collection of its subsets has a maximum element ⟺ every collection of subsets has a minimal element

Ia-finite -- not the union of two sets which are not I-finite

II-finite -- every possible way of finding within the set a chain of nested subsets always contains a maximum element (Tarski finite) ⟺ equivalently every such chain contains its intersection ⟺ (Linearly Finite) ⟺ (Stäckel Finite)

|||-finite -- It's powerset is |V-finite finite, (weakly Dedekind finite) ⟺ cannot be mapped onto ordinal ω ⟺ doesn't contain a chain of subsets which can be placed in order with ω

|V-finite -- doesn't have a proper subset which is equinumerous to itself. (Dedekind finite) ⟺ there is no 1-1 map from ordinal ω to it ⟺ it is strictly dominated by the disjoint sum of it and a singleton (acts finite under successor)

V-finite -- it is either empty or strictly dominated by the disjoint sum of it with itself (acts finite under addition)

VI-finite -- it is either empty, a singleton or strictly dominated by the Cartesian product of it with itself (acts finite under multiplication)

VII-finite -- it cannot be infinitely well-ordered (not equiinumerous with the ordinal ω or any larger ordinal)

Keeping busy with Sudoku, Wordle, Crosswords, and mining textbooks for statements to practice proving in #Metamath ...

Inspired by this video playlist on #RealAnalysis #Math

https://www.youtube.com/playlist?list=PLYPU-RArkLZNwMZ55-y8FZZEeY7zCJX59

Which was created from https://www.jirka.org/ra/ _Basic Analysis:
Introduction to Real Analysis_ by Jiří Lebl. #JiříLebl a #CreativeCommons (4.0) free #math #textbook

In Metamath's compressed format, my proofs of strong induction over the natural numbers took 316 bytes, well-ordering of the natural numbers: 376 bytes, 2ⁿ⁻¹ ≤ n! : 1204 bytes, formula for finite sum of geometric series: 2566 bytes. The last has 147 steps, some of which are reused and some of which depend on up to 8 prior steps and on a truncated library of 30477 statements of syntax, axioms, definitions, and theorems from the wider world of Metamath.

Keeping busy with Sudoku, Wordle, Crosswords, and mining textbooks for statements to practice proving in #Metamath ...

Inspired by this video playlist on #RealAnalysis #Math

https://www.youtube.com/playlist?list=PLYPU-RArkLZNwMZ55-y8FZZEeY7zCJX59

Which was created from https://www.jirka.org/ra/ _Basic Analysis:
Introduction to Real Analysis_ by Jiří Lebl. #JiříLebl a #CreativeCommons (4.0) free #math #textbook

In Metamath's compressed format, my proofs of strong induction over the natural numbers took 316 bytes, well-ordering of the natural numbers: 376 bytes, 2ⁿ⁻¹ ≤ n! : 1204 bytes, formula for finite sum of geometric series: 2566 bytes. The last has 147 steps, some of which are reused and some of which depend on up to 8 prior steps and on a truncated library of 30477 statements of syntax, axioms, definitions, and theorems from the wider world of Metamath.

Keeping busy with Sudoku, Wordle, Crosswords, and mining textbooks for statements to practice proving in #Metamath ...

Inspired by this video playlist on #RealAnalysis #Math

https://www.youtube.com/playlist?list=PLYPU-RArkLZNwMZ55-y8FZZEeY7zCJX59

Which was created from https://www.jirka.org/ra/ _Basic Analysis:
Introduction to Real Analysis_ by Jiří Lebl. #JiříLebl a #CreativeCommons (4.0) free #math #textbook

In Metamath's compressed format, my proofs of strong induction over the natural numbers took 316 bytes, well-ordering of the natural numbers: 376 bytes, 2ⁿ⁻¹ ≤ n! : 1204 bytes, formula for finite sum of geometric series: 2566 bytes. The last has 147 steps, some of which are reused and some of which depend on up to 8 prior steps and on a truncated library of 30477 statements of syntax, axioms, definitions, and theorems from the wider world of Metamath.

Keeping busy with Sudoku, Wordle, Crosswords, and mining textbooks for statements to practice proving in #Metamath ...

Inspired by this video playlist on #RealAnalysis #Math

https://www.youtube.com/playlist?list=PLYPU-RArkLZNwMZ55-y8FZZEeY7zCJX59

Which was created from https://www.jirka.org/ra/ _Basic Analysis:
Introduction to Real Analysis_ by Jiří Lebl. #JiříLebl a #CreativeCommons (4.0) free #math #textbook

In Metamath's compressed format, my proofs of strong induction over the natural numbers took 316 bytes, well-ordering of the natural numbers: 376 bytes, 2ⁿ⁻¹ ≤ n! : 1204 bytes, formula for finite sum of geometric series: 2566 bytes. The last has 147 steps, some of which are reused and some of which depend on up to 8 prior steps and on a truncated library of 30477 statements of syntax, axioms, definitions, and theorems from the wider world of Metamath.

Keeping busy with Sudoku, Wordle, Crosswords, and mining textbooks for statements to practice proving in #Metamath ...

Inspired by this video playlist on #RealAnalysis #Math

https://www.youtube.com/playlist?list=PLYPU-RArkLZNwMZ55-y8FZZEeY7zCJX59

Which was created from https://www.jirka.org/ra/ _Basic Analysis:
Introduction to Real Analysis_ by Jiří Lebl. #JiříLebl a #CreativeCommons (4.0) free #math #textbook

In Metamath's compressed format, my proofs of strong induction over the natural numbers took 316 bytes, well-ordering of the natural numbers: 376 bytes, 2ⁿ⁻¹ ≤ n! : 1204 bytes, formula for finite sum of geometric series: 2566 bytes. The last has 147 steps, some of which are reused and some of which depend on up to 8 prior steps and on a truncated library of 30477 statements of syntax, axioms, definitions, and theorems from the wider world of Metamath.

Readings shared October 29, 2025. https://jaalonso.github.io/vestigium/posts/2025/10/30-readings_shared_10-29-25 #AI #CompSci #FunctionalProgramming #GenAI #HOL_Light #Haskell #ITP #IsabelleHOL #LeanProver #Logic #Math #Metamath #Mizar #Programming #Python #Rocq

Readings shared October 29, 2025. https://jaalonso.github.io/vestigium/posts/2025/10/30-readings_shared_10-29-25 #AI #CompSci #FunctionalProgramming #GenAI #HOL_Light #Haskell #ITP #IsabelleHOL #LeanProver #Logic #Math #Metamath #Mizar #Programming #Python #Rocq

Readings shared October 29, 2025. https://jaalonso.github.io/vestigium/posts/2025/10/30-readings_shared_10-29-25 #AI #CompSci #FunctionalProgramming #GenAI #HOL_Light #Haskell #ITP #IsabelleHOL #LeanProver #Logic #Math #Metamath #Mizar #Programming #Python #Rocq

Readings shared October 29, 2025. https://jaalonso.github.io/vestigium/posts/2025/10/30-readings_shared_10-29-25 #AI #CompSci #FunctionalProgramming #GenAI #HOL_Light #Haskell #ITP #IsabelleHOL #LeanProver #Logic #Math #Metamath #Mizar #Programming #Python #Rocq

Readings shared October 29, 2025. https://jaalonso.github.io/vestigium/posts/2025/10/30-readings_shared_10-29-25 #AI #CompSci #FunctionalProgramming #GenAI #HOL_Light #Haskell #ITP #IsabelleHOL #LeanProver #Logic #Math #Metamath #Mizar #Programming #Python #Rocq

1000+ theorems (The spiritual successor of Freek’s list of 100 theorems. Now with more than 1000 theorems!). ~ Katja Berčič et als. https://1000-plus.github.io/all #Math #ITP #IsabelleHOL #HOL_Light #Rocq #LeanProver #Metamath #Mizar

1000+ theorems (The spiritual successor of Freek’s list of 100 theorems. Now with more than 1000 theorems!). ~ Katja Berčič et als. https://1000-plus.github.io/all #Math #ITP #IsabelleHOL #HOL_Light #Rocq #LeanProver #Metamath #Mizar

1000+ theorems (The spiritual successor of Freek’s list of 100 theorems. Now with more than 1000 theorems!). ~ Katja Berčič et als. https://1000-plus.github.io/all #Math #ITP #IsabelleHOL #HOL_Light #Rocq #LeanProver #Metamath #Mizar

1000+ theorems (The spiritual successor of Freek’s list of 100 theorems. Now with more than 1000 theorems!). ~ Katja Berčič et als. https://1000-plus.github.io/all #Math #ITP #IsabelleHOL #HOL_Light #Rocq #LeanProver #Metamath #Mizar

1000+ theorems (The spiritual successor of Freek’s list of 100 theorems. Now with more than 1000 theorems!). ~ Katja Berčič et als. https://1000-plus.github.io/all #Math #ITP #IsabelleHOL #HOL_Light #Rocq #LeanProver #Metamath #Mizar

Lecturas compartidas el 25 de junio de 2024. https://jaalonso.github.io/vestigium/posts/2024/06/26-lecturas_compartidas_el_25-jun-24 #ITP #Agda #Coq #HOL #HOL_Light #IsabelleHOL #Lean4 #Metamath #Math #Haskell #Python #Algorithms #AI #MachineLearning

Lecturas compartidas el 25 de junio de 2024. https://jaalonso.github.io/vestigium/posts/2024/06/26-lecturas_compartidas_el_25-jun-24 #ITP #Agda #Coq #HOL #HOL_Light #IsabelleHOL #Lean4 #Metamath #Math #Haskell #Python #Algorithms #AI #MachineLearning

Lecturas compartidas el 25 de junio de 2024. https://jaalonso.github.io/vestigium/posts/2024/06/26-lecturas_compartidas_el_25-jun-24 #ITP #Agda #Coq #HOL #HOL_Light #IsabelleHOL #Lean4 #Metamath #Math #Haskell #Python #Algorithms #AI #MachineLearning

Lecturas compartidas el 25 de junio de 2024. https://jaalonso.github.io/vestigium/posts/2024/06/26-lecturas_compartidas_el_25-jun-24 #ITP #Agda #Coq #HOL #HOL_Light #IsabelleHOL #Lean4 #Metamath #Math #Haskell #Python #Algorithms #AI #MachineLearning

Lecturas compartidas el 25 de junio de 2024. https://jaalonso.github.io/vestigium/posts/2024/06/26-lecturas_compartidas_el_25-jun-24 #ITP #Agda #Coq #HOL #HOL_Light #IsabelleHOL #Lean4 #Metamath #Math #Haskell #Python #Algorithms #AI #MachineLearning