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 63 · Answer 1 · 2018-11-08 03:06:56

63

Views

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 08 Nov 2018 - 3:06

score 44 · Answer 2 · 2018-11-08 10:03:20

44

Views

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 08 Nov 2018 - 10:03

score 413 · Answer 3 · 2018-11-10 22:24:55

413

Views

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

Published on 10 Nov 2018 - 22:24

score 223 · Answer 4 · 2026-03-25 06:00:18

223

Views

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

Published on 25 Mar 2026 - 6:00

score 116 · Answer 5 · 2018-11-15 22:37:41

116

Views

Compositions of ordinal numbers

Published on 15 Nov 2018 - 22:37

score 164 · Answer 6 · 2018-11-17 09:21:37

164

Views

Burali-Forti paradox for cardinals

Published on 17 Nov 2018 - 9:21

score 196 · Answer 7 · 2018-11-18 05:58:21

196

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 18 Nov 2018 - 5:58

score 143 · Answer 8 · 2018-11-18 10:49:35

143

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 18 Nov 2018 - 10:49

score 174 · Answer 9 · 2018-11-19 12:22:21

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 490 · Answer 10 · 2026-03-25 04:07:22

490

Views

Conway Notation for Large Countable Ordinals

Published on 25 Mar 2026 - 4:07

score 545 · Answer 11 · 2018-11-21 21:18:39

545

Views

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

Published on 21 Nov 2018 - 21:18

score 341 · Answer 12 · 2018-11-22 05:19:37

341

Views

A weak Goodstein sequence will eventually terminate

Published on 22 Nov 2018 - 5:19

score 688 · Answer 13 · 2026-03-25 22:10:03

688

Views

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

Published on 25 Mar 2026 - 22:10

score 362 · Answer 14 · 2018-11-23 04:07:54

362

Views

Proof that an ordinal number does not contain itself

Published on 23 Nov 2018 - 4:07

score 79 · Answer 15 · 2018-11-24 15:07:19

79

Views

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

Published on 24 Nov 2018 - 15:07

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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