Question

How do I proceed to prove $\tau<\alpha\cdot\beta\iff \tau=\alpha\cdot\eta+\zeta$ for a unique $\eta<\beta,\zeta<\alpha$?

score 66 · Answer 1 · 2026-04-15 14:03:50

66

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How do I proceed to prove $\tau<\alpha\cdot\beta\iff \tau=\alpha\cdot\eta+\zeta$ for a unique $\eta<\beta,\zeta<\alpha$?

Published on 15 Apr 2026 - 14:03

score 47 · Answer 2 · 2026-04-16 09:20:59

47

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Let $f$ be a mapping, $\beta$ be an ordinal, $X=\{\alpha\mid f(\alpha)\le \beta\}$, and $\gamma=\sup X$. Is $\gamma\in X$?

Published on 16 Apr 2026 - 9:20

score 416 · Answer 3 · 2026-04-09 01:49:44

416

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Is $\omega + \omega^2 = \omega^2$ true?

Published on 09 Apr 2026 - 1:49

score 229 · Answer 4 · 2026-05-10 12:32:24

229

Views

Could there be an $\omega_1^{CK}$th hyperoperation?

Published on 10 May 2026 - 12:32

score 119 · Answer 5 · 2026-04-08 17:18:38

119

Views

Compositions of ordinal numbers

Published on 08 Apr 2026 - 17:18

score 167 · Answer 6 · 2026-04-12 18:52:53

167

Views

Burali-Forti paradox for cardinals

Published on 12 Apr 2026 - 18:52

score 199 · Answer 7 · 2026-04-16 13:30:05

199

Views

Let $\alpha,\gamma$ be ordinals such that $0<\alpha\le\gamma$. Then there is a greatest ordinal $\beta$ such that $\alpha\cdot\beta\le\gamma$

Published on 16 Apr 2026 - 13:30

score 146 · Answer 8 · 2026-04-16 07:55:19

146

Views

Let $\gamma,\alpha\neq 0$ be ordinals. Then there exists a unique ordinal $\beta$ and a unique $\rho<\alpha$ such that $\gamma=\alpha\cdot\beta+\rho$

Published on 16 Apr 2026 - 7:55

score 177 · Answer 9 · 2026-04-16 22:47:27

Every ordinal $\alpha>0$ can be expressed uniquely as $\alpha=\omega^{\beta_1}\cdot k_1+\omega^{\beta_2}\cdot k_2+\cdots+\omega^{\beta_n}\cdot k_n$

score 495 · Answer 10 · 2026-05-10 11:39:50

495

Views

Conway Notation for Large Countable Ordinals

Published on 10 May 2026 - 11:39

score 548 · Answer 11 · 2026-04-15 10:54:18

548

Views

Why doesn't ZFC "know about" all of the ordinals?

Published on 15 Apr 2026 - 10:54

score 344 · Answer 12 · 2026-04-12 16:40:23

344

Views

A weak Goodstein sequence will eventually terminate

Published on 12 Apr 2026 - 16:40

score 692 · Answer 13 · 2026-05-10 16:01:06

692

Views

Showing that $[0, \omega_1]$ is compact.

Published on 10 May 2026 - 16:01

score 365 · Answer 14 · 2026-04-11 03:46:58

365

Views

Proof that an ordinal number does not contain itself

Published on 11 Apr 2026 - 3:46

score 82 · Answer 15 · 2026-04-13 00:32:15

82

Views

Is it true that for any cardinal $\kappa$ that in injects into $\aleph_{\omega}$, $(\kappa)^{+}$ injects into $\aleph_{\omega}$?

Published on 13 Apr 2026 - 0:32

List Question

How do I proceed to prove $\tau<\alpha\cdot\beta\iff \tau=\alpha\cdot\eta+\zeta$ for a unique $\eta<\beta,\zeta<\alpha$?

Let $f$ be a mapping, $\beta$ be an ordinal, $X=\{\alpha\mid f(\alpha)\le \beta\}$, and $\gamma=\sup X$. Is $\gamma\in X$?

Is $\omega + \omega^2 = \omega^2$ true?

Could there be an $\omega_1^{CK}$th hyperoperation?

Compositions of ordinal numbers

Burali-Forti paradox for cardinals

Let $\alpha,\gamma$ be ordinals such that $0<\alpha\le\gamma$. Then there is a greatest ordinal $\beta$ such that $\alpha\cdot\beta\le\gamma$

Let $\gamma,\alpha\neq 0$ be ordinals. Then there exists a unique ordinal $\beta$ and a unique $\rho<\alpha$ such that $\gamma=\alpha\cdot\beta+\rho$

Every ordinal $\alpha>0$ can be expressed uniquely as $\alpha=\omega^{\beta_1}\cdot k_1+\omega^{\beta_2}\cdot k_2+\cdots+\omega^{\beta_n}\cdot k_n$

Conway Notation for Large Countable Ordinals

Why doesn't ZFC "know about" all of the ordinals?

A weak Goodstein sequence will eventually terminate

Showing that $[0, \omega_1]$ is compact.

Proof that an ordinal number does not contain itself

Is it true that for any cardinal $\kappa$ that in injects into $\aleph_{\omega}$, $(\kappa)^{+}$ injects into $\aleph_{\omega}$?

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