Aleph

[Warman]

I would like to know how to go from one aleph to another, for example (correct me if I'm wrong) aleph 0 + 1 = aleph 1 . so logically aleph 1 + 1 = aleph 2 my question would be that this system would be valid for any aleph? so aleph 1028829201927392019283 + 1 = aleph 4 1028829201927392019284 ? thank you in advance

 

[ActuallySpaceMan42]

Uh...

Aleph 0 + 1 is Aleph 0.

Aleph 0 + Aleph 0 is also Aleph 0.

Even Aleph 0 * Aleph 0 is Aleph 0.

 

[Warman]

thank you ! I was talking about it with someone on discord and she told me that aleph 0 + 1 = aleph 1 thank you! so to go from aleph 0 to aleph 1 would that be the cardinality of aleph 0 or 2^aleph 0?

 

[sen_]

thank you ! I was talking about it with someone on discord and she told me that aleph 0 + 1 = aleph 1 thank you! so to go from aleph 0 to aleph 1 would that be the cardinality of aleph 0 or 2^aleph 0?

the next smallest Aleph is defined by the successor operation, or basically the cardinality of the smallest Ordinal Number into which you can make an injective map, but cannot map it back into the set. Aleph-1 in this case is the smallest uncountable Infinite Cardinal greater than Aleph-0.

 

[Warman]

the next smallest Aleph is defined by the successor operation, or basically the cardinality of the smallest Ordinal Number into which you can make an injective map, but cannot map it back into the set. Aleph-1 in this case is the smallest uncountable Infinite Cardinal greater than Aleph-0.

Thanks so 2 power aleph 0 ≠ aleph 1 ?

 

[Reiner04]

Thanks so 2 power aleph 0 ≠ aleph 1 ?

2^aleph0 = aleph 1.

 

[sen_]

Thanks so 2 power aleph 0 ≠ aleph 1 ?

It is both consistent that 2^Aleph-0 = Aleph-1 and 2^Aleph-0 = Aleph-n for any n > 0 (except some special cases like aleph-ω and cardinals with cofinality ω), the assertion that 2^Aleph-0 = Aleph-1 is what we call the Continuum Hypothesis, which can be further generalized to GCH (Generalized Continuum Hypothesis), which says that 2^Aleph-n = Aleph-(n+1), the truth of the Continuum Hypothesis is independent from ZF (Zermelo-Fraenkel Set theory, the most commonly used axiomatic system of set theory). So the answer to your question is that it cannot be proven in ZFC or ZF.

 

[Reiner04]

It is both consistent that 2^Aleph-0 = Aleph-1 and 2^Aleph-0 = Aleph-n for any n > 0 (except some special cases like aleph-ω and cardinals with cofinality ω), the assertion that 2^Aleph-0 = Aleph-1 is what we call the Continuum Hypothesis, which can be further generalized to GCH (Generalized Continuum Hypothesis), which says that 2^Aleph-n = Aleph-(n+1), the truth of the Continuum Hypothesis is independent from ZF (Zermelo-Fraenkel Set theory, the most commonly used axiomatic system of set theory). So the answer to your question is that it cannot be proven in ZFC or ZF.

Nerd

 

[sen_]

Nerd

Ratio.

 

[Warman]

2^aleph0 = aleph 1

It is both consistent that 2^Aleph-0 = Aleph-1 and 2^Aleph-0 = Aleph-n for any n > 0 (except some special cases like aleph-ω and cardinals with cofinality ω), the assertion that 2^Aleph-0 = Aleph-1 is what we call the Continuum Hypothesis, which can be further generalized to GCH (Generalized Continuum Hypothesis), which says that 2^Aleph-n = Aleph-(n+1), the truth of the Continuum Hypothesis is independent from ZF (Zermelo-Fraenkel Set theory, the most commonly used axiomatic system of set theory). So the answer to your question is that it cannot be proven in ZFC or ZF.

So 2^aleph 1665 = aleph 1666 ?

 

[sen_]

So 2^aleph 1665 = aleph 1666 ?

If you assume GCH to be true, then yes. Though not necessarily.

 

[Reiner04]

So 2^aleph 1665 = aleph 1666 ?

As for our standards we assume continuum hypothesis to be true, so yes.

 

[Warman]

En ce qui concerne nos normes, nous supposons que l'hypothèse du continuum est vraie, donc oui.

Si vous supposez que GCH est vrai, alors oui. Bien que pas nécessairement.

Thanks

 

[Warman]

If you assume GCH to be true, then yes. Though not necessarily.

As for our standards we assume continuum hypothesis to be true, so yes.

I have a last question that will remove my remaining doubts (I know that cardinal = what constitutes the set) so my question would be the cardinal of aleph 0 would be greater than or equal to aleph 0?

 

[Reiner04]

I have a last question that will remove my remaining doubts (I know that cardinal = what constitutes the set) so my question would be the cardinal of aleph 0 would be greater than or equal to aleph 0?

Equal.

 

[sen_]

I have a last question that will remove my remaining doubts (I know that cardinal = what constitutes the set) so my question would be the cardinal of aleph 0 would be greater than or equal to aleph 0?

A cardinal number is a number which measures the cardinality of sets, the cardinality of a set is what you're talking about, and Aleph-0 = Aleph-0 of course. Though i think your question is more "If we look at Aleph-0 in terms of sets, then what is its cardinality?", and the answer is, still Aleph-0.

 

[Achille]

It is both consistent that 2^Aleph-0 = Aleph-1 and 2^Aleph-0 = Aleph-n for any n > 0 (except some special cases like aleph-ω and cardinals with cofinality ω), the assertion that 2^Aleph-0 = Aleph-1 is what we call the Continuum Hypothesis, which can be further generalized to GCH (Generalized Continuum Hypothesis), which says that 2^Aleph-n = Aleph-(n+1), the truth of the Continuum Hypothesis is independent from ZF (Zermelo-Fraenkel Set theory, the most commonly used axiomatic system of set theory). So the answer to your question is that it cannot be proven in ZFC or ZF.

Now, let's address the accuracy of the statements made in the article:

The statement "It is both consistent that 2^Aleph-0 = Aleph-1 and 2^Aleph-0 = Aleph-n for any n > 0 (except some special cases like aleph-ω and cardinals with cofinality ω)" is not completely correct. It is consistent with ZFC that 2^ℵ0 = ℵ1, as this is equivalent to the Continuum Hypothesis (CH). However, it is also consistent with ZFC that 2^ℵ0 > ℵ1, which means that the size of the power set of the set of natural numbers is strictly larger than the size of the set of real numbers. This is known as the negation of CH.
The statement "the truth of the Continuum Hypothesis is independent from ZF" is correct. In other words, it is possible to construct models of set theory based on ZF in which CH is true, and it is also possible to construct models of set theory based on ZF in which CH is false. This means that CH cannot be proven or disproven using the axioms of ZF alone.
The statement "So the answer to your question is that it cannot be proven in ZFC or ZF" is not completely correct. While it is true that CH cannot be proven or disproven within ZF or ZFC, it is possible to prove or disprove CH using additional axioms beyond ZF or ZFC. For example, the axiom of determinacy (AD) implies...


The continuum hypothesis can be formally stated as follows:

CH: ℵ0 < |R| = ℵ1

Here, |R| denotes the cardinality of the set of real numbers and ℵ0 and ℵ1 denote the cardinalities of the sets of natural numbers and the first uncountable cardinal, respectively.

The continuum hypothesis is a statement about the size of sets and is independent of the axioms of set theory. In other words, it is possible to construct a model of set theory in which the continuum hypothesis is true and a model in which it is false.

In 1940, Kurt Gödel showed that the continuum hypothesis is consistent with the axioms of Zermelo-Fraenkel set theory (ZFC), the most widely used axiomatic system for set theory. In 1963, Paul Cohen showed that if ZFC is consistent, then so is the theory obtained by adding the negation of the continuum hypothesis as an axiom (ZFC+ ¬CH). This result, known as Cohen's forcing theorem, established the independence of the continuum hypothesis from ZFC.

In addition to the continuum hypothesis, there are several other related statements that have been studied in set theory. The generalized continuum hypothesis (GCH) is a generalization of the continuum hypothesis that states that for any infinite cardinal λ, there is no cardinal number strictly between λ and 2^λ. The GCH can be formally stated as follows:

GCH: ℵ0 < |R| = ℵ1 ∧ ∀ λ ∈ Card, ℵ0 < λ < 2^λ ⇒ λ = ℵn for some n ∈ ℕ

Here, Card denotes the class of all infinite cardinal numbers and ℵn denotes the nth aleph number. The GCH is also independent of the axioms of set theory and can be either true or false in different models of set theory.

 

[XXKINGXX69]

Now, let's address the accuracy of the statements made in the article:

The statement "It is both consistent that 2^Aleph-0 = Aleph-1 and 2^Aleph-0 = Aleph-n for any n > 0 (except some special cases like aleph-ω and cardinals with cofinality ω)" is not completely correct. It is consistent with ZFC that 2^ℵ0 = ℵ1, as this is equivalent to the Continuum Hypothesis (CH). However, it is also consistent with ZFC that 2^ℵ0 > ℵ1, which means that the size of the power set of the set of natural numbers is strictly larger than the size of the set of real numbers. This is known as the negation of CH.
The statement "the truth of the Continuum Hypothesis is independent from ZF" is correct. In other words, it is possible to construct models of set theory based on ZF in which CH is true, and it is also possible to construct models of set theory based on ZF in which CH is false. This means that CH cannot be proven or disproven using the axioms of ZF alone.
The statement "So the answer to your question is that it cannot be proven in ZFC or ZF" is not completely correct. While it is true that CH cannot be proven or disproven within ZF or ZFC, it is possible to prove or disprove CH using additional axioms beyond ZF or ZFC. For example, the axiom of determinacy (AD) implies...


The continuum hypothesis can be formally stated as follows:

CH: ℵ0 < |R| = ℵ1

Here, |R| denotes the cardinality of the set of real numbers and ℵ0 and ℵ1 denote the cardinalities of the sets of natural numbers and the first uncountable cardinal, respectively.

The continuum hypothesis is a statement about the size of sets and is independent of the axioms of set theory. In other words, it is possible to construct a model of set theory in which the continuum hypothesis is true and a model in which it is false.

In 1940, Kurt Gödel showed that the continuum hypothesis is consistent with the axioms of Zermelo-Fraenkel set theory (ZFC), the most widely used axiomatic system for set theory. In 1963, Paul Cohen showed that if ZFC is consistent, then so is the theory obtained by adding the negation of the continuum hypothesis as an axiom (ZFC+ ¬CH). This result, known as Cohen's forcing theorem, established the independence of the continuum hypothesis from ZFC.

In addition to the continuum hypothesis, there are several other related statements that have been studied in set theory. The generalized continuum hypothesis (GCH) is a generalization of the continuum hypothesis that states that for any infinite cardinal λ, there is no cardinal number strictly between λ and 2^λ. The GCH can be formally stated as follows:

GCH: ℵ0 < |R| = ℵ1 ∧ ∀ λ ∈ Card, ℵ0 < λ < 2^λ ⇒ λ = ℵn for some n ∈ ℕ

Here, Card denotes the class of all infinite cardinal numbers and ℵn denotes the nth aleph number. The GCH is also independent of the axioms of set theory and can be either true or false in different models of set theory.

Scans?

 

[Achille]

Scans?

Continuum hypothesis - Wikipedia

en.wikipedia.org

2^ℵ0 = ℵ1

|R| = 2^ℵ0 so |R| = ℵ1

This is for some explanations;

classification theory

classification theory, principles governing the organization of objects into groups according to their similarities and differences or their relation to a set of criteria. Classification theory has applications in all branches of knowledge, especially the biological and social sciences. Its...

www.britannica.com

+ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287

and https://plato.stanford.edu/entries/continuum-hypothesis/

 

[sen_]

Now, let's address the accuracy of the statements made in the article:

The statement "It is both consistent that 2^Aleph-0 = Aleph-1 and 2^Aleph-0 = Aleph-n for any n > 0 (except some special cases like aleph-ω and cardinals with cofinality ω)" is not completely correct. It is consistent with ZFC that 2^ℵ0 = ℵ1, as this is equivalent to the Continuum Hypothesis (CH). However, it is also consistent with ZFC that 2^ℵ0 > ℵ1, which means that the size of the power set of the set of natural numbers is strictly larger than the size of the set of real numbers. This is known as the negation of CH.

The bolded part is incorrect, it can be proved that the cardinality of the power set of N is exactly equal to |R|, the rest is correct (Hint: to prove the bijection between P(N) and R, either consider the function f(A) = ∑2^{-r} for r ∈ A and A ⊆ N which pretty much constructs a bijection between P(N) and [0, 1], or construct an injection f: P(N) → R and another injection g: R → P(N), then use the Schröder–Bernstein theorem to prove a bijection).

The statement "So the answer to your question is that it cannot be proven in ZFC or ZF" is not completely correct. While it is true that CH cannot be proven or disproven within ZF or ZFC, it is possible to prove or disprove CH using additional axioms beyond ZF or ZFC. For example, the axiom of determinacy (AD) implies...

I never said anything about additional axioms, i said that it cannot be proven nor disproven within ZF or ZFC.

In addition to the continuum hypothesis, there are several other related statements that have been studied in set theory. The generalized continuum hypothesis (GCH) is a generalization of the continuum hypothesis that states that for any infinite cardinal λ, there is no cardinal number strictly between λ and 2^λ. The GCH can be formally stated as follows:

GCH: ℵ0 < |R| = ℵ1 ∧ ∀ λ ∈ Card, ℵ0 < λ < 2^λ ⇒ λ = ℵn for some n ∈ ℕ

Here, Card denotes the class of all infinite cardinal numbers and ℵn denotes the nth aleph number. The GCH is also independent of the axioms of set theory and can be either true or false in different models of set theory.

The |R| = ℵ1 is redunant here and it isn't necessary to include it, since |P(N)| = 2^|N| = |R| which directly implies |R| = Aleph-1 assuming GCH.

 

[Shmeatywerbenmanjenson]

How does one begin to acquire this level of understanding of Set theory?

 

[Achille]

The bolded part is incorrect

I said, "Which means that the size of the power set of the set of natural numbers is strictly larger than the size of the set of real numbers."

Actually, when I said so, I emphasized "2^א0 > א1".

I apologize for misrepresenting it at the very beginning.

then use the Schröder–Bernstein theorem to prove a bijection

Let me even correct my mistake using the Schröder–Bernstein theorem.



It is true that the cardinality of the power set of the set of natural numbers (denoted P(N)) and the cardinality of the set of real numbers (denoted |R|) are equal. This can be proven using the Schröder-Bernstein theorem, which states that if there exist injections (functions that are both one-to-one and onto) f: A -> B and g: B -> A between two sets A and B, then there exists a bijection (a function that is both one-to-one and onto) between A and B.

To prove that P(N) and |R| are bijective, we can first construct an injection f: P(N) -> |R|. One way to do this is by using the function f(A) = ∑ 2^(-r) for r ∈ A and A ⊆ N, which maps each subset of N to a unique real number in the interval [0,1]. This function is injective because for any two distinct subsets A and B of N, there must exist an element r ∈ A that is not in B, or an element s ∈ B that is not in A. This means that f(A) and f(B) will differ by at least 2^(-r) or 2^(-s), and therefore cannot be equal.

We can then construct an injection g: |R| -> P(N) by mapping each real number x in |R| to the subset of N containing all the positive integers n such that x > 1/n. This function is also injective, because for any two distinct real numbers x and y, there must exist a positive integer n such that x > 1/n and y <= 1/n, or y > 1/n and x <= 1/n. This means that g(x) and g(yy) will differ by the presence or absence of the integer n, and therefore cannot be equal.

Finally, we can use the Schröder-Bernstein theorem to conclude that there exists a bijection between P(N) and |R|. This means that the two sets have the same cardinality, and the statement "the size of the power set of the set of natural numbers is strictly larger than the size of the set of real numbers" is incorrect.

The |R| = ℵ1 is redunant here and it isn't necessary to include it, since |P(N)| = 2^|N| = |R| which directly implies |R| = Aleph-1 assuming GCH.

Eh... Not exactly superfluous. we explain the same thing in two different ways, just that's it.

It is true that the cardinality of the power set of the natural numbers (i.e. the set of all subsets of the natural numbers) is equal to the cardinality of the real numbers, denoted as |P(N)| = |R|. This can be proven using the fact that the cardinality of the power set of a set is equal to 2^|S|, where |S| denotes the cardinality of set S. Therefore, |P(N)| = 2^|N| = 2^ℵ0 = ℵ1.

The assumption of the Generalized Continuum Hypothesis (GCH) is not necessary to prove that |P(N)| = |R|. In fact, the Continuum Hypothesis (CH), which states that |P(N)| = ℵ1, is independent of the standard axioms of set theory (such as ZFC). This means that it is possible to construct a model of set theory in which CH holds, and another model in which CH is false.

Therefore, while it is true that |P(N)| = |R| and |R| = ℵ1, the assumption of GCH is not necessary to prove these statements.

 

[Achille]

How does one begin to acquire this level of understanding of Set theory?

You need to know what the mathematical symbols used in set theory's mathematical formulas and equations mean.

Then, if you have a good foundation, you can read academic articles (available from google scholar, sciencedirect, etc.).

 

[sen_]

I said, "Which means that the size of the power set of the set of natural numbers is strictly larger than the size of the set of real numbers."

Actually, when I said so, I emphasized "2^א0 > א1".

I apologize for misrepresenting it at the very beginning.

No problem at all.

It is true that the cardinality of the power set of the set of natural numbers (denoted P(N)) and the cardinality of the set of real numbers (denoted |R|) are equal. This can be proven using the Schröder-Bernstein theorem, which states that if there exist injections (functions that are both one-to-one and onto) f: A -> B and g: B -> A between two sets A and B, then there exists a bijection (a function that is both one-to-one and onto) between A and B.

To prove that P(N) and |R| are bijective, we can first construct an injection f: P(N) -> |R|. One way to do this is by using the function f(A) = ∑ 2^(-r) for r ∈ A and A ⊆ N, which maps each subset of N to a unique real number in the interval [0,1]. This function is injective because for any two distinct subsets A and B of N, there must exist an element r ∈ A that is not in B, or an element s ∈ B that is not in A. This means that f(A) and f(B) will differ by at least 2^(-r) or 2^(-s), and therefore cannot be equal.

We can then construct an injection g: |R| -> P(N) by mapping each real number x in |R| to the subset of N containing all the positive integers n such that x > 1/n. This function is also injective, because for any two distinct real numbers x and y, there must exist a positive integer n such that x > 1/n and y <= 1/n, or y > 1/n and x <= 1/n. This means that g(x) and g(yy) will differ by the presence or absence of the integer n, and therefore cannot be equal.

Finally, we can use the Schröder-Bernstein theorem to conclude that there exists a bijection between P(N) and |R|. This means that the two sets have the same cardinality, and the statement "the size of the power set of the set of natural numbers is strictly larger than the size of the set of real numbers" is incorrect.

The proof works, just a small correction, the functions should be denoted f: P(N) → R and g: R → P(N) since |R| denotes the cardinality of R while this is a function between sets.

Eh... Not exactly superfluous. we explain the same thing in two different ways, just that's it.

It is true that the cardinality of the power set of the natural numbers (i.e. the set of all subsets of the natural numbers) is equal to the cardinality of the real numbers, denoted as |P(N)| = |R|. This can be proven using the fact that the cardinality of the power set of a set is equal to 2^|S|, where |S| denotes the cardinality of set S. Therefore, |P(N)| = 2^|N| = 2^ℵ0 = ℵ1.

The assumption of the Generalized Continuum Hypothesis (GCH) is not necessary to prove that |P(N)| = |R|. In fact, the Continuum Hypothesis (CH), which states that |P(N)| = ℵ1, is independent of the standard axioms of set theory (such as ZFC). This means that it is possible to construct a model of set theory in which CH holds, and another model in which CH is false.

Therefore, while it is true that |P(N)| = |R| and |R| = ℵ1, the assumption of GCH is not necessary to prove these statements.

You might have misunderstood what i said a little bit, i mean that it is redunant to include |R| = ℵ1 in your expression of GCH, because it is already included in the second part of your statement, which is also very slightly incorrect because the assumption λ < 2^λ has little to do with GCH and is already proved by Cantor's Theorem. I think what you meant to express is: ∀λ ∈ Card, ¬(∃κ : λ < κ < 2^λ). Which is also equivalent to: ∀α ∈ ORD, ℵ(α+1) = 2^(ℵα).

 

[Kowalski]

I would like to know how to go from one aleph to another, for example (correct me if I'm wrong) aleph 0 + 1 = aleph 1 . so logically aleph 1 + 1 = aleph 2 my question would be that this system would be valid for any aleph? so aleph 1028829201927392019283 + 1 = aleph 4 1028829201927392019284 ? thank you in advance

I suggest you read: https://en.wikipedia.org/wiki/Aleph_number