#metamath — Public Fediverse posts

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

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

@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

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

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 11 de mayo de 2024. https://jalonso.substack.com/lecturas-compartidas-el-11-de-mayo #ITP #Lean4 #IsabelleHOL #Agda #Coq #HOL_Light #Metamath #Mizar #Math #Programming #CommonLisp

Algorithm and abstraction in formal mathematics. ~ Heather Macbeth. https://arxiv.org/abs/2405.04699 #ITP #Agda #Coq #Lean4 #HOL_Light #IsabelleHOL #Metamath #Mizar #Math

@seano @amy

Also, Gottlob Frege, Begriffsschrift (1879).
Also, https://us.metamath.org/index.html
Also, https://web.ics.purdue.edu/~dulrich/Description-of-condensed-detachment.htm

#CondensedDetachment #metamath #Logic

@seano @amy

Also, Gottlob Frege, Begriffsschrift (1879).
Also, https://us.metamath.org/index.html
Also, https://web.ics.purdue.edu/~dulrich/Description-of-condensed-detachment.htm

#CondensedDetachment #metamath #Logic

@seano @amy

Also, Gottlob Frege, Begriffsschrift (1879).
Also, https://us.metamath.org/index.html
Also, https://web.ics.purdue.edu/~dulrich/Description-of-condensed-detachment.htm

#CondensedDetachment #metamath #Logic

@seano @amy

Also, Gottlob Frege, Begriffsschrift (1879).
Also, https://us.metamath.org/index.html
Also, https://web.ics.purdue.edu/~dulrich/Description-of-condensed-detachment.htm

#CondensedDetachment #metamath #Logic