The Axiom of Choice (AC) has very bizarre consequences. Doubling a ball is one, but a paradox involving an infinity of prisoners guessing their hats’ colors is a much simpler application where AC gives an absurd result.
But a better example for your question is the Axiom of Determinacy (AD) which describes a seemingly trivial fact about simple 2-person win/loss games. But AD cannot apply to all topological games of length ω unless AC is false.
I was surprised to see the fraction ↉ available in Unicode. Is that the only zero-numerator fraction that has a glyph? What’s so special about 3? Of course there are lots of one-numerator fractions available: ⅟[sub]29[/sub], ⅟[sub]42[/sub], etc.
Or do you want to write numbers like this?: ❶ ❷ ❸ ❹ ❺ ❻ ❼ ❽ ❾ ❿ Or this?: ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ⑪ ⑫ ⑬ ⑭ ⑮ ⑯ ⑰ ⑱ ⑲ ⑳
∜81 = ∛27
It seems strange to me, but some Unicode characters are colored: 
When you want to to snip text in a quote box, why not use a snipper: ✀
And of course the following glyph pair is shorthand for a certain man with short orange fingers who likes to play golf: 
(Please tell me if some of these characters don’t render on your machine.)
septimus, I seem to see everything but there are references to colors in your post and I only see B&W.
A followup “color test” post with the color words next to the symbols could be useful.
purple umbrella with blue raindrops: 
white coffee cup with brown coffee: 
gold star: 
yellow hand: 
scissors, although it only shows numbers in a box for me on Firefox 66 + straight dope theme: ✀
yellow flag-stick with a red flag and green grass: 
Here’s a larger (but non-exhaustive) collection of colored Unicode glyphs.
- Zodiac symbols with indigo background.
- Blue Handicapped, Yellow Lightning?
- Little red or yellow
- Single non-B/W color
- Two or more non-B/W colors
- Zodiac symbols with indigo background.
- Blue Handicapped, Yellow Lightning?
- Little red or yellow
- Single non-B/W color
- Two or more non-B/W colors
I see all these colored on my default Windows Firefox environment, trying both the default font and Courier.
All are colored with Chrome also.
The following aren’t colored but might come in handy when transcribing some weird language.
ꀀ ꀁ ꀂ ꀃ ꀄ ꀅ ꀆ ꀇ ꀈ ꀉ ꀊ ꀋ ꀌ ꀍ ꀎ ꀏ ꀐ ꀑ ꀒ ꀓ ꀔ ꀕ ꀖ ꀗ ꀘ ꀙ ꀚ ꀛ ꀜ ꀝ ꀞ ꀟ ꀠ ꀡ ꀢ ꀣ ꀤ ꀥ ꀦ ꀧ
ꀨ ꀩ ꀪ ꀫ ꀬ ꀭ ꀮ ꀯ ꀰ ꀱ ꀲ ꀳ ꀴ ꀵ ꀶ ꀷ ꀸ ꀹ ꀺ ꀻ ꀼ ꀽ ꀾ ꀿ ꁀ ꁁ ꁂ ꁃ ꁄ ꁅ ꁆ ꁇ ꁈ ꁉ ꁊ ꁋ ꁌ ꁍ ꁎ ꁏ
ꁐ ꁑ ꁒ ꁓ ꁔ ꁕ ꁖ ꁗ ꁘ ꁙ ꁚ ꁛ ꁜ ꁝ ꁞ ꁟ ꁠ ꁡ ꁢ ꁣ ꁤ ꁥ ꁦ ꁧ ꁨ ꁩ ꁪ ꁫ ꁬ ꁭ ꁮ ꁯ ꁰ ꁱ ꁲ ꁳ ꁴ ꁵ ꁶ ꁷ
ꁸ ꁹ ꁺ ꁻ ꁼ ꁽ ꁾ ꁿ ꂀ ꂁ ꂂ ꂃ ꂄ ꂅ ꂆ ꂇ ꂈ ꂉ ꂊ ꂋ ꂌ ꂍ ꂎ ꂏ ꂐ ꂑ ꂒ ꂓ ꂔ ꂕ ꂖ ꂗ ꂘ ꂙ ꂚ ꂛ ꂜ ꂝ ꂞ ꂟ
ꂠ ꂡ ꂢ ꂣ ꂤ ꂥ ꂦ ꂧ ꂨ ꂩ ꂪ ꂫ ꂬ ꂭ ꂮ ꂯ ꂰ ꂱ ꂲ ꂳ ꂴ ꂵ ꂶ ꂷ ꂸ ꂹ ꂺ ꂻ ꂼ ꂽ ꂾ ꂿ ꃀ ꃁ ꃂ ꃃ ꃄ ꃅ ꃆ ꃇ
ꃈ ꃉ ꃊ ꃋ ꃌ ꃍ ꃎ ꃏ ꃐ ꃑ ꃒ ꃓ ꃔ ꃕ ꃖ ꃗ ꃘ ꃙ ꃚ ꃛ ꃜ ꃝ ꃞ ꃟ ꃠ ꃡ ꃢ ꃣ ꃤ ꃥ ꃦ ꃧ ꃨ ꃩ ꃪ ꃫ ꃬ ꃭ ꃮ ꃯ
ꃰ ꃱ ꃲ ꃳ ꃴ ꃵ ꃶ ꃷ ꃸ ꃹ ꃺ ꃻ ꃼ ꃽ ꃾ ꃿ ꄀ ꄁ ꄂ ꄃ ꄄ ꄅ ꄆ ꄇ ꄈ ꄉ ꄊ ꄋ ꄌ ꄍ ꄎ ꄏ ꄐ ꄑ ꄒ ꄓ ꄔ ꄕ ꄖ ꄗ
ꄘ ꄙ ꄚ ꄛ ꄜ ꄝ ꄞ ꄟ ꄠ ꄡ ꄢ ꄣ ꄤ ꄥ ꄦ ꄧ ꄨ ꄩ ꄪ ꄫ ꄬ ꄭ ꄮ ꄯ ꄰ ꄱ ꄲ ꄳ ꄴ ꄵ ꄶ ꄷ ꄸ ꄹ ꄺ ꄻ ꄼ ꄽ ꄾ ꄿ
ꅀ ꅁ ꅂ ꅃ ꅄ ꅅ ꅆ ꅇ ꅈ ꅉ ꅊ ꅋ ꅌ ꅍ ꅎ ꅏ ꅐ ꅑ ꅒ ꅓ ꅔ ꅕ ꅖ ꅗ ꅘ ꅙ ꅚ ꅛ ꅜ ꅝ ꅞ ꅟ ꅠ ꅡ ꅢ ꅣ ꅤ ꅥ ꅦ ꅧ
ꅨ ꅩ ꅪ ꅫ ꅬ ꅭ ꅮ ꅯ ꅰ ꅱ ꅲ ꅳ ꅴ ꅵ ꅶ ꅷ ꅸ ꅹ ꅺ ꅻ ꅼ ꅽ ꅾ ꅿ ꆀ ꆁ ꆂ ꆃ ꆄ ꆅ ꆆ ꆇ ꆈ ꆉ ꆊ ꆋ ꆌ ꆍ ꆎ ꆏ
ꆐ ꆑ ꆒ ꆓ ꆔ ꆕ ꆖ ꆗ ꆘ ꆙ ꆚ ꆛ ꆜ ꆝ ꆞ ꆟ ꆠ ꆡ ꆢ ꆣ ꆤ ꆥ ꆦ ꆧ ꆨ ꆩ ꆪ ꆫ ꆬ ꆭ ꆮ ꆯ ꆰ ꆱ ꆲ ꆳ ꆴ ꆵ ꆶ ꆷ
ꆸ ꆹ ꆺ ꆻ ꆼ ꆽ ꆾ ꆿ ꇀ ꇁ ꇂ ꇃ ꇄ ꇅ ꇆ ꇇ ꇈ ꇉ ꇊ ꇋ ꇌ ꇍ ꇎ ꇏ ꇐ ꇑ ꇒ ꇓ ꇔ ꇕ ꇖ ꇗ ꇘ ꇙ ꇚ ꇛ ꇜ ꇝ ꇞ ꇟ
ꇠ ꇡ ꇢ ꇣ ꇤ ꇥ ꇦ ꇧ ꇨ ꇩ ꇪ ꇫ ꇬ ꇭ ꇮ ꇯ ꇰ ꇱ ꇲ ꇳ ꇴ ꇵ ꇶ ꇷ ꇸ ꇹ ꇺ ꇻ ꇼ ꇽ ꇾ ꇿ ꈀ ꈁ ꈂ ꈃ ꈄ ꈅ ꈆ ꈇ
ꈈ ꈉ ꈊ ꈋ ꈌ ꈍ ꈎ ꈏ ꈐ ꈑ ꈒ ꈓ ꈔ ꈕ ꈖ ꈗ ꈘ ꈙ ꈚ ꈛ ꈜ ꈝ ꈞ ꈟ ꈠ ꈡ ꈢ ꈣ ꈤ ꈥ ꈦ ꈧ ꈨ ꈩ ꈪ ꈫ ꈬ ꈭ ꈮ ꈯ
ꈰ ꈱ ꈲ ꈳ ꈴ ꈵ ꈶ ꈷ ꈸ ꈹ ꈺ ꈻ ꈼ ꈽ ꈾ ꈿ ꉀ ꉁ ꉂ ꉃ ꉄ ꉅ ꉆ ꉇ ꉈ ꉉ ꉊ ꉋ ꉌ ꉍ ꉎ ꉏ ꉐ ꉑ ꉒ ꉓ ꉔ ꉕ ꉖ ꉗ
ꉘ ꉙ ꉚ ꉛ ꉜ ꉝ ꉞ ꉟ ꉠ ꉡ ꉢ ꉣ ꉤ ꉥ ꉦ ꉧ ꉨ ꉩ ꉪ ꉫ ꉬ ꉭ ꉮ ꉯ ꉰ ꉱ ꉲ ꉳ ꉴ ꉵ ꉶ ꉷ ꉸ ꉹ ꉺ ꉻ ꉼ ꉽ ꉾ ꉿ
ꊀ ꊁ ꊂ ꊃ ꊄ ꊅ ꊆ ꊇ ꊈ ꊉ ꊊ ꊋ ꊌ ꊍ ꊎ ꊏ ꊐ ꊑ ꊒ ꊓ ꊔ ꊕ ꊖ ꊗ ꊘ ꊙ ꊚ ꊛ ꊜ ꊝ ꊞ ꊟ ꊠ ꊡ ꊢ ꊣ ꊤ ꊥ ꊦ ꊧ
ꊨ ꊩ ꊪ ꊫ ꊬ ꊭ ꊮ ꊯ ꊰ ꊱ ꊲ ꊳ ꊴ ꊵ ꊶ ꊷ ꊸ ꊹ ꊺ ꊻ ꊼ ꊽ ꊾ ꊿ ꋀ ꋁ ꋂ ꋃ ꋄ ꋅ ꋆ ꋇ ꋈ ꋉ ꋊ ꋋ ꋌ ꋍ ꋎ ꋏ
ꋐ ꋑ ꋒ ꋓ ꋔ ꋕ ꋖ ꋗ ꋘ ꋙ ꋚ ꋛ ꋜ ꋝ ꋞ ꋟ ꋠ ꋡ ꋢ ꋣ ꋤ ꋥ ꋦ ꋧ ꋨ ꋩ ꋪ ꋫ ꋬ ꋭ ꋮ ꋯ ꋰ ꋱ ꋲ ꋳ ꋴ ꋵ ꋶ ꋷ
ꋸ ꋹ ꋺ ꋻ ꋼ ꋽ ꋾ ꋿ ꌀ ꌁ ꌂ ꌃ ꌄ ꌅ ꌆ ꌇ ꌈ ꌉ ꌊ ꌋ ꌌ ꌍ ꌎ ꌏ ꌐ ꌑ ꌒ ꌓ ꌔ ꌕ ꌖ ꌗ ꌘ ꌙ ꌚ ꌛ ꌜ ꌝ ꌞ ꌟ
ꌠ ꌡ ꌢ ꌣ ꌤ ꌥ ꌦ ꌧ ꌨ ꌩ ꌪ ꌫ ꌬ ꌭ ꌮ ꌯ ꌰ ꌱ ꌲ ꌳ ꌴ ꌵ ꌶ ꌷ ꌸ ꌹ ꌺ ꌻ ꌼ ꌽ ꌾ ꌿ ꍀ ꍁ ꍂ ꍃ ꍄ ꍅ ꍆ ꍇ
ꍈ ꍉ ꍊ ꍋ ꍌ ꍍ ꍎ ꍏ ꍐ ꍑ ꍒ ꍓ ꍔ ꍕ ꍖ ꍗ ꍘ ꍙ ꍚ ꍛ ꍜ ꍝ ꍞ ꍟ ꍠ ꍡ ꍢ ꍣ ꍤ ꍥ ꍦ ꍧ ꍨ ꍩ ꍪ ꍫ ꍬ ꍭ ꍮ ꍯ
ꍰ ꍱ ꍲ ꍳ ꍴ ꍵ ꍶ ꍷ ꍸ ꍹ ꍺ ꍻ ꍼ ꍽ ꍾ ꍿ ꎀ ꎁ ꎂ ꎃ ꎄ ꎅ ꎆ ꎇ ꎈ ꎉ ꎊ ꎋ ꎌ ꎍ ꎎ ꎏ ꎐ ꎑ ꎒ ꎓ ꎔ ꎕ ꎖ ꎗ
ꎘ ꎙ ꎚ ꎛ ꎜ ꎝ ꎞ ꎟ ꎠ ꎡ ꎢ ꎣ ꎤ ꎥ ꎦ ꎧ ꎨ ꎩ ꎪ ꎫ ꎬ ꎭ ꎮ ꎯ ꎰ ꎱ ꎲ ꎳ ꎴ ꎵ ꎶ ꎷ ꎸ ꎹ ꎺ ꎻ ꎼ ꎽ ꎾ ꎿ
ꏀ ꏁ ꏂ ꏃ ꏄ ꏅ ꏆ ꏇ ꏈ ꏉ ꏊ ꏋ ꏌ ꏍ ꏎ ꏏ ꏐ ꏑ ꏒ ꏓ ꏔ ꏕ ꏖ ꏗ ꏘ ꏙ ꏚ ꏛ ꏜ ꏝ ꏞ ꏟ ꏠ ꏡ ꏢ ꏣ ꏤ ꏥ ꏦ ꏧ
ꏨ ꏩ ꏪ ꏫ ꏬ ꏭ ꏮ ꏯ ꏰ ꏱ ꏲ ꏳ ꏴ ꏵ ꏶ ꏷ ꏸ ꏹ ꏺ ꏻ ꏼ ꏽ ꏾ ꏿ ꐀ ꐁ ꐂ ꐃ ꐄ ꐅ ꐆ ꐇ ꐈ ꐉ ꐊ ꐋ ꐌ ꐍ ꐎ ꐏ
ꐐ ꐑ ꐒ ꐓ ꐔ ꐕ ꐖ ꐗ ꐘ ꐙ ꐚ ꐛ ꐜ ꐝ ꐞ ꐟ ꐠ ꐡ ꐢ ꐣ ꐤ ꐥ ꐦ ꐧ ꐨ ꐩ ꐪ ꐫ ꐬ ꐭ ꐮ ꐯ ꐰ ꐱ ꐲ ꐳ ꐴ ꐵ ꐶ ꐷ
ꐸ ꐹ ꐺ ꐻ ꐼ ꐽ ꐾ ꐿ ꑀ ꑁ ꑂ ꑃ ꑄ ꑅ ꑆ ꑇ ꑈ ꑉ ꑊ ꑋ ꑌ ꑍ ꑎ ꑏ ꑐ ꑑ ꑒ ꑓ ꑔ ꑕ ꑖ ꑗ ꑘ ꑙ ꑚ ꑛ ꑜ ꑝ ꑞ ꑟ
ꑠ ꑡ ꑢ ꑣ ꑤ ꑥ ꑦ ꑧ ꑨ ꑩ ꑪ ꑫ ꑬ ꑭ ꑮ ꑯ ꑰ ꑱ ꑲ ꑳ ꑴ ꑵ ꑶ ꑷ ꑸ ꑹ ꑺ ꑻ ꑼ ꑽ ꑾ ꑿ ꒀ ꒁ ꒂ ꒃ ꒄ ꒅ ꒆ ꒇ
Those are after Chinese, but before some upside-down Roman letters. A bit later is an even weirder alphabet:
ꔀ ꔁ ꔂ ꔃ ꔄ ꔅ ꔆ ꔇ ꔈ ꔉ ꔊ ꔋ ꔌ ꔍ ꔎ ꔏ ꔐ ꔑ ꔒ ꔓ ꔔ ꔕ ꔖ ꔗ ꔘ ꔙ ꔚ ꔛ ꔜ ꔝ ꔞ ꔟ ꔠ ꔡ ꔢ ꔣ ꔤ ꔥ ꔦ ꔧ
ꔨ ꔩ ꔪ ꔫ ꔬ ꔭ ꔮ ꔯ ꔰ ꔱ ꔲ ꔳ ꔴ ꔵ ꔶ ꔷ ꔸ ꔹ ꔺ ꔻ ꔼ ꔽ ꔾ ꔿ ꕀ ꕁ ꕂ ꕃ ꕄ ꕅ ꕆ ꕇ ꕈ ꕉ ꕊ ꕋ ꕌ ꕍ ꕎ ꕏ
ꕐ ꕑ ꕒ ꕓ ꕔ ꕕ ꕖ ꕗ ꕘ ꕙ ꕚ ꕛ ꕜ ꕝ ꕞ ꕟ ꕠ ꕡ ꕢ ꕣ ꕤ ꕥ ꕦ ꕧ ꕨ ꕩ ꕪ ꕫ ꕬ ꕭ ꕮ ꕯ ꕰ ꕱ ꕲ ꕳ ꕴ ꕵ ꕶ ꕷ
ꕸ ꕹ ꕺ ꕻ ꕼ ꕽ ꕾ ꕿ ꖀ ꖁ ꖂ ꖃ ꖄ ꖅ ꖆ ꖇ ꖈ ꖉ ꖊ ꖋ ꖌ ꖍ ꖎ ꖏ ꖐ ꖑ ꖒ ꖓ ꖔ ꖕ ꖖ ꖗ ꖘ ꖙ ꖚ ꖛ ꖜ ꖝ ꖞ ꖟ
ꖠ ꖡ ꖢ ꖣ ꖤ ꖥ ꖦ ꖧ ꖨ ꖩ ꖪ ꖫ ꖬ ꖭ ꖮ ꖯ ꖰ ꖱ ꖲ ꖳ ꖴ ꖵ ꖶ ꖷ ꖸ ꖹ ꖺ ꖻ ꖼ ꖽ ꖾ ꖿ ꗀ ꗁ ꗂ ꗃ ꗄ ꗅ ꗆ ꗇ
ꗈ ꗉ ꗊ ꗋ ꗌ ꗍ ꗎ ꗏ ꗐ ꗑ ꗒ ꗓ ꗔ ꗕ ꗖ ꗗ ꗘ ꗙ ꗚ ꗛ ꗜ ꗝ ꗞ ꗟ ꗠ ꗡ ꗢ ꗣ ꗤ ꗥ ꗦ ꗧ ꗨ ꗩ ꗪ ꗫ ꗬ ꗭ ꗮ ꗯ
ꗰ ꗱ ꗲ ꗳ ꗴ ꗵ ꗶ ꗷ ꗸ ꗹ ꗺ ꗻ ꗼ ꗽ ꗾ ꗿ ꘀ ꘁ ꘂ ꘃ ꘄ ꘅ ꘆ ꘇ ꘈ ꘉ ꘊ ꘋ ꘌ ꘍ ꘎ ꘏ ꘐ ꘑ ꘒ ꘓ ꘔ ꘕ ꘖ ꘗ
ꘘ ꘙ ꘚ ꘛ ꘜ ꘝ ꘞ ꘟ ꘠ ꘡ ꘢ ꘣ ꘤ ꘥ ꘦ ꘧ ꘨ ꘩ ꘪ ꘫ
Nope, symbols but no colors. (Classic Opera.)
Checking good old IE, no colors and really terrible formatting.
Let’s see, do I still have Netscape Navigator still installed … ? No. But I did have Mosaic around not long ago.
The former chunk of characters are from of Yi script, an ancient language of the Yi ethnic group around the southern Sichuan province, China (Nanzhong in ancient times).
The latter characters are from the language of the Vai ethnic group in Liberia, one of the few African scripts that is not based on the latin or arabic alphabet.
~Max
All of those sample characters appear to render properly for me (Mac Firefox, 3.7.3 theme), except for the snipper, which is a box containing the numbers 2 7 0 0.
Of note, most of the colored ones, including the ones septimus described as having only one color, have multiple shades of color. For instance, the yellow star is more yellow at the bottom, but shades to more white at the top, and the zodiac symbols have a light-colored highlight at the top.
**Max S **Very impressive and thorough, thanks. If it’s all the same to everyone else, I’ll just post links. I think I finally have that down.
That’s an excellent example–thanks. I was aware of some of the “paradoxical” results of the AoC, though I hadn’t known of the prisoner’s game solution. Still, those could be argued to be features, not bugs. But the AoD seems like it’s also something we want to be true, and almost as “obvious” as the AoC itself.
Personally, I think that the Axiom of Choice is completely unobvious, and have no idea why anyone would take it seriously.
Mathematician: And now, I will choose one element from each of these sets…
Me: OK, do it.
Mathematician: Done. And now…
Me: So which one did you choose?
Mathematician: Um, I don’t know…
Me: So what do you mean that you chose one, then? What kind of choice is it when you don’t even know what you’ve chosen?
As for being able to prove a lot of things using the AoC, well, I can prove a lot of things using 1 = 0. That doesn’t mean that it makes a good axiom.
Well, 1=0 is inconsistent with the rest of mathematics. AoC+ and AoC- are both consistent with most math that we care about.
To me, it feels “obvious” because for any specific example, it seems easy to pick a rule. Sets of natural numbers? Pick the smallest absolute value, favoring positive. Closed intervals of reals? Ok, smallest absolute value doesn’t work–instead pick the number with the shortest binary representation. Closed/open regions of complex numbers? Same deal, just come up with a real/imaginary tiebreaking rule. Fractal sets, like Cantor dust or a Sierpinski gasket? There’s gonna be some base element that everything’s repeated from; pick that. Etc.
Then again, we could have a set of, say, incomputable numbers. How can we specify a member if we can’t even compute it? But maybe the problem is more declaring the existence of the set in the first place. Maybe we should only allow sets when there is a “pick an element” function.
Sorry, TLDR.
That said, congrats on deciphering the Voynich Manuscript!