ultrafinitism

mathematics

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July 14, 2026

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Ultrafinitism is a philosophical stance in mathematics that questions the existence of even very large numbers, suggesting that numbers beyond a certain, extremely large threshold (e.g., (10^{(10^{10})})) do not exist. This view is a more extreme form of finitism, which itself posits that infinite sets do not exist, except potentially.1

While most mathematicians accept the existence of large numbers and infinite sets, ultrafinitists employ subtle arguments to support their position. This perspective contrasts with mainstream mathematical thought, which readily utilizes concepts of infinity, such as in the comparison of different sizes of infinite sets (transfinite cardinals) and the exploration of continuous lines.12

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foundations of mathematics
Intuitionistic logic
…even more extreme position, called ultrafinitism, maintains that even very large numbers do not exist, say numbers greater than 10^(10¹⁰). Of course, the vast majority of mathematicians reject this view by referring to 10^(10¹⁰) + 1, but the true believers have subtle ways of getting around this objection, which, however,…
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…even more extreme position, called ultrafinitism, maintains that even very large numbers do not exist, say numbers greater than 10^(10¹⁰). Of course, the vast majority of mathematicians reject this view by referring to 10^(10¹⁰) + 1, but the true believers have subtle ways of getting around this objection, which, however,…
Read More