"met" means meeting all dense sets of the partial order. Adjoin G to V to form V[G]. In V[G], said functions can be used to show that $2^{\aleph_0} = \aleph_2$, showing that there is a model of ZFC in which CH holds. But CH is also false under another ZFC model, therefore CH is independent from ZFC.
June 25, 2025 at 9:02 AM
"met" means meeting all dense sets of the partial order. Adjoin G to V to form V[G]. In V[G], said functions can be used to show that $2^{\aleph_0} = \aleph_2$, showing that there is a model of ZFC in which CH holds. But CH is also false under another ZFC model, therefore CH is independent from ZFC.
writing numbers in base 0 roughly gives us a bijection between R and 2^{aleph_0}, but ofc you need to shift some stuff around and correct it to account for numbers outside of [ 0, 1) and nonunique representations. But it basically does work
March 24, 2024 at 5:45 AM
writing numbers in base 0 roughly gives us a bijection between R and 2^{aleph_0}, but ofc you need to shift some stuff around and correct it to account for numbers outside of [ 0, 1) and nonunique representations. But it basically does work
over } M \iff \operatorname{cf}(\delta_1) \geq \kappa\]
Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the $\aleph_0$-tameness assumption and assuming the independence relation is defined only on [5/7 of https://arxiv.org/abs/2503.11605v1]
Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the $\aleph_0$-tameness assumption and assuming the independence relation is defined only on [5/7 of https://arxiv.org/abs/2503.11605v1]
March 17, 2025 at 6:04 AM
over } M \iff \operatorname{cf}(\delta_1) \geq \kappa\]
Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the $\aleph_0$-tameness assumption and assuming the independence relation is defined only on [5/7 of https://arxiv.org/abs/2503.11605v1]
Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the $\aleph_0$-tameness assumption and assuming the independence relation is defined only on [5/7 of https://arxiv.org/abs/2503.11605v1]
wild if it is not an $n$-truncation for any ${n \in \mathbb{Z}}$ and we prove that, under Martin's Axiom, any finitary matroid of infinite rank and size of less than continuum admits ${2^{2^{\aleph_0}}}$ wild generalised truncations. [4/4 of https://arxiv.org/abs/2504.05064v1]
April 8, 2025 at 6:06 AM
wild if it is not an $n$-truncation for any ${n \in \mathbb{Z}}$ and we prove that, under Martin's Axiom, any finitary matroid of infinite rank and size of less than continuum admits ${2^{2^{\aleph_0}}}$ wild generalised truncations. [4/4 of https://arxiv.org/abs/2504.05064v1]
An aleph_1 of $1 bills would be worth more than an aleph_0 of $20 bills, at least
May 2, 2024 at 12:17 AM
An aleph_1 of $1 bills would be worth more than an aleph_0 of $20 bills, at least
Continuum Hypothesis loses its force in this context: between aleph_0 and c lies not a single cardinal jump, but a stratified sequence of definitional stages, each forming a countable yet irreducible approximation to the continuum. We argue that the [4/6 of https://arxiv.org/abs/2504.04637v1]
April 8, 2025 at 6:05 AM
Continuum Hypothesis loses its force in this context: between aleph_0 and c lies not a single cardinal jump, but a stratified sequence of definitional stages, each forming a countable yet irreducible approximation to the continuum. We argue that the [4/6 of https://arxiv.org/abs/2504.04637v1]
Why do you say that? Just because all of the modern "alpha masculinity" thing (a la Peterson and Tate) is aimed at impressing other "alpha men"?
(Ok, so now I have to go invent "aleph masculinity". Not sure what that looks like, tho. Ooh, "aleph_0 masculinity"!)
(Ok, so now I have to go invent "aleph masculinity". Not sure what that looks like, tho. Ooh, "aleph_0 masculinity"!)
September 25, 2024 at 8:25 PM
Why do you say that? Just because all of the modern "alpha masculinity" thing (a la Peterson and Tate) is aimed at impressing other "alpha men"?
(Ok, so now I have to go invent "aleph masculinity". Not sure what that looks like, tho. Ooh, "aleph_0 masculinity"!)
(Ok, so now I have to go invent "aleph masculinity". Not sure what that looks like, tho. Ooh, "aleph_0 masculinity"!)
$\kappa>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $\kappa$ in $Add(M)$ is free for all $\kappa>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $(\Vert [3/5 of https://arxiv.org/abs/2502.19641v1]
February 28, 2025 at 6:05 AM
$\kappa>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $\kappa$ in $Add(M)$ is free for all $\kappa>\Vert R\Vert+\aleph+\aleph_0$; if and only if every $R$-module of cardinal $(\Vert [3/5 of https://arxiv.org/abs/2502.19641v1]
不正確な文章を書いてしまった。ω じゃなくて |ω| (とか \aleph_0 とか) ですね。
February 23, 2025 at 4:31 PM
不正確な文章を書いてしまった。ω じゃなくて |ω| (とか \aleph_0 とか) ですね。
idk if this makes the CH true or false but i feeel that 2^{\aleph_0} is obviously |R|. like this doesn't need a proof bcuz we all know it's true
March 24, 2024 at 5:38 AM
idk if this makes the CH true or false but i feeel that 2^{\aleph_0} is obviously |R|. like this doesn't need a proof bcuz we all know it's true
arXiv:2503.24207v1 Announce Type: new
Abstract: Let $E$ be a vector space over a countable field of dimension $\aleph_0$. Two infinite-dimensional subspaces $V,W \subseteq E$ are almost disjoint if $V \cap W$ is finite-dimensional. This paper [1/4 of https://arxiv.org/abs/2503.24207v1]
Abstract: Let $E$ be a vector space over a countable field of dimension $\aleph_0$. Two infinite-dimensional subspaces $V,W \subseteq E$ are almost disjoint if $V \cap W$ is finite-dimensional. This paper [1/4 of https://arxiv.org/abs/2503.24207v1]
April 1, 2025 at 6:04 AM
arXiv:2503.24207v1 Announce Type: new
Abstract: Let $E$ be a vector space over a countable field of dimension $\aleph_0$. Two infinite-dimensional subspaces $V,W \subseteq E$ are almost disjoint if $V \cap W$ is finite-dimensional. This paper [1/4 of https://arxiv.org/abs/2503.24207v1]
Abstract: Let $E$ be a vector space over a countable field of dimension $\aleph_0$. Two infinite-dimensional subspaces $V,W \subseteq E$ are almost disjoint if $V \cap W$ is finite-dimensional. This paper [1/4 of https://arxiv.org/abs/2503.24207v1]
@saegge.bsky.social magst du einen Thread über Aleph_0 machen? 😬😬😬
April 26, 2024 at 12:12 PM
@saegge.bsky.social magst du einen Thread über Aleph_0 machen? 😬😬😬
if people are wondering, these are both aleph_0, the smallest infinite cardinal; the infinity of the coubting numbers. we know Real Numbers have a higher cardinality from cantor’s diagnolization argument. we know the Power Set of the integers is larger
we don’t know if those infinities are the same
we don’t know if those infinities are the same
May 2, 2024 at 12:09 AM
if people are wondering, these are both aleph_0, the smallest infinite cardinal; the infinity of the coubting numbers. we know Real Numbers have a higher cardinality from cantor’s diagnolization argument. we know the Power Set of the integers is larger
we don’t know if those infinities are the same
we don’t know if those infinities are the same
R\Vert+\aleph+\aleph_0)^{+}$ in $Add(M)$ is free.
As an application, we show that the class of pure-projective $R$-modules is categorical in some (all) big cardinal if and only if $R\cong M_n(D)$ where $D$ is a division [4/5 of https://arxiv.org/abs/2502.19641v1]
As an application, we show that the class of pure-projective $R$-modules is categorical in some (all) big cardinal if and only if $R\cong M_n(D)$ where $D$ is a division [4/5 of https://arxiv.org/abs/2502.19641v1]
February 28, 2025 at 6:05 AM
R\Vert+\aleph+\aleph_0)^{+}$ in $Add(M)$ is free.
As an application, we show that the class of pure-projective $R$-modules is categorical in some (all) big cardinal if and only if $R\cong M_n(D)$ where $D$ is a division [4/5 of https://arxiv.org/abs/2502.19641v1]
As an application, we show that the class of pure-projective $R$-modules is categorical in some (all) big cardinal if and only if $R\cong M_n(D)$ where $D$ is a division [4/5 of https://arxiv.org/abs/2502.19641v1]
"Sib To Sib"
Ft. Kurohebi & ALEPH_0 (my children!!!)
Ft. Kurohebi & ALEPH_0 (my children!!!)
May 15, 2025 at 2:52 AM
"Sib To Sib"
Ft. Kurohebi & ALEPH_0 (my children!!!)
Ft. Kurohebi & ALEPH_0 (my children!!!)
Honestly, I have seen just as much debate about whether ℤ⁺ contains 0 as I have about N.
I'll leave ω and aleph_0 to the set theorists.
I'll leave ω and aleph_0 to the set theorists.
November 20, 2024 at 2:41 AM
Honestly, I have seen just as much debate about whether ℤ⁺ contains 0 as I have about N.
I'll leave ω and aleph_0 to the set theorists.
I'll leave ω and aleph_0 to the set theorists.
> A little game of... taking care of you and that plant. A win for both. ...A win for you.
> You matter.
> Have a nice day.
[ SIGN — ALEPH_0 (_NULLITY) ]
> You matter.
> Have a nice day.
[ SIGN — ALEPH_0 (_NULLITY) ]
July 22, 2025 at 3:06 AM
> A little game of... taking care of you and that plant. A win for both. ...A win for you.
> You matter.
> Have a nice day.
[ SIGN — ALEPH_0 (_NULLITY) ]
> You matter.
> Have a nice day.
[ SIGN — ALEPH_0 (_NULLITY) ]
space $X$ with $\dim X = n \geq 0$ we have $1 \leq \mathrm{Sn}^{M_{dim}}X \leq n+1$;
(b) for any countable-dimensional metrizable space $Y$ we have $1 \leq \mathrm{Sn}^{M_{dim}}Y \leq \aleph_0$,
where $ M_{dim}$ is the [3/4 of https://arxiv.org/abs/2502.16354v1]
(b) for any countable-dimensional metrizable space $Y$ we have $1 \leq \mathrm{Sn}^{M_{dim}}Y \leq \aleph_0$,
where $ M_{dim}$ is the [3/4 of https://arxiv.org/abs/2502.16354v1]
February 25, 2025 at 6:03 AM
space $X$ with $\dim X = n \geq 0$ we have $1 \leq \mathrm{Sn}^{M_{dim}}X \leq n+1$;
(b) for any countable-dimensional metrizable space $Y$ we have $1 \leq \mathrm{Sn}^{M_{dim}}Y \leq \aleph_0$,
where $ M_{dim}$ is the [3/4 of https://arxiv.org/abs/2502.16354v1]
(b) for any countable-dimensional metrizable space $Y$ we have $1 \leq \mathrm{Sn}^{M_{dim}}Y \leq \aleph_0$,
where $ M_{dim}$ is the [3/4 of https://arxiv.org/abs/2502.16354v1]
Nathan Bowler, Henri Ortm\"uller: $\mathcal{R}(K_{\aleph_0}, \hat{K}_{2,3})$ is a win for Player 1 https://arxiv.org/abs/2512.03664 https://arxiv.org/pdf/2512.03664 https://arxiv.org/html/2512.03664
December 4, 2025 at 6:37 AM
Nathan Bowler, Henri Ortm\"uller: $\mathcal{R}(K_{\aleph_0}, \hat{K}_{2,3})$ is a win for Player 1 https://arxiv.org/abs/2512.03664 https://arxiv.org/pdf/2512.03664 https://arxiv.org/html/2512.03664
Writing at the start of my thesis "let κ be a strong limit cardinal with cofinalty > aleph_0" and then going back and updating the bound every time I need to take a larger colimit
March 25, 2024 at 1:08 AM
Writing at the start of my thesis "let κ be a strong limit cardinal with cofinalty > aleph_0" and then going back and updating the bound every time I need to take a larger colimit
2^aleph_0 wrongs make a right but it’s an open question if aleph_1 wrongs make a right; obviously the continuum hypothesis implies it so it’s consistent and independent of the theory of real numbers
May 10, 2025 at 12:25 PM
2^aleph_0 wrongs make a right but it’s an open question if aleph_1 wrongs make a right; obviously the continuum hypothesis implies it so it’s consistent and independent of the theory of real numbers
Metroidvanias at a continuum price?
Diabolical
Diabolical
September 3, 2025 at 9:14 PM
Metroidvanias at a continuum price?
Diabolical
Diabolical
Calor \Aleph_0
February 17, 2025 at 6:04 PM
Calor \Aleph_0