{"history":[],"article":{"id":"bcc2435f-f7a4-41c4-a1ec-4a1e42be1ad8","title":"Normal function","slug":"normal-function","content":":::summary\n:::\n\n# \nIn [axiomatic set theory](https://bliptext.com/articles/axiomatic-set-theory), a function\nf : Ord → Ord\nis called **normal** (or a **normal** function) if it is [continuous](https://bliptext.com/articles/continuous-function) (with respect to the [order topology](https://bliptext.com/articles/order-topology)) and [strictly monotonically increasing](https://bliptext.com/articles/monotonic-function).\nThis is equivalent to the following two conditions:\nf (γ) = sup\n.\nα < β\n, it is the case that\nf (α) < f (β)\n.\n\n# Examples\nA simple normal function is given by\nf (α) = 1 + α\n(see [ordinal arithmetic](https://bliptext.com/articles/ordinal-arithmetic)).\nBut\nf (α) = α + 1\nis not normal because it is not continuous at any limit ordinal; that is, the [inverse image](https://bliptext.com/articles/inverse-image) of the one-point open set\nis the set\n, which is not open when λ is a limit ordinal.\nIf β is a fixed ordinal, then the functions\nf (α) = β + α\n,\nf (α) = β × α\n(for\nβ ≥ 1\n), and\nf (α) = βα\n(for\nβ ≥ 2\n) are all normal.\nMore important examples of normal functions are given by the [aleph numbers](https://bliptext.com/articles/aleph-number), which connect ordinal and [cardinal numbers](https://bliptext.com/articles/cardinal-number), and by the [beth numbers](https://bliptext.com/articles/beth-number).\n\n# Properties\nIf f is normal, then for any ordinal α,\nf (α) ≥ α\n. **Proof**: If not, choose γ minimal such that\nf (γ) < γ\n. Since f is strictly monotonically increasing,\nf (f (γ)) < f (γ)\n, contradicting minimality of γ.\nFurthermore, for any [non-empty](https://bliptext.com/articles/empty-set) set S of ordinals, we have\nf (sup S) = sup f (S)\n. **Proof**: \"≥\" follows from the monotonicity of f and the definition of the [supremum](https://bliptext.com/articles/supremum).\nFor \"\n≤\n\", set\nδ = sup S\nand consider three cases:\nδ = 0\n, then\nS =\nand\nsup f (S) = f (0)\nδ = ν + 1\nis a successor, then there exists s in S with\nν < s\n, so that\nδ ≤ s\n. Therefore,\nf (δ) ≤ f (s)\n, which implies\nf (δ) ≤ sup f (S)\nν < δ\n, and an s in S such that\nν < s\n(possible since\nδ = sup S\n). Therefore,\nf (ν) < f (s)\nso that\nf (ν) < sup f (S)\n, yielding\nf (δ) = sup ≤ sup f (S)\n, as desired.\nEvery normal function f has arbitrarily large fixed points; see the [fixed-point lemma for normal functions](https://bliptext.com/articles/fixed-point-lemma-for-normal-functions) for a proof.\nOne can create a normal function\nf ′ : Ord → Ord\n, called the **derivative** of f, such that\nf ′(α)\nis the α-th fixed point of f. For a hierarchy of normal functions, see [Veblen functions](https://bliptext.com/articles/veblen-function).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:28:44.815Z","updated_at":"2025-01-25T19:28:44.815Z","search_vector":"'function':2 'normal':1","search_content":" \n \n\n# \nIn [axiomatic set theory](https://bliptext.com/articles/axiomatic-set-theory), a function\nf : Ord → Ord\nis called **normal** (or a **normal** function) if it is [continuous](https://bliptext.com/articles/continuous-function) (with respect to the [order topology](https://bliptext.com/articles/order-topology)) and [strictly monotonically increasing](https://bliptext.com/articles/monotonic-function).\nThis is equivalent to the following two conditions:\nf (γ) = sup\n.\nα < β\n, it is the case that\nf (α) < f (β)\n.\n\n# Examples\nA simple normal function is given by\nf (α) = 1 + α\n(see [ordinal arithmetic](https://bliptext.com/articles/ordinal-arithmetic)).\nBut\nf (α) = α + 1\nis not normal because it is not continuous at any limit ordinal; that is, the [inverse image](https://bliptext.com/articles/inverse-image) of the one-point open set\nis the set\n, which is not open when λ is a limit ordinal.\nIf β is a fixed ordinal, then the functions\nf (α) = β + α\n,\nf (α) = β × α\n(for\nβ ≥ 1\n), and\nf (α) = βα\n(for\nβ ≥ 2\n) are all normal.\nMore important examples of normal functions are given by the [aleph numbers](https://bliptext.com/articles/aleph-number), which connect ordinal and [cardinal numbers](https://bliptext.com/articles/cardinal-number), and by the [beth numbers](https://bliptext.com/articles/beth-number).\n\n# Properties\nIf f is normal, then for any ordinal α,\nf (α) ≥ α\n. **Proof**: If not, choose γ minimal such that\nf (γ) < γ\n. Since f is strictly monotonically increasing,\nf (f (γ)) < f (γ)\n, contradicting minimality of γ.\nFurthermore, for any [non-empty](https://bliptext.com/articles/empty-set) set S of ordinals, we have\nf (sup S) = sup f (S)\n. **Proof**: \"≥\" follows from the monotonicity of f and the definition of the [supremum](https://bliptext.com/articles/supremum).\nFor \"\n≤\n\", set\nδ = sup S\nand consider three cases:\nδ = 0\n, then\nS =\nand\nsup f (S) = f (0)\nδ = ν + 1\nis a successor, then there exists s in S with\nν < s\n, so that\nδ ≤ s\n. Therefore,\nf (δ) ≤ f (s)\n, which implies\nf (δ) ≤ sup f (S)\nν < δ\n, and an s in S such that\nν < s\n(possible since\nδ = sup S\n). Therefore,\nf (ν) < f (s)\nso that\nf (ν) < sup f (S)\n, yielding\nf (δ) = sup ≤ sup f (S)\n, as desired.\nEvery normal function f has arbitrarily large fixed points; see the [fixed-point lemma for normal functions](https://bliptext.com/articles/fixed-point-lemma-for-normal-functions) for a proof.\nOne can create a normal function\nf ′ : Ord → Ord\n, called the **derivative** of f, such that\nf ′(α)\nis the α-th fixed point of f. For a hierarchy of normal functions, see [Veblen functions](https://bliptext.com/articles/veblen-function).","revisionCount":0},"hasMore":false,"nextCursor":null}

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