Lagrange number

In set theory and computability theory, Kleene's $\mathcal{𝒪}$ is a canonical subset of the natural numbers when regarded as ordinal notations. It contains ordinal notations for every recursive ordinal, that is, ordinals below Church–Kleene ordinal, ${\omega }_1^{CK}$. Since ${\omega }_1^{CK}$ is the first ordinal not representable in a computable system of ordinal notations the elements of $\mathcal{𝒪}$ can be regarded as the canonical ordinal notations.

Kleene (1938) described a system of notation for all recursive ordinals (those less than the Church–Kleene ordinal). It uses a subset of the natural numbers instead of finite strings of symbols. Unfortunately, there is in general no effective way to tell whether some natural number represents an ordinal, or whether two numbers represent the same ordinal. However, one can effectively find notations which represent the ordinal sum, product, and power (see ordinal arithmetic) of any two given notations in Kleene's $\mathcal{𝒪}$; and given any notation for an ordinal, there is a recursively enumerable set of notations which contains one element for each smaller ordinal and is effectively ordered.

Kleene's $\mathcal{𝒪}$

The basic idea of Kleene's system of ordinal notations is to build up ordinals in an effective manner. For members $p$ of $\mathcal{𝒪}$, the ordinal for which $p$ is a notation is $\left|p\right|$. The standard definition proceeds via transfinite induction and the ordering ${<}_{\mathcal{𝒪}}$ (a partial ordering of Kleene's $\mathcal{𝒪}$) is defined simultaneously.

• The natural number 0 belongs to Kleene's $\mathcal{𝒪}$ and $\left|0\right|=0$.
• If $i$ belongs to Kleene's $\mathcal{𝒪}$ and $\left|i\right|=\alpha$, then $2^i$ belongs to Kleene's $\mathcal{𝒪}$ and $\left|2^i\right|=\alpha +1$ and $i{<}_{\mathcal{𝒪}}{2}^i$.
• Suppose $\{e\}$ is the $e$-th partial recursive function. If $e$ is total, with range contained in $\mathcal{𝒪}$, and for every natural number $n$, we have $\{e\}(n){<}_{\mathcal{𝒪}}\{e\}(n+1)$, then $3\cdot 5^e$ belongs to Kleene's $\mathcal{𝒪}$, $\{e\}(n){<}_{\mathcal{𝒪}}3\cdot 5^e$ for each $n$ and $|3\cdot 5^e|=\underset{k}{\lim }|\{e\}(k)|$, i.e. $3\cdot 5^e$ is a notation for the limit of the ordinals ${\gamma }_k$ where $|\{e\}(k)|={\gamma }_k$ for every natural number $k$.
• $p{<}_{\mathcal{𝒪}}q$ and $q{<}_{\mathcal{𝒪}}r$ imply $p{<}_{\mathcal{𝒪}}r$ (this guarantees that ${<}_{\mathcal{𝒪}}$ is transitive.)

This definition has the advantages that one can recursively enumerate the predecessors of a given ordinal (though not in the ${<}_{\mathcal{𝒪}}$ ordering) and that the notations are downward closed, i.e., if there is a notation for $\gamma$ and $\alpha <\gamma$ then there is a notation for $\alpha$.

Basic properties of ${<}_{\mathcal{𝒪}}$

• If $\left|i\right|=\alpha$ and $\left|j\right|=\beta$ and $i{<}_{\mathcal{𝒪}}j,$ then $\alpha <\beta$; but the converse may fail to hold.
• ${<}_{\mathcal{𝒪}}$ induces a tree structure on $\mathcal{𝒪}$, so $\mathcal{𝒪}$ is well-founded.
• $\mathcal{𝒪}$ only branches at limit ordinals; and at each notation of a limit ordinal, $\mathcal{𝒪}$ is infinitely branching.
• Since every recursive function has countably many indices, each infinite ordinal receives countably many notations; the finite ordinals have unique notations, $n$ usually denoted $n_{\mathcal{𝒪}}$.
• The first ordinal that doesn't receive a notation is called the Church–Kleene ordinal and is denoted by ${\omega }_1^{CK}$. Since there are only countably many recursive functions, the ordinal ${\omega }_1^{CK}$ is evidently countable.
• The ordinals with a notation in Kleene's $\mathcal{𝒪}$ are exactly the recursive ordinals. (The fact that every recursive ordinal has a notation follows from the closure of this system of ordinal notations under successor and effective limits.)
• ${<}_{\mathcal{𝒪}}$ is not recursively enumerable, but there is a recursively enumerable relation which agrees with ${<}_{\mathcal{𝒪}}$ precisely on members of $\mathcal{𝒪}$.
• For any notation $p$, the set $\{q\mid q{<}_{\mathcal{𝒪}}p\}$ of notations below $p$ is recursively enumerable. However, Kleene's $\mathcal{𝒪}$, when taken as a whole, is ${\mathrm{\Pi }}_1^1$ (see analytical hierarchy).
• In fact, $\mathcal{𝒪}$ is ${\mathrm{\Pi }}_1^1$-complete and every ${\mathrm{\Sigma }}_1^1$ subset of $\mathcal{𝒪}$ is effectively bounded in $\mathcal{𝒪}$ (a result of Spector).
• $\mathcal{𝒪}$ is the universal system of ordinal notations in the sense that any specific set of ordinal notations can be mapped into it in a straightforward way. More precisely, there is a recursive function $f$ such that if $e$ is an index for a recursive well-ordering, then $f(e)$ is a member of $\mathcal{𝒪}$ and ${<}_e$ is order-isomorphic to an initial segment of the set $\{p\mid p{<}_{\mathcal{𝒪}}f(e)\}$.
• There is a recursive function ${+}_{\mathcal{𝒪}}$, which, for members of $\mathcal{𝒪}$, mimics ordinal addition and has the property that $\max \{p,q\}\le p{+}_{\mathcal{𝒪}}q$. (Jockusch)

Properties of paths in $\mathcal{𝒪}$

A path in $\mathcal{𝒪}$ is a subset $\mathcal{𝒫}$ of $\mathcal{𝒪}$ which is totally ordered by ${<}_{\mathcal{𝒪}}$ and is closed under predecessors, i.e. if $p$ is a member of a path $\mathcal{𝒫}$ and $q{<}_{\mathcal{𝒪}}p$ then $q$ is also a member of $\mathcal{𝒫}$. A path $\mathcal{𝒫}$ is maximal if there is no element of $\mathcal{𝒪}$ which is above (in the sense of ${<}_{\mathcal{𝒪}}$) every member of $\mathcal{𝒫}$, otherwise $\mathcal{𝒫}$ is non-maximal.

• A path $\mathcal{𝒫}$ is non-maximal if and only if $\mathcal{𝒫}$ is recursively enumerable (r.e.). It follows by remarks above that every element $p$ of $\mathcal{𝒪}$ determines a non-maximal path $\mathcal{𝒫}$; and every non-maximal path is so determined.
• There are $2^{{\mathrm{\aleph }}_0}$ maximal paths through $\mathcal{𝒪}$; since they are maximal, they are non-r.e.
• In fact, there are $2^{{\mathrm{\aleph }}_0}$ maximal paths within $\mathcal{𝒪}$ of length ${\omega }^2$. (Crossley, Schütte)
• For every non-zero ordinal $\lambda <{\omega }_1^{CK}$, there are $2^{{\mathrm{\aleph }}_0}$ maximal paths within $\mathcal{𝒪}$ of length ${\omega }^2\cdot \lambda$. (Aczel)
• Further, if $\mathcal{𝒫}$ is a path whose length is not a multiple of ${\omega }^2$ then $\mathcal{𝒫}$ is not maximal. (Aczel)
• For each r.e. degree $d$, there is a member $e_d$ of $\mathcal{𝒪}$ such that the path $\mathcal{𝒫}=\{p\mid p{<}_{\mathcal{𝒪}}{e}_d\}$ has many-one degree $d$. In fact, for each recursive ordinal $\alpha \ge {\omega }^2$, a notation $e_d$ exists with $\left|e_d\right|=\alpha$. (Jockusch)
• There exist ${\mathrm{\aleph }}_0$ paths through $\mathcal{𝒪}$ which are ${\mathrm{\Pi }}_1^1$. Given a progression of recursively enumerable theories based on iterating Uniform Reflection, each such path is incomplete with respect to the set of true ${\mathrm{\Pi }}_1^0$ sentences. (Feferman & Spector)
• There exist ${\mathrm{\Pi }}_1^1$ paths through $\mathcal{𝒪}$ each initial segment of which is not merely r.e., but recursive. (Jockusch)
• Various other paths in $\mathcal{𝒪}$ have been shown to exist, each with specific kinds of reducibility properties. (See references below)

See also

• Recursive ordinal
• Large countable ordinal
• Ordinal notation

References

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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010