在 Hacker News 上看到選擇公理

沒想到會在 Hacker News 的首頁上看到這麼硬核的主題,選擇公理 (Axiom of choice,通常縮寫成 AC):「What is the Axiom of Choice?」,對應的討論在「What is the Axiom of Choice? (jaydaigle.net)」。

出自「xkcd: Set Theory

應該是大一教集合論的時候學到的,算是一個非常重要的公設,雖然的確有些數學系統是可以假定 AC 不成立,但用起來會不太好用,主要是因為「對於集合 S,取出任意一個元素」這類用法太常出現,在沒有 AC 的情況下這件事情就不一定能操作了...

我們目前常用的數學一般是建立在 Zermelo-Fraenkel Set Theory (ZF) 這個公理系統加上 AC,簡寫變成 ZFC。而 AC 在集合論常常會被拿出來說明,主要還是因為在歷史上花了不少力氣才證明 ZF 與 AC 的相對協調性 (ZF 與 AC 不衝突),以及 ZF 與 AC 獨立性 (ZF 無法推導出 AC)。

有了 AC 後就會再解釋連續統假設 (Continuum hypothesis,簡稱 CH),也就是 \mathbb{N}2^{\mathbb{N}} 之間存不存在一個集合 S 使得 |\mathbb{N}| < |S| < |2^{\mathbb{N}}|

然後再打臉一次,說明 ZFC 與 CH 的協調性 (ZFC 與 CH 不衝突),與獨立性 (ZFC 無法推導出 CH)。

當時學的時候的確是很頭痛,不過現在回頭看倒是覺得很有趣:在數學上你可以證明「某個敘述無法被證明」,這點應該是以前沒想過的...

T-Mobile 提供國際漫遊單一費率...

T-Mobile 將要推出國際漫遊單一費率的資費方案:「T-Mobile to offer free unlimited international data, texts」。

引用裡面的文字:

T-Mobile's latest shake-up puts international roaming rates in its sights, with the carrier eliminating fees for data and text messages entirely in more than 100 countries. It will also simplify the calling rates, charging a fixed rate of 20 cents per minute.

語音的部份,每分鐘新台幣 6 元?

不過 unlimited international data 是有速度限制的:

Chief Marketing Officer Mike Sievert said the average speed customers would get would be around 128 kilobits a second.

如果要速度,多付錢可以解決問題:

T-Mobile hopes to make some of its money back with "speed packs" that customers can purchase on the fly to boost their connection speed temporarily. For $15, a customer gets a single day's worth of high-speed data up to 100 megabytes, while $20 gets one week at 200 megabytes and $50 gets two weeks and 500 megabytes. The speeds are more accustomed to T-Mobile's HSPA+ networks, since international roaming with LTE isn't broadly available.

在官方網站的「Simple Choice International Plans | Unlimited Data & Text | T-Mobile」上面寫的更清楚:

Only T-Mobile’s network can give you unlimited everything for everyone. Other carriers may even make you share minutes, messages, and data between you and your family. With the Simple Choice Plan, each line comes with unlimited talk, text, and data while on our home network—and starting October 20, unlimited data and text in over 100 countries at no extra charge. That means no overages just about anywhere you go. Plus, there's no annual service contract and it's easy to add additional lines.

現有用戶在漫遊時直接受益...

不知道外國人辦這個划不划算,尤其是常常跑去國外的...