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超限數.
數學嘅數
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基本
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延伸
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其他
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圓周率 π = 3.141592653…
自然對數嘅底 e = 2.718281828…
虛數單位 i =
無窮大量 ∞
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超限數係大過晒所有嘅有限數、仍然唔需要定義做絕對無窮嘅基數或者序數。術語「超限」(transfinite)係由康托爾提出,佢希望避免詞語無限(infinite)同嗰啲只不過唔係有限(finite)嘅嗰啲義象有關嘅某啲暗含。
當期時其他作者都好少有呢啲疑惑;依家俾人所接受嘅用法係假定超限基數或者序數係無限嘅,但即使係咁,而家術語「超限」一樣重有人用緊。
對於有限數,有兩種方式考慮超限數,作為基數同作為序數。不似得有限基數同序數咁樣,超限基數同超限序數定義咗唔同類別嘅數。
- 第一個超限基數係 aleph-0 ,整數嘅無限集合嘅勢。如果選擇公理成立,下一個更高嘅基數就係 aleph-1 。如果唔成立,就會出現好多唔可以同 aleph-1 比較並且大過 aleph-0 嘅其他基數。但係喺任何情況之下,點都唔會有基數係大過 aleph-0 並且細過 aleph-1。
連續統假設聲稱係 aleph-0 同連續統(實數嘅集合)嘅勢之間係冇任何中間基數:即係話 aleph-1 係額數集合嘅勢。已經喺數學上証實咗連續統假設係証實唔到係真定係假,咁係由於不完備性嘅影響。
某啲作者,比如 Suppes、Rubin 用術語超限基數嚟稱呼戴德金無限集合嘅勢,係可以唔等於無限基數嘅上下文中;即係話係唔假定可數選擇公理成立嘅上下文中。
假設呢個定義係啱,下面四項就係等價嘅:
- 係超限基數。就係話有一個戴德金無限集合 A 令到 A 嘅勢係 。
- 。
- 。
- 有一個基數 使到 。
- O'Connor, J. J. and E. F. Robertson (1998) , MacTutor History of Mathematics archive.
- Patrick Suppes, "Axiomatic Set Theory", Dover, 1972, ISBN 0-486-61630-4
- Jean E. Rubin, "Set Theory for the Mathematician", Holden-Day (San Francisico, 1967)
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常用大數 | |
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其他漢字大數 | |
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佛教大數 | |
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超級大數 | |
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超越有限數 | |
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超越無窮級 | |
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記數方法 | 記數法 | 科學記數法、Knuth上箭嘴記號、Conway鏈式箭嘴記號、Steinhaus-Moser記號 |
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運算 | 超運算(四迭次方、五迭次方)、Ackermann函數、Grzegorczyk階層、快速成長階層 |
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