The Independence of) The Continuum Hypothesis
In the late 19th century, a German mathematician named Georg Cantor blew everyone’s minds by figuring out that infinities come in different sizes, called cardinalities. He proved the foundational theorems about cardinality, which modern day math majors
tend to learn in their Discrete Math classes.
Cantor proved that the set of real numbers is larger than the set of natural numbers, which we write as |ℝ|>|ℕ|. It was easy to establish that the size of the natural numbers, |ℕ|, is the first infinite size; no infinite set is smaller than ℕ.
Now, the real numbers are larger, but are they the second infinite size? This turned out to be a much harder question, known as The Continuum Hypothesis (CH).
If CH is true, then |ℝ| is the second infinite size, and no infinite sets are smaller than ℝ, yet larger than ℕ. And if CH is false, then there is at least one size in between.
So what’s the answer? This is where things take a turn.
CH has been proven independent, relative to the baseline axioms of math. It can be true, and no logical contradictions follow, but it can also be false, and no logical contradictions will follow.
It’s a weird state of affairs, but not completely uncommon in modern math. You may have heard of the Axiom of Choice, another independent statement. The proof of this outcome spanned decades and, naturally, split into two major parts: the proof that CH
is consistent, and the proof that the negation of CH is consistent.
The first half is thanks to Kurt Gödel, the legendary Austro-Hungarian logician. His 1938 mathematical construction, known as Gödel’s Constructible Universe, proved CH compatible with the baseline axioms, and is still a cornerstone of Set Theory
classes. The second half was pursued for two more decades until Paul Cohen, a mathematician at Stanford, solved it by inventing an entire method of proof in Model Theory known as “forcing.”
Gödel’s and Cohen’s halves of the proof each take a graduate level of Set Theory to approach, so it’s no wonder this unique story has been esoteric outside mathematical circles.
(Wikimedia Commons)

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In article <b96733a2-5f6b-4882-88d3-fa97fcc4b0fen@googlegroups.com>, Mathematician Numbers <mhewordsearch@gmail.com> wrote:

The Independence of) The Continuum Hypothesis
In the late 19th century, a German mathematician named Georg Cantor blew ev= >eryone=E2=80=99s minds by figuring out that infinities come in different si= >zes, called cardinalities. He proved the foundational theorems about cardin= >ality, which modern day math majors tend to learn in their Discrete Math cl= >asses.
Cantor proved that the set of real numbers is larger than the set of natura= >l numbers, which we write as |=E2=84=9D|>|=E2=84=95|. It was easy to establ= >ish that the size of the natural numbers, |=E2=84=95|, is the first infinit= >e size; no infinite set is smaller than =E2=84=95.
Now, the real numbers are larger, but are they the second infinite size? Th= >is turned out to be a much harder question, known as The Continuum Hypothes= >is (CH).
If CH is true, then |=E2=84=9D| is the second infinite size, and no infinit= >e sets are smaller than =E2=84=9D, yet larger than =E2=84=95. And if CH is = >false, then there is at least one size in between.
So what=E2=80=99s the answer? This is where things take a turn.
CH has been proven independent, relative to the baseline axioms of math. It=
can be true, and no logical contradictions follow, but it can also be fals=
e, and no logical contradictions will follow.
It=E2=80=99s a weird state of affairs, but not completely uncommon in moder= >n math. You may have heard of the Axiom of Choice, another independent stat= >ement. The proof of this outcome spanned decades and, naturally, split into=
two major parts: the proof that CH is consistent, and the proof that the n=
egation of CH is consistent.
The first half is thanks to Kurt G=C3=B6del, the legendary Austro-Hungarian=
logician. His 1938 mathematical construction, known as G=C3=B6del=E2=80=99=
s Constructible Universe, proved CH compatible with the baseline axioms, an= >d is still a cornerstone of Set Theory classes. The second half was pursued=
for two more decades until Paul Cohen, a mathematician at Stanford, solved= it by inventing an entire method of proof in Model Theory known as =E2=80=
=9Cforcing.=E2=80=9D
G=C3=B6del=E2=80=99s and Cohen=E2=80=99s halves of the proof each take a gr= >aduate level of Set Theory to approach, so it=E2=80=99s no wonder this uniq= >ue story has been esoteric outside mathematical circles.
(Wikimedia Commons)

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