Disordered Infinity

[Jarhyn]

So, I was thinking about infinities

Countable infinity. You can sequentially name ever number

Uncountable infinity. You can sequentially place any number you can name.

Disordered Infinity. The set of graphs created by real arrangements, impossible.tomorder and distinctly different. They have infinitesimal and even trivial definitions. You can name them and they have a logical identity.

Is this set proven equal in size to uncountable infinity?

 

[Swammerdami]

There is no single "uncountable infinity." There are an infinite number of distinct infinities (non-finite cardinalities), along with bizarre theorems about some that are bizarrely large. All but one of these infinities is "uncountable."

The three simplest infinities are

• ℵ₀ = |{0, 1, 2, 3, 4, 5, 6, ...}|
• P(ℵ₀) = 2^ℵ₀ = C
• P(P(ℵ₀)) = 2^2ℵ₀

It's not clear what you mean by "disordered infinity." P(P(ℵ₀)) is already big enough for many purposes! (That cardinal number is equal to ℵ₂ IF GCH is true. GCH, the Generalized Continuum Hypothesis, can be considered a strong version of the Axiom of Choice.)

 

[steve_bank]

Bounded and infinite are mutually exclusive. Infinity is not a number.

There are work arounds. Given the sets of real numbers a from 1 to 2 and b 2 to 3 both are infinitely divisible. Each set has an infinite amount of numbers. You can add a and b and say you have added two infinites and the size of the two together are twice the infinities of either one alone. You can manipulate infinite sets.

To me that is mathematical sophistry.

If by some method you have found a countable l infinity then the test is to put a number to it.

 

[Jarhyn]

Bounded and infinite are mutually exclusive. Infinity is not a number.

There are work arounds. Given the sets of real numbers a from 1 to 2 and b 2 to 3 both are infinitely divisible. Each set has an infinite amount of numbers. You can add a and b and say you have added two infinites and the size of the two together are twice the infinities of either one alone. You can manipulate infinite sets.

To me that is mathematical sophistry.

If by some method you have found a countable l infinity then the test is to put a number to it.

You're not reading the post, and it shows...

It's one of the first topics discussed in "foundations of mathematics".

Uncountable infinity is a different set than uncountable infinity. I am not asking your approval or ascertainment on that fact. It is proven that the size of the set of real numbers from 1 to 2 is = to the size of the set of real numbers from 2 to 3. But it has also been proven that the set of all integers is smaller than the set of real numbers from 1 to 2, despite the fact they are both infinite.

Look it up then come back.

As to swammerdami, The question is if unbounded noncardinalities may represent a larger set than nonbounded cardinalities

 

[Swammerdami]

As to swammerdami, The question is if unbounded noncardinalities may represent a larger set than nonbounded cardinalities

How can you speak of the size of a "non-cardinality"? Cardinal numbers are precisely what are used to describe the sizes of sets.

Are you interested in measure? (Different from "size.") Can you give a specific example of the sort of set of interest? You write "The set of graphs created by real arrangements." The set of all possible sets of points on the real line (or plane) has cardinality 2^2ℵ₀.

 

[Bomb#20]

Infinity is not a number.
...
If by some method you have found a countable l infinity then the test is to put a number to it.

ℵ₀

"Countable infinity" is technical jargon; it isn't a claim that you can count all the things in the infinite set. What it means is that you can set up a "one-to-one and onto" mapping between the things in the set and the natural numbers. The decision to use the word "countable" for that purpose was unfortunate and leads to misunderstandings.

 

[steve_bank]

Bounded and infinite are mutually exclusive. Infinity is not a number.

There are work arounds. Given the sets of real numbers a from 1 to 2 and b 2 to 3 both are infinitely divisible. Each set has an infinite amount of numbers. You can add a and b and say you have added two infinites and the size of the two together are twice the infinities of either one alone. You can manipulate infinite sets.

To me that is mathematical sophistry.

If by some method you have found a countable l infinity then the test is to put a number to it.

You're not reading the post, and it shows...

It's one of the first topics discussed in "foundations of mathematics".

Uncountable infinity is a different set than uncountable infinity. I am not asking your approval or ascertainment on that fact. It is proven that the size of the set of real numbers from 1 to 2 is = to the size of the set of real numbers from 2 to 3. But it has also been proven that the set of all integers is smaller than the set of real numbers from 1 to 2, despite the fact they are both infinite.

Look it up then come back.

As to swammerdami, The question is if unbounded noncardinalities may represent a larger set than nonbounded cardinalities

Apologies, should not have posted. I have no real interest.

 

[Swammerdami]

So, I was thinking about infinities

Countable infinity. You can sequentially name ever number

Uncountable infinity. You can sequentially place any number you can name.

Disordered Infinity. The set of graphs created by real arrangements, impossible.to []order ...
Is this set proven equal in size to uncountable infinity?

I'm still not sure what OP's exact question is, but it's about "ordering" and "Foundations" so is probably related to "Well-Ordering" and ordinal numbers.

A set is well-ordered if it has a smallest element AND each of its subsets has a smallest element (i.e. an element x with x < y for every other y in the set or subset). The set of positive integers is well-ordered. The set of negative integers is NOT well-ordered, but it can be made well-ordered by imposing a different definition of '<' (e.g. simply reversing the usual order).

Any well-ordered set has an ordinal number. The ordinal number of the positive integers is ω. The set {6, 6.6, 6.66, 6.666, 6.6666, ..., 66, 66.6, 66.66, 66.666, ..., } — the set of all-6 numbers whose whole part is either 6 or 66 — has ordinal number ω+ω. The set {6, 6.6, 6.66, 6.666, 6.6666, ..., 66, 66.6, 66.66, 66.666, ..., 666, 666.6, ..., 6666, 6666.6, ..., 66666, ..., 666666, ..., ... } — the set of all positive rationals expressed in decimal only with 6's — has ordinal number ω+ω+ω+ω+... = ω².

The positive rational numbers (fractions) are not well-ordered — there is no smallest fraction — but they can be made well-ordered by inventing a different ordering relation '<'. Let the fraction p/q (expressed in lowest terms) be "less than" r/s (again in lowest terms) if q < s. For tie-breaker (when q = s) use p < r. This ordering gives the set of positive rational numbers the ordinal number ω². (Left as exercise: Find a different ordering which gives that set ordinal number ω.)

Every ordinal number has a corresponding cardinal number. The well-ordered sets we've considered above have ordinal numbers ω, ω+ω, or ω² but they all have cardinal number ℵ₀.

How about the set of real numbers x in a closed interval, 1 ≤ x ≤ 2 ? They have a smallest element, but no 2nd smallest element.

We used a trick to impose a well-ordering on the rational numbers; is there a similar trick to well-order the reals? Nobody knows!! Or rather, they know they cannot find such an ordering, but happily assume one exists anyway! The famous Axiom of Choice is simply the assumption that any set can be well-ordered whether you can find a specific ordering trick or not.

The Axiom of Choice is one of the great "mysteries" of mathematical foundations. It may deserve its own thread here. Using that Axiom, all sorts of paradoxical and unbelievable theorems can be derived. Banach's splitting a ball into parts and them reassembling them to form TWO complete balls, each with the same size as the original ball is one example, but there are other paradoxical results which are much simpler and even more unbelievable. Yet the Axiom is so convenient it is frequently used in mathematical proofs; sometimes the prover doesn't even notice he's relying on that Axiom.

So, is the Axiom of Choice true or false? Again, nobody knows. Or better, following David Hilbert's view that mathematics is just an intricate game played with paper and pencil, any mathematician can accept or reject the Axiom as he wishes, and continue playing his games!

If a set X can be well-ordered, then Hartogs' 1915 proof establishes the existence of  Hartogs number, a larger ordinal number than X. This leads to the apparent conclusion that there are specific smallest cardinal numbers ℵ₀ < ℵ₁ < ℵ₂ < ℵ₃ < ...

Nobody has ever found^* a set larger than the integers but smaller than the reals, so it is natural to assume that 2^ℵ₀ — the cardinal number of the reals — is equal to ℵ₁, but nobody knows if this true! (* - Actually, some mathematicians have written papers about sets bigger than the integers and smaller than the reals, but just by playing games with pencil and paper! )

... The three simplest infinities are

• ℵ₀ = |{0, 1, 2, 3, 4, 5, 6, ...}|
• P(ℵ₀) = 2^ℵ₀ = C
• P(P(ℵ₀)) = 2^2ℵ₀

... P(P(ℵ₀)) is ... equal to ℵ₂ IF GCH is true. GCH, the Generalized Continuum Hypothesis, can be considered a strong version of the Axiom of Choice.)

I wanted to close this post with a VERY crude outline of the proof that GCH implies that any set can be well-ordered. But I will be satisfied to link to one of many simplified(!?!) discussions on the 'Net.

 

[Swammerdami]

The Axiom of Choice is one of the great "mysteries" of mathematical foundations.

If you doubt this, consider this famous mathematical joke: "The axiom of choice is obviously true, the well-ordering theorem obviously false, and who can tell about Zorn's lemma?"

The punchline is this. The three named axiom/theorem/lemma are equivalent to each other. Given any one of them, you can prove the other two!

Using that Axiom, all sorts of paradoxical and unbelievable theorems can be derived. Banach's splitting a ball into parts and them reassembling them to form TWO complete balls, each with the same size as the original ball is one example, but there are other paradoxical results which are much simpler and even more unbelievable.

... If a set X can be well-ordered, then Hartogs' 1915 proof establishes the existence of  Hartogs number, a larger ordinal number than X. This leads to the apparent conclusion that there are specific smallest cardinal numbers ℵ₀ < ℵ₁ < ℵ₂ < ℵ₃ < ...

Nobody has ever found^* a clear-cut set larger than the integers but smaller than the reals, so it is natural to assume that 2^ℵ₀ — the cardinal number of the reals — is equal to ℵ₁, but nobody knows if this is "true"! (Many mathematicians would shrug and say "Take your pick.")
...
I wanted to close this post with a VERY crude outline of the proof that GCH implies that any set can be well-ordered. But I will be satisfied to link to one of many simplified(!?!) discussions on the 'Net.

N = {0, 1, 2, 3, 4, ... }
ℵ₀ = | N |
R = {x | x is a real number }
C = | R |

It is a theorem first proved by Cantor that
C = 2^ℵ₀ = | {x | x ⊆ N } |

The Continuum Hypothesis asserts that C is the 2nd smallest infinite cardinal number. (The Generalized CH asserts that ℵ_z+1 = 2^ℵ_z for any z.

Finally recall that any subset of a well-ordered set is itself well-ordered. (This is part of the definition of 'well-ordered.')

Von Neumann Ordinal Numbers
If you didn't already know that John von Neumann was one of the greatest geniuses in all of history, these ordinal numbers might persuade you ... due to their utter simplicity!

First, what is the difference between an ordinal number and a cardinal number? The set {0, 1, 2, 3, 4} is an instance of cardinal number 5, and is precisely equal to (Definition) the ordinal number 5. Von Neumann used a little trick to reduce the notation to nothing but {} braces.
By increasing size, some of the ordinal numbers are

• oridnal number y = { x | x is any ordinal number smaller than y}
• 0 = {} // no ordinals are smaller than 0.
• 1 = {0} = 0 ∪ {0} = {{{}}, {}} // { 0 } is the set of all ordinals smaller than 1.
• 2 = {0, 1} = 1 ∪ {1}= {{{{}}, {}}, {{}}, {}}
• 3 = {0, 1, 2} = 2 ∪ {2}
• ...
• ω = {0, 1, 2, 3, 4, ... }
• ...
• 2ω + 3 is the ordinal number exemplified by {0, 1, 2, 3, 4, ..., A0, A1, A2, A3, ..., B0, B1, B2}
  But the canonical exempler (indeed definition) of that ordinal number is the particular set
  {0, 1, 2, 3, 4, ..., ω, ω+1, ω+2, ..., 2ω, 2ω + 1, 2ω + 2 }
  Note that 2ω + 3 cannot be an element of 2ω + 3 — that would violate Russell's antinomy x∈x.
• ...
• 9·ω² + 8ω+ 7
• ω^ωω // Gak!
• Ω₁ = {x | x is any countable ordinal number }
• The set Ω₁ just mentioned has, by definition, cardinal number ℵ₁
• Ω₂ = {x | x is any ordinal number whose cardinal number is ℵ₁ or smaller}

Does the set Ω₁ even exist? Its definition seems VERY straightforward, but I am no expert and even the most "obvious" can cause trouble.
The set Ω₁ cannot be "countable" (i.e. its cardinal number must be greater than ℵ₀). If Ω₁ were countable, it would be a set which includes itself, in violation of Russell's antinomy.

The Continuum Hypothesis is simply that ℵ₁ = 2^ℵ₀. Ω₁ is well-ordered by Hartogs' Theorem (and seems to be seen directly if you believe in von Neumann ordinals!) Any set smaller than that well-ordered set can be mapped (injected) into the larger (well-ordered) set and this would impose a well-ordering on that image of the real numbers. So the reals can be well-ordered. Q.E.D. But obviously this argument is flawed since the previously linked-to discussion requires more.

Left as an exercise is to find the flaws in this proof and correct them. We've got to watch out for the axioms of set construction. We've got to be sure that any ≤ we introduce is well defined and trichotomy holds. Some mathematicians, to prove their versions of the axiom of choice or its contrary, introduce ridiculously large cardinals, larger than ℵ_ℵℵℵ,,,. I just post to celebrate this great genius Johnny von Neumann, inventor of the modern ordinal numbers. This invention hardly scratches the surface of his resume bullet-points. Indeed it hardly makes the top ten of his pure mathematics discoveries, even ignoring his many contributions to physics, engineering, game theory and computer science. (Some of his contributions may be impolitic to mention in our Woke era. He designed the implosion geometry for the 'Fat Man' device, without which Nagasaki's destruction would have required another Hiroshima-type 'Little Boy' device.)

Von Neumann modestly described himself as "only the third best mathematician of my time." Assuming Carl Siegel was #1, who was #2? (Presumably not David Hilbert, 41 years older than von Neumann — their careers hardly overlapped so Hilbert was hardly "of [Johnny's] time"?)