#axiomofchoice — Public Fediverse posts

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.

#AxiomOfChoice #OrdinalNumbers #SetTheory #FiniteSet

@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-Elements

Prior work:
(𝔖 or ZF) + AC ⊢ Fin₁ = Fin₁ₐ = Fin₂ = Fin₃ = Fin₄ = Fin₅ = Fin₆ = Fin₇

EScbO = the axiom "Every set can be (linearly, strictly) ordered"

Theorem 1 is
(𝔖 or ZF) ⊢ Fin₁ ⊆ Fin₁ₐ ⊆ Fin₂ ⊆ Fin₃ ⊆ Fin₄ ⊆ Fin₅ ⊆ Fin₆ ⊆ Fin₇

Theorem 2 is
(𝔖 or ZF) ⊢ ( R Orders A ∧ A ∈ Fin₂ ) → A ∈ Fin₁

Theorem 3 is
(𝔖 or ZF) + EScbO ⊢ Fin₁ = Fin₁ₐ = Fin₂

Theorem 4 is
𝔖 + EScbO ⊢ Fin₁ ≠ Fin₃

Theorem 5 is
𝔖 + EScbO ⊢ Fin₃ ≠ Fin₄

Theorem 6 is
𝔖 + EScbO ⊢ Fin₄ ≠ Fin₅

Theorem 7 is
𝔖 + EScbO ⊢ Fin₅ ≠ Fin₆

Theorem 8 is
𝔖 + EScbO ⊢ Fin₆ ≠ Fin₇

Theorem 9 is
𝔖 ⊢ Fin₁ ≠ Fin₁ₐ

Theorem 10 is
(𝔖 or ZF) ⊢ ( Fin₁ₐ = Fin₂ → Fin₁ = Fin₁ₐ )

Theorem 11 is
𝔖 ⊢ Fin₁ₐ ≠ Fin₂

As it so happens, I know of a proof:
ZF + CC ⊢ Fin₁ = Fin₄

Where 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

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)

Foundations of mathematics: I'd be curious to know, in some detail, which topics in present-day undergraduate mathematics (in a scientific study programme) would have to be eliminated or drastically changed if one didn't want to use theorems that need the axiom of choice or the continuum hypothesis. Can anyone suggest works that discuss this? **Preferably works that examine this matter in a neutral, rational, non-sensationalistic way.**

Thank you!

#mathematics #axiomofchoice #continuumhypothesis #foundationsofmathematics

Yet another academic paper that dives into the rabbit hole of Zermelo’s "Axiom of Choice" 🐇🎩, as if it was the Da Vinci Code of set theory. 😴🔍 100 years and counting, and we're still trying to decipher what the heck the problem was—spoiler alert, it's probably just a bunch of mathematicians arguing over who gets the last slice of infinity pie. 🍰♾️
https://research.mietek.io/mi.MartinLof2006.html #AxiomOfChoice #SetTheory #InfinityPie #MathHumor #AcademicPapers #HackerNews #ngated

The Man Who Almost Broke Math (And Himself...)
#Veritasium #DerekMuller #VideoEssay #History #Culture #Science #ArtScience #Maths #Math #Infinities #AC #AoC #AxiomOfChoice
https://www.youtube.com/watch?v=_cr46G2K5Fo

گر من ز می مُغانه مستم هستم
گر عاشق و رند و می پرستم هستم
هر طایفه‌ای ز من گمانی دارد
من زان خودم چُنان که هستم هستم

چنان که هستم هستم

[با صدای مامک خادم از آلبوم گشایش (Unfolding) از گروه Axiom of Choice بشنوید.]

#np #AxiomOfChoice #خیام

Whenever I read mathematics, no matter what kind it is, I always think that I don't understand it unless I could, in principle, teach a computer how to perform it.

I'm not exactly an intuitionist, but this attitude of mine does mean that I don't like the #AxiomOfChoice very much. What good are objects that are literally undescribable?

The only solution around that is to mechanise the #metamathematics instead of the mathematics and that always feels so unsatisfying and distanced to me.