超有限論，一種否定無限的數學思想

失去無限性我們能獲得什么？

格雷戈里·巴伯

2026年4月29日

超有限論，一種否定無限的哲學，長期以來被視為數學異端邪說。但它也在數學及其他領域產生了新的見解。

多倫·澤爾伯格是一位數學家，他認為萬物終有盡頭。正如我們是有限的存在，自然也有其邊界——因此，數字亦是如此。看看窗外，在其他人眼中，現實是一片連續不斷的廣闊，一刻接一刻地不可阻擋地向前流動，而澤爾伯格卻認為并非如此。（在新標簽頁中打開）他看到的是一個運轉的宇宙，一個獨立的機器。在他周圍世界流暢的運動中，他捕捉到了翻頁書的細微模糊。

在澤爾伯格看來，相信無窮大就像相信上帝一樣。這是一個誘人的想法，它迎合了我們的直覺，幫助我們理解各種各樣的現象。但問題在于，我們無法真正觀測到無窮大，因此也無法真正定義它。方程式定義了延伸到黑板之外的線條，但它們延伸到哪里呢？證明中充斥著意味深長的省略號。澤爾伯格——羅格斯大學的資深教授，組合數學領域的著名人物——認為，這些方程式和證明既“丑陋不堪”，又是錯誤的。他用沙啞的聲音喘息著，仿佛已經筋疲力盡，說道：“這完全是胡說八道�！�

他認為，從實際角度來看，無窮大是可以剔除的。“你其實并不需要它�！崩�，數學家可以構建一種不涉及無窮大的微積分形式，徹底排除無窮小極限。曲線看起來或許光滑，但其中隱藏著細微的粗糙；計算機在有限的數字范圍內也能很好地處理數學運算。（澤爾伯格將他自己的計算機——他稱之為“Shalosh B. Ekhad”——列為論文的合作者。）澤爾伯格說，剔除無窮大之后，唯一失去的只是“根本不值得做的”數學。

大多數數學家會持相反的觀點——認為澤爾伯格才是胡說八道。這不僅是因為無窮的概念對于我們描述宇宙非常有用且自然，還因為將數集（例如整數集）視為實際的、無限的對象，正是數學的核心所在，它根植于數學最基本的規則和假設之中。

至少，即便數學家們不愿將無窮視為一個實際存在的實體，他們也承認數列、形狀和其他數學對象具有無限增長的潛力。理論上，兩條平行線可以無限延伸；數軸的末端總可以添加另一個數字。

多倫·澤爾伯格或許是主張將無窮從數學中剔除的最積極倡導者。“無窮可能存在，也可能不存在；上帝可能存在，也可能不存在，”他說�！暗跀祵W中，無論是無窮還是上帝，都不應該有任何容身之地�！�

承蒙多倫·澤爾伯格惠允

澤爾伯格對此持不同意見。對他而言，重要的不是某件事在理論上是否可能，而是它在實踐中是否可行。這意味著，不僅無窮大值得懷疑，極其巨大的數字也同樣如此。以“斯奎斯數”為例，$latex e^{e^{e^{79}}}$。這是一個極其巨大的數字，至今無人能夠將其寫成十進制形式。那么，我們究竟能對它做出怎樣的判斷呢？它是整數嗎？它是質數嗎？我們能在自然界中找到這樣的數字嗎？我們有可能把它寫出來嗎？或許，它根本就不是一個數字。

這就引出了一個顯而易見的問題，比如我們究竟會在哪里找到終點。澤爾伯格無法回答。事實上，沒有人能回答。這正是許多人對他的哲學——被稱為“超有限論”——嗤之以鼻的首要原因�！爱斈愕谝淮蜗騽e人介紹超有限論時，聽起來就像是江湖騙術——比如‘我認為存在一個最大數’之類的，”賈斯汀·克拉克-多恩說道。他是哥倫比亞大學的一位哲學家

“很多數學家都覺得這個提議荒謬至極，”喬爾·戴維·漢金斯說道。（在新標簽頁中打開）他是圣母大學的一位集合論學家。超有限論在數學學會的晚宴上可不是個好話題。研究它的人寥寥無幾（或許可以說少得可憐）。像澤爾伯格這樣愿意公開表達自己觀點的正式會員更是鳳毛麟角。這不僅是因為超有限論本身就帶有反主流色彩，更是因為它提倡一種本質上更小的數學，在這種數學中，某些重要的問題將無法再被提出。

然而，這確實給漢金斯等人提供了許多值得思考的問題。從某種角度來看，超有限論可以被視為一種更現實的數學。它更好地反映了人類創造和驗證能力的局限性；它甚至可能更好地反映了物理宇宙。雖然我們可能傾向于認為空間和時間是永恒擴張且可分割的，但超有限論者會認為，這些假設正日益受到科學的質疑——正如澤爾伯格可能會說的那樣，科學甚至讓對上帝的懷疑也受到了質疑。

“我們所描述的世界必須從頭到尾都誠實可靠，”克拉克-多恩說道。他于2025年4月召集了一批專家，罕見地探討了超有限論的思想。“如果事物的數量可能只有有限個，那么我們最好也使用一種不會一開始就假定事物數量無限多的數學�！� 在他看來，“這當然應該成為數學哲學的一部分�！�

賈斯汀·克拉克-多恩最近組織了一場會議，讓超有限論者們能夠討論和辯論他們的觀點。他認為超有限論“應該成為數學哲學的一部分”。詹妮弗·麥克唐納

然而，要想讓數學家們認真對待超有限論，首先超有限論者需要就他們所討論的內容達成共識——正如漢金斯所說，他們需要將那些聽起來像是“虛張聲勢”的論點轉化為正式的理論。數學深諳形式體系和通用框架之道，而超有限論卻缺乏這樣的結構。

逐個解決問題是一回事，而改寫數學本身的邏輯基礎則是另一回事�！拔艺J為超有限論被摒棄的原因并非人們提出了有力的反駁論點，”克拉克-多恩說，“而是人們覺得，唉，這根本沒希望�！�

這是某些極端有限論者仍在試圖解決的問題。

與此同時，澤爾伯格準備放棄數學理想，轉而接受一種本質上混亂的數學——就像世界本身一樣。與其說他是一位基礎理論家，不如說他是一位觀點家，他在自己的網站上列出了195條觀點。（在新標簽頁中打開）“如果不做這些異想天開的事，我就當不了終身教授，”他說。但他補充道，總有一天，數學家們會回過頭來看，發現這個異想天開的人，就像過去那些質疑神明和迷信的人一樣，是對的。“幸運的是，異端分子不再會被綁在火刑柱上燒死了�！�

異議數學

亞里士多德認為無限是你可以努力接近卻永遠無法企及的東西。他寫道：“分割的過程永無止境，這保證了這種活動的潛在存在，但這并不意味著無限本身獨立存在�！� 數千年來，這種“潛在”的無限概念一直占據主導地位。

但在19世紀末，格奧爾格·康托爾和其他數學家證明了無窮確實存在。康托爾的方法是將一系列數字（例如整數）視為一個完備的無窮集合。這種方法對于創建數學基礎理論——策梅洛-弗蘭克爾集合論——至關重要，而數學家們至今仍在運用這一理論。他證明，無窮是一個真實存在的對象。此外，它可以呈現多種不同的大��；通過操縱和比較這些不同的無窮，數學家們可以證明一些令人驚訝的真理，而這些真理表面上看起來似乎與無窮毫無關系。雖然很少有數學家會花費大量時間研究高階無窮，但漢金斯說：“如今，幾乎每個數學家都是一個實數主義者。”無窮的存在已成為默認的假設。

但自現代數學的這一基礎被提出以來，它就引發了激烈的爭論。原因之一是，接受關于無窮的核心假設會讓人構建出奇特的悖論：例如，可以將一個球切成五份，然后用這五份再制作五個新球，每個新球的體積都與第一個球的體積相等。

另一種反對意見更偏向哲學層面。在康托爾揭示其原理后的幾十年里，一些數學家認為，你不能簡單地斷言某種數學結構的存在——你必須通過一種思維建構的過程來證明它的存在。例如，在這種“直覺主義”哲學中，π與其說是一個具有無限不循環小數展開式的數字，不如說是一個代表生成數字的算法過程的符號。

“如果事物的數量可能只有有限的，那么我們最好也使用一種數學方法，而不是一開始就假設事物的數量是無限的。”賈斯汀·克拉克-多恩，哥倫比亞大學

但直覺主義只要求某種心理構造在理論上可行：它禁止實際的無窮大，但允許潛在的無窮大。一些數學家對此仍不滿意。他們仍然對斯基維斯數以及其他一些大到無法用文字表達的數值感到困惑。于是，他們試圖將直覺主義的思想推向極致。

“如果你在思考，在這種觀點下哪些數字會存在，那么這些數字必須是我們能夠在實踐中構建的數字，”而不僅僅是理論上構建的數字，牛津大學的一位哲學家奧弗拉·馬吉多爾說道。

20 世紀 60 年代和 70 年代，蘇聯數學家兼詩人亞歷山大·葉賽寧-沃爾平的著作使一種新的直覺主義——一種將這些實際限制銘記于心的直覺主義——逐漸形成。

紐約市立大學的一位邏輯學家指出，葉賽寧-沃爾平移民美國后，曾接待過他，是一位古怪的房客，他會在帕里克家的閣樓里整夜踱步，用他妻子心愛的陶瓷器皿當煙灰缸，同時研究一種奇怪的理論，這種理論不僅否定了潛在的無窮大，甚至否定了極其巨大的數字——那些無法在人腦中構建的數字。

亞歷山大·葉賽寧-沃爾平是一位蘇聯持不同政見者、數學家和詩人。

邏輯學家哈維·弗里德曼曾請葉賽寧-沃爾平指出一個截止點。（在新標簽頁中打開）什么才算太大？給定一個表達式 2^ n ， n取何值時，數字就停止存在了？2^ 0真的是個數字嗎？2 ^1、2 ^ 2等等，直到 2^ 100呢？葉賽寧-沃爾平逐一回答了這個問題。是的，2^ 1存在。是的，2^ 2也存在。但每次他都等待更長時間才作答。對話很快變得沒完沒了。

埃賽寧-沃爾平已經闡明了他的觀點。正如帕里克等人后來所言，數字的局限性根植于證明其存在所需的有限資源，例如時間、計算機內存或證明過程的物理長度�！按蠖鄶党邢拚撜哒J為，有限與無限之間的區別本質上是模糊的，”克拉克-多恩說道。

在埃賽寧-沃爾平理論中，一個條件可能對n成立，也可能對n + 1 成立——直到它不再成立為止。孩子不斷成長，直到有一天他們不再是孩子。我們不必指定一個具體的終點。重要的是，終點存在于其中，在某個地方。

埃賽寧-沃爾平的工作呼吁建立一種新的數學，這種數學在某種意義上能夠容忍模糊性。此后，超有限主義者們繼承了他的事業，探索如何使他那些模糊不清、近乎荒謬的數學變得嚴謹可靠。

危機控制

有一天，愛德華·納爾遜醒來后意識到，無限可能并不存在。這讓他陷入了存在論危機。

1976 年的一個早晨，普林斯頓大學數學家愛德華·納爾遜（在新標簽頁中打開）他醒來后經歷了一場信仰危機�！拔腋械揭凰查g，一種強大的存在感撲面而來，讓我意識到自己因為相信存在一個無限的數字世界而顯得傲慢自大，”幾十年后他回憶道。（在新標簽頁中打開）“把我像個嬰兒一樣留在搖籃里，只能掰著手指頭數數�！�

數學有一些基本規則，或者說公理。納爾遜知道，即使是那些使我們能夠進行簡單算術運算的最基本公理，也包含著關于無窮的假設——例如，我們總能通過給一個數加1來得到新的數。他想推翻之前的規則，構建一套完全禁止無窮的規則。如果數學完全由這些新的公理構成，它會是什么樣子呢？

事實證明，這些公理體系極其脆弱。納爾遜研究了各種排除無窮的公理體系，發現如果他用其中任何一套來嘗試進行基本的算術運算，就連證明“ a + b永遠等于b + a”這樣簡單的命題都變得不可能。像乘方這樣的基本運算也不再總是可行：你或許可以構造出數字 100，或者數字 1000，但卻無法構造出數字 100 × 1000。數學家工具箱中最強大的技巧之一——歸納法（它指出，如果你能證明某個命題對一個數成立，那么它對所有數也成立）——就此徹底失效了。

在納爾遜看來，這種弱點代表著真理的一絲曙光。他希望證明，數學家們習以為常的那些更強大的算術公理（允許無窮存在的“皮亞諾公理”）從根本上存在缺陷——它們會導致矛盾。“我相信，許多我們視為數學既定事實的東西終將被推翻，”他曾這樣說道。

然而，納爾遜未能推翻他們。2003年，他宣布自己利用較弱的公理找到了皮亞諾公理中的矛盾之處，但這一轟動性的結果很快就被駁斥了。

羅希特·帕里克的超有限主義思想已在理論計算機科學中得到應用。勞倫·弗萊什曼

納爾遜更為有限的算術——以及帕里克等人發展的相關非標準算術形式——在計算機領域確實發揮了作用。研究人員希望了解算法能夠高效地證明哪些結論，又不能證明哪些結論。這些超有限主義的數學方法已被轉化為計算效率的語言，并用于探索算法能力的極限。

在納爾遜看來，數學的本質在于“你選擇相信的真理”——你認定哪些公理是正確的。即便你選擇相信的是默認公理，這個道理依然成立。當然，作為缺乏穩固根基的異端，極端有限論者還有更多需要證明的地方。

耐心練習

2025年4月，一群形形色色的人齊聚紐約，參加哥倫比亞大學舉辦的一場關于“廢除無限”的會議。他們中有物理學家、哲學家、邏輯學家和數學家。其中既有像澤爾伯格那樣堅定的超有限論者；也有相信各種無限存在的集合論學者；還有一些純粹出于好奇的人。會議組織者克拉克-多恩回憶說，結果是“對每個人來說都是一次耐心考驗”。一般來說，哲學家們習慣于在課堂上激烈辯論，然后聚在一起喝啤酒。數學家們則不然。通常情況下，如果他們意見不合，那就意味著有人犯了嚴重的錯誤。

顯而易見的是，構建普適的超有限論理論進展緩慢，部分原因是該運動缺乏明確的動機，也缺乏確定其基本邏輯的統一方法。因此，像納爾遜那樣執著于基本規則或許并非正確的方法�！拔艺J為這是浪費時間，”帕里克告訴我，“你必須把形式主義當作雙筒望遠鏡，更關注你所看到的。如果你開始研究雙筒望遠鏡本身，你就輸了。”

其他人將現實視為一個連續的廣闊空間，一刻又一刻地不可阻擋地向前流動，而澤爾伯格則將現實視為一個滴答作響的宇宙。

澤爾伯格樂于透過（或許扭曲的）鏡子來看待事物，即便他必須在一個無窮概念依然鮮活存在的世界里這樣做。他并不希望從零開始重建一套沒有無窮概念的數學體系。他可以自上而下地進行研究。以實分析為例，實分析研究的是實數和實函數的性質。澤爾伯格稱之為“退化情形” 。（在新標簽頁中打開）離散分析（研究的是離散對象而非連續對象的行為）的原理是，你可以用一串由微�。ǖ⒎菬o窮大）數值差異構成的“離散數字項鏈”來取代連續的實數域。他說道，你可以利用這個項鏈重寫微積分和微分方程（現在稱為“差分”方程）的規則，從而消除其中哪怕是最細微的無窮大運算。他承認，這條路走得很艱難，但并非不可能，尤其是在計算機的幫助下。他認為，雖然最終的結果可能不如經典數學那樣優雅，但卻更加優美，因為它反映了他所認為的物理現實的真實面貌。

作為布魯塞爾自由大學的一位數學哲學家，讓·保羅·范·本德根對超有限論的探索并非始于數字，而是始于小學幾何。他看著數學老師在黑板上畫了一條線，這條線似乎無限延伸�！把由斓侥睦�？”他回憶起自己當時的提問。如果右邊向一個方向無限延伸，左邊向另一個方向無限延伸，它們最終會到達同一個地方嗎？還是說，黑板邊緣隱藏著不同的無窮？他的老師讓他別再問問題了。

讓-保羅·范·本德根發展出一種有限幾何學，其中點和曲線都有寬度。

范·本德根后來成為超有限論邏輯領域的領軍學者，他通過構建一種幾何學來解決這些問題。在這種幾何學中，直線或曲線具有寬度，并且既是有限的，又是有限可分的。它可以被分割成一系列點，這些點雖然極其微小，但并非無限小。任何用這些點、直線和曲線構建的結構也必須是有限的，從而提供了一種離散的經典幾何學類比。盡管這些工具仍然存在局限性，但在過去的幾十年里，人們對它們進行了深入的研究——這不僅是為了超有限論本身，更是因為厘清事物的形狀對于發展有限物理學至關重要。

我們常常將物理宇宙想象成無限廣闊且無限可分割，但物理學家們自己卻對這種假設提出質疑。宇宙存在一些根本性的極限，例如普朗克尺度（有時被稱為宇宙的像素大�。^這個尺度，距離的概念就失去了意義。當無窮大出現在物理學家的方程式中時，它會帶來問題，而這正是他們想要避免的�！耙A測一個無限增長、不斷重復的宇宙會發生什么，真的非常非常難，”肖恩·卡羅爾說道。（在新標簽頁中打開）約翰·霍普金斯大學的一位物理學家，曾對量子力學的有限模型進行過實驗。（在新標簽頁中打開）“大多數宇宙學家處理這個問題的方式就是假裝它不存在。”

不來梅康斯特拉特大學和日內瓦大學的量子物理學家尼古拉斯·吉辛認為，直覺主義數學提供了一種思考物理學核心謎題之一的方法：在大尺度上，物理系統的行為是確定性的、可預測的。但在量子領域，隨機性占據主導地位；一個粒子具有多個量子態，并以不可預測的方式坍縮到其中之一。過去一個世紀以來，物理學家一直在試圖理解這種不匹配的根源。

尼古拉斯·吉辛提出，物理學中最大的謎團之一可能是由于對無窮大的錯誤假設造成的。

吉辛認為這是由于一個錯誤的假設造成的。他指出，研究人員默認認為，從宇宙誕生之初，粒子的量子態就可以用無限多位的實數來無限精確地定義。但吉辛認為，使用實數是錯誤的。如果改用直覺主義數學，就會發現決定論不過是擁有不切實際的完美信息所造成的假象。物理系統的大規模確定性行為自然會變得不精確且不可預測，從而消弭了經典領域和量子領域之間的界限。吉辛的理論引起了其他物理學家的興趣，部分原因是它有助于解決諸如宇宙大爆炸等現象的悖論。

但值得注意的是，他的研究并非否定了潛在的無限性，即亞里士多德意義上的無限性，也就是某種潛在可達到的事物。吉辛秉承直覺主義數學家通過時間和努力計算更大或更精確數字的傳統，允許創造越來越多的信息�？傆幸惶�，宇宙將包含完美且無限精確的信息。但這無關緊要，因為那一天永遠不會到來�！斑@里的潛在無限性實際上是指無限的等待時間，這與現實無關，”吉辛說道。重要的是，無限性不再是默認的假設。

物理學家肖恩·卡羅爾對宇宙可能是有限的這種可能性很感興趣。

這些基于物理學的對無限性的挑戰往往令那些極端有限論的數學家們欣喜不已，他們以此為證，證明他們的數學理論更能真實地描述現實。在2025年的會議上，卡羅爾關于宇宙究竟是真正的無限，還是如他所說“僅僅相當大”的演講，使他在哥倫比亞大學的校園里聲名鵲起。但他告誡說，舉證的責任仍然在于那些對無限性持懷疑態度的人。如果你能通過實驗證明物理宇宙實際上是有限的，即使是最堅定的“更高層次的無限”支持者也可能會停下來思考片刻�？紤]到集合論允許存在如此之多的實際無限，他們甚至可能會質疑集合論的自洽性。無論如何，時不時地進行這樣的思考是件好事。

即便這種情況真的發生，研究和運用無窮概念的集合論學者仍然有權繼續他們的工作，而不會受到影響——他們甚至可以說，這或許正是物理學和數學必須分道揚鑣的地方。數學和物理學并不一定描述相同的事物（盡管許多人認為它們必須如此），而無窮概念或許會在某種更廣義的柏拉圖式意義上繼續存在下去。

但如果這些實驗證明結果恰恰相反——無限確實存在于自然界——那么極端有限論者的回旋余地就小得多�！叭绻鎸嵉奈锢硎澜缯娴拇嬖跓o限，那么極端有限論者就很難立足了，”卡羅爾說道。

重新命名 Ultrafinite

“我為那些極端有限論者感到難過，因為人們在不了解他們的情況下就對他們不屑一顧，”卡羅爾后來告訴我。“但另一方面，極端有限論者在推銷他們的理論方面做得不夠好�！�

在數學領域，更好的營銷活動可能看起來像一個連貫的理論，就像納爾遜所追求的那種理論——一套形式化的規則，就像現代數學的基礎規則一樣，它排除了無窮大，但又足夠強大，可以進行有用的數學運算。

克拉克-多恩說，想法并不匱乏，但或許缺乏愿意將職業生涯早期投入到這些想法發展中的研究生。在他看來，紐約的這次聚會標志著一種轉變，表明人們足夠好奇，愿意重新審視這個問題，并且并不懼怕潛在的負面影響�！叭藗冋谟懻撨@種觀點，并積極思考如何將其建立在堅實的基礎之上，”他說。

大多數數學家都置身事外。涵蓋數學全貌的形式理論與他們無關。他們感興趣的是行之有效的方法，是解決具體問題并構建證明�；A性問題——數字是否存在于物理現實之外？數學是一個發明的過程還是一個發現的過程？——對他們來說可能有點尷尬，只有當數學家某天突然陷入危機時才會去思考這類問題。

有關的：

1. 我們信任電腦嗎？
2. 數學本質上是混亂的還是有序的？
3. 巴拿赫-塔斯基與無限克隆悖論

What Can We Gain by Losing Infinity?

By GREGORY BARBER

April 29, 2026

Ultrafinitism, a philosophy that rejects the infinite, has long been dismissed as mathematical heresy. But it is also producing new insights in math and beyond.

Kristina Armitage/Quanta Magazine

Doron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger(opens a new tab) sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book.

To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses. These equations and proofs are, according to Zeilberger — a longtime professor at Rutgers University and a famed figure in combinatorics — both “very ugly” and false. It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.

As a matter of practicality, infinity can be scrubbed out, he contends. “You don’t really need it.” Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits. (Zeilberger lists his own computer, which he named “Shalosh B. Ekhad,” as a collaborator on his papers.) With infinity eliminated, the only thing lost is mathematics that was “not worth doing at all,” Zeilberger said.

Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense. Not just because infinity is so useful and so natural to our descriptions of the universe, but because treating sets of numbers (like the integers) as actual, infinite objects is at the very core of mathematics, embedded in its most fundamental rules and assumptions.

At the very least, even if mathematicians don’t want to think about infinity as an actual entity, they acknowledge that sequences, shapes, and other mathematical objects have the potential to grow indefinitely. Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.

Doron Zeilberger is perhaps the most vocal proponent of banishing infinity from mathematics. “Infinity may or may not exist; God may or may not exist,” he said. “But in mathematics, there should not be any place, neither for infinity nor God.”

Courtesy of Doron Zeilberger

Zeilberger disagrees. To him, what matters is not whether something is possible in principle, but whether it is actually feasible. What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well. Consider “Skewes’ number,” eee79. This is an exceptionally large number, and no one has ever been able to write it out in decimal form. So what can we really say about it? Is it an integer? Is it prime? Can we find such a number anywhere in nature? Could we ever write it down? Perhaps, then, it is not a number at all.

This raises obvious questions, such as where, exactly, we will find the end point. Zeilberger can’t say. Nobody can. Which is the first reason that many dismiss his philosophy, known as ultrafinitism. “When you first pitch the idea of ultrafinitism to somebody, it sounds like quackery — like ‘I think there’s a largest number’ or something,” said Justin Clarke-Doane(opens a new tab), a philosopher at Columbia University.

“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins(opens a new tab), a set theorist at the University of Notre Dame. Ultrafinitism is not polite talk at a mathematical society dinner. Few (one might say an ultrafinite number) work on it. Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void. That’s not just because ultrafinitism is contrarian, but because it advocates for a mathematics that is fundamentally smaller, one where certain important questions can no longer be asked.

And yet it gives Hamkins and others a good deal to think about. From one angle, ultrafinitism can be seen as a more realistic mathematics. It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe. While we might be inclined to think of space and time as eternally expansive and divisible, the ultrafinitist would argue that these are assumptions that science has increasingly brought into question — much as, Zeilberger might say, science brought doubt to God’s doorstep.

“The world that we’re describing needs to be honest through and through,” said Clarke-Doane, who in April 2025 convened a rare gathering of experts to explore ultrafinitist ideas. “If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.” To him, “it sure seems like that should be part of the menu in the philosophy of math.”

Justin Clarke-Doane recently organized a conference where ultrafinitists could discuss and debate their ideas. He thinks that ultrafinitism “should be part of the menu in the philosophy of math.”

Jennifer McDonald

For mathematicians to take it seriously, though, ultrafinitists first need to agree on what they’re talking about — to turn arguments that sound like “bluster,” as Hamkins puts it, into an official theory. Mathematics is steeped in formal systems and common frameworks. Ultrafinitism, meanwhile, lacks such structure.

It is one thing to tackle problems piecemeal. It is quite another to rewrite the logical foundations of mathematics itself. “I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said. “The feeling is that, oh, well, it’s hopeless.”

That’s a problem that some ultrafinitists are still trying to address.

Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is. He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website(opens a new tab). “I cannot be a tenured professor without doing this crackpot stuff,” he said. But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”

Dissident Mathematics

Aristotle saw infinity as something that you could move toward but never reach. “The fact that the process of dividing never comes to an end ensures that this activity exists potentially,” he wrote. “But not that the infinite exists separately.” For millennia, this “potential” version of infinity reigned supreme.

Why Math’s Final Axiom Proved So Controversial

SET THEORY

Why Math’s Final Axiom Proved So Controversial

APRIL 29, 2026

But in the late 1800s, Georg Cantor and other mathematicians showed that the infinite really can exist. Cantor’s approach was to treat a series of numbers, such as the integers, as a complete infinite set. This approach would become essential in the creation of the foundational theory of mathematics, known as Zermelo-Fraenkel set theory, that mathematicians still use today. Infinity, he showed, is an actual object. Moreover, it can come in many different sizes; by manipulating and comparing these different infinities, mathematicians can prove surprising truths that on their face seem to have nothing to do with infinity at all. While few mathematicians spend much time in the realm of the higher infinite, “nowadays, almost every mathematician is an actualist,” Hamkins said. Infinity is assumed by default.

But this foundation of modern math has inspired fierce arguments since it was first proposed. One reason is that accepting a core assumption about infinity allows you to construct strange paradoxes: It becomes possible, for instance, to carve a ball into five parts and use them to create five new balls, each with a volume equal to that of the first.

Another objection is more philosophical. In the decades after Cantor’s revelations, some mathematicians argued that you cannot simply assert the existence of a mathematical structure — you must prove that it exists through a process of mental construction. In this “intuitionist” philosophy, for example, pi is less a number with an infinite non-repeating decimal expansion, and more a symbol that represents an algorithmic process for generating digits.

If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.

Justin Clarke-Doane, Columbia University

But intuitionism only requires that a given mental construction be possible in theory: It prohibits actual infinity but permits potential infinity. Some mathematicians still weren’t satisfied with this. They remained troubled by Skewes’ number and other values so large they could never be written down. And so they sought to take intuitionist ideas to an extreme.

“If you’re thinking, which numbers are going to exist in this view, those are going to have to be numbers that we can in practice construct,” not just theoretically construct, said Ofra Magidor(opens a new tab), a philosopher at the University of Oxford.

A new version of intuitionism — one that took these practical constraints to heart — crystallized in the 1960s and ’70s, with the work of Alexander Esenin-Volpin, a Soviet mathematician and poet.

Esenin-Volpin was known first and foremost as a political dissident. For leading protests and spreading anti-Soviet rhetoric and poetry, he was institutionalized. “He said, ‘I’m a human being. I have fundamental rights,’” said Rohit Parikh(opens a new tab), a logician at the City University of New York who hosted Esenin-Volpin in his home after the Soviets forced him to emigrate in the 1970s. Esenin-Volpin was a strange houseguest, who would pace around Parikh’s attic all night and use his wife’s beloved ceramics as an ashtray while working on a strange theory that rejected not only potential infinity but even extremely large numbers — those that couldn’t be constructed in a person’s mind.

Alexander Esenin-Volpin was a Soviet dissident, mathematician, and poet who was imprisoned several times for his human rights activism.

Irene Caesar

The logician Harvey Friedman once asked Esenin-Volpin to pinpoint a cutoff(opens a new tab) for what makes a number too large. Given an expression like 2n, at what value of n do numbers stop? Was 20 actually a number? What about 21, 22, and so on, up to 2100? Esenin-Volpin responded to each number in turn. Yes, 21 existed. Yes, 22 did. But each time, he waited longer to reply. The dialogue soon grew interminable.

Esenin-Volpin had made his point. As Parikh and others would later put it, the limits of numbers were rooted in the limited resources needed to demonstrate their existence, like time. Or available computer memory, or the physical length of a proof. “Most ultrafinitists have the view that the distinction between the finite and infinite is inherently vague,” Clarke-Doane said.

For Esenin-Volpin, a condition may be true for n, and for n + 1 — until it is not. A child grows and grows, until one day they’re no longer a child. One need not specify a specific end point. The important thing is that the end is in there, somewhere.

Esenin-Volpin’s work was a call for a new kind of mathematics that could, in some sense, tolerate vagueness. Ultrafinitists have since picked up where he left off, exploring how to make his vague, borderline-nonsensical mathematics solid.

Crisis Control

One day, Edward Nelson woke up and realized that infinity might not be real. It left him in an existential crisis.

Mariana Cook

One morning in 1976, the Princeton mathematician Edward Nelson(opens a new tab) woke up and experienced a crisis of faith. “I felt the momentary overwhelming presence of one who convicted me of arrogance for my belief in the real existence of an infinite world of numbers,” he reflected decades later(opens a new tab), “leaving me like an infant in my crib reduced to counting on my fingers.”

Mathematics has basic rules, or axioms. Nelson knew that even the bare-bones axioms that make it possible to do simple arithmetic contain assumptions about infinity — for instance, that we can always add 1 to a number to create new numbers. He wanted to start over, to construct a new set of rules that would forbid infinity entirely. What would mathematics look like if it could be built up from only these new axioms?

Remarkably weak, it turned out. Nelson studied various sets of axioms that banished infinity and found that if he used any of them to try to do basic arithmetic, it became impossible to prove something as simple as the statement that a + balways equals b + a. Elementary operations like exponentiation were no longer always possible: You might be able to construct the number 100, or the number 1,000, but not the number 1001,000. One of the most powerful techniques in a mathematician’s tool kit — a method known as induction, which says that if you can prove that a statement is true for one number, then it must be true for them all — was lost entirely.

To Nelson, this weakness represented a glimmer of truth. He hoped to show that the more powerful axioms of arithmetic that mathematicians took for granted (the infinity-permitting “Peano axioms”) were fundamentally flawed — that they could lead to contradictions. “I believe that many of the things we regard as being established in mathematics will be overthrown,” he once said.

Nelson was unable to overthrow them, however. In 2003, he announced that he’d used his weaker axioms to find an inconsistency in the Peano axioms, but the splashy result was quickly debunked.

Rohit Parikh’s ultrafinitist ideas have had applications in theoretical computer science.

Lauren Fleishman

Nelson’s more limited arithmetic — as well as related forms of nonstandard arithmetic developed by Parikh and others — did prove useful in the realm of computers, where researchers want to understand what algorithms can efficiently prove and what they can’t. These ultrafinitist approaches to mathematics have been translated into the language of computational efficiency and used to probe the limits of algorithms’ capabilities.

To Nelson, mathematics is all about “the truth you choose to believe” — the axioms that you decide are the right ones. That’s true even if you’ve chosen to believe the default axioms. Of course, the ultrafinitist, as the heretic without stable foundations, has a lot more to prove.

Exercises in Patience

In April 2025, a motley crew gathered in New York City for a conference at Columbia University on abolishing the infinite. They included physicists, philosophers, logicians, and mathematicians. There were card-carrying ultrafinitists like Zeilberger; set theorists, who believe in all sorts of infinity; and the merely curious. The result was, recalled Clarke-Doane, the conference organizer, “an exercise in patience for everyone.” Philosophers, in general, are used to disagreeing vehemently in the classroom and then gathering over a beer. Mathematicians aren’t. Usually, if they disagree, it means somebody royally messed up.

What was clear was that progress toward a universal theory of ultrafinitism has been halting in part because there has been no one clear motivation for the movement, or any singular approach to deciding what its underlying logic should look like. Perhaps, then, fixating on the ground rules, like Nelson did, isn’t the right approach. “I think it’s a waste,” Parikh told me. “You have to use the formalism as a binocular and pay more attention to what you are seeing. If you start studying the binoculars themselves, you’ve lost the game.”

Where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks.

Zeilberger is happy to see things through the (possibly distorted) looking glass, even if he must do so in a world where infinity is very much alive and present. He doesn’t hope to rebuild a mathematics without infinity from scratch. He can work from the top down instead. Take, for example, real analysis, which deals with how real numbers and functions behave. Zeilberger calls it a “degenerate case(opens a new tab)” of discrete analysis (which studies the behavior of distinct objects rather than continuous ones). You can replace the continuous landscape of the reals, he says, with a “discrete necklace” of numbers, separated by tiny — but not infinitesimal — differences in value. You can then use this to rewrite the rules of calculus and differential equations (now called “difference” equations) to remove even subtle uses of infinity from them. The going is tough, he acknowledges, but doable, especially with the help of a computer. And while the result may look less elegant than classical math, it is more beautiful, he says, because it reflects physical reality as he believes it to truly be.

For Jean Paul Van Bendegem(opens a new tab), a philosopher of mathematics at the Free University of Brussels, the journey into ultrafinitism began not with numbers, but with elementary school geometry. He watched his math teacher draw a line on the chalkboard that supposedly extended infinitely. “To where?” he recalled asking. If the right-hand side went infinitely far in one direction and the left-hand side in another, did they arrive at the same place? Or did different infinities lurk off the edges of the board? His teacher told him to stop asking questions.

Jean-Paul van Bendegem developed a finite version of geometry in which points and curves have width.

Inge Kinnet

Van Bendegem, who would become a leading scholar on ultrafinitist logic, later addressed these concerns by considering a geometry in which a line or curve has width and is both finite and finitely divisible. It can be broken up into an array of points that, though incredibly small, are not infinitely so. Any structure one then builds with these points, lines, and curves must also be finite, providing a discrete analogue of classical geometry. While these tools remain limited, they have been explored deeply over the past few decades — not just for the sake of ultrafinitism, but because sorting out the shape of things is important for developing a finite physics.

While we often imagine the physical universe as both endlessly vast and endlessly divisible, physicists themselves question this assumption. There are fundamental limits, such as the Planck scale (sometimes called the pixel size of the universe), beyond which the very idea of distance loses meaning. And when infinity does crop up in physicists’ equations, it can be problematic, something they want to avoid. “To make predictions about what to expect in a universe that grows without bounds and repeats itself and things like that turns out to be really, really hard,” said Sean Carroll(opens a new tab), a physicist at Johns Hopkins University who has experimented with finitistic models of quantum mechanics(opens a new tab). “The way that most cosmologists deal with that problem is by pretending it’s not there.”

For Nicolas Gisin(opens a new tab), a quantum physicist at Constructor University in Bremen, Germany, and the University of Geneva, intuitionist mathematics provides a way to think about one of the core mysteries in physics: At large scales, the behavior of physical systems is deterministic, predictable. But in the quantum realm, randomness reigns; a particle comes with multiple quantum states, collapsing to just one of them in unpredictable ways. Physicists have been trying to understand the source of this mismatch for the past century.

Nicolas Gisin proposed that one of the greatest mysteries in physics might be due to incorrect assumptions about infinity.

Carole Parodi

Gisin posits that it’s due to a faulty assumption. Researchers implicitly believe, he says, that from the start of the universe, a particle’s quantum state can be defined with infinite precision, by real numbers with infinitely many digits. But, according to Gisin, using the real numbers is a mistake. If you use intuitionist mathematics instead, then it becomes clear that determinism is but an artifact of having unrealistically perfect information. The large-scale, deterministic behavior of physical systems naturally becomes imprecise and unpredictable, dissolving the divide between the classical and quantum realms. Gisin’s theory has proved intriguing to other physicists, in part because it could help resolve paradoxes about phenomena like the Big Bang.

But it’s important to note that his work does not abolish potential infinity, in the Aristotelian sense of something that can potentially be reached. In the tradition of the intuitionist mathematician calculating larger or more precise numbers with time and effort, Gisin allows for more and more information to be created. Someday, the universe will contain perfect, infinitely precise information. But it doesn’t matter, because that someday will never come. “The potential infinity here is really waiting infinite time, which has nothing to do with reality,” Gisin said. The important thing is that infinity is no longer the default assumption.

The physicist Sean Carroll is intrigued by the possibility that the universe might be finite.

Larry Canner/Johns Hopkins

These physics-based challenges to the infinite tend to delight ultrafinitist mathematicians, who hold them up as evidence that their mathematics is a truer description of reality. At the 2025 conference, Carroll’s talk on whether the universe is truly infinite or “merely quite large,” as he put it, made him something of a celebrity in the Columbia University halls. But the burden of proof, he cautions, remains with the infinity doubters. If you could somehow prove experimentally that the physical universe is indeed finite, even the most ardent backers of the higher infinite would likely take a moment to pause and reflect. They would probably even wonder about the consistency of set theory, given the towers of actual infinities that it allows. That’s a healthy thing to do from time to time, anyway.

Even if this were to happen, set theorists who study and use infinity would still be within their rights to continue their work unfazed — to say that perhaps this is where physics and math must branch off from each other. It is no requirement that math and physics describe the same things (though many believe it is), and infinity might live on in some larger Platonic sense.

But if those experiments proved the opposite — that infinity does exist in nature — the ultrafinitist would have far less room to negotiate. “It would be hard to be an ultrafinitist if the actual physical world had infinities in it,” Carroll said.

Rebranding the Ultrafinite

“I feel bad for the ultrafinitists because people dismiss it without understanding it,” Carroll told me later. “But on the other hand, the ultrafinitists don’t do a good enough job of marketing their product.”

Within mathematics, a better marketing campaign would probably look like a coherent theory, the kind Nelson sought — a set of formal rules, like those underlying modern math, that excludes infinity but is powerful enough to do useful mathematics.

There is no shortage of ideas, Clarke-Doane said — though there’s perhaps a shortage of graduate students willing to stake their early careers on developing them. To him, the gathering in New York was a sign of change, that people are curious enough to give it another look, and not too scared of the potential backlash. “People are talking about the view and actively trying to think about how to put the view on a serious foundation,” he said.

Most mathematicians live outside all this. Formal theories encompassing the totality of mathematics do not concern them. They are interested in what works, in solving specific problems and building proofs. Foundational questions — do numbers exist beyond physical reality? Is math a process of invention or discovery? — can feel a little cringe, the sort of thing mathematicians only do when they wake up one day in a state of crisis.

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