Emmy Noether

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. Template:Harvnb showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second order arithmetic). This was the third "natural" example of a true statement that is unprovable in Peano arithmetic (after Gerhard Gentzen's 1943 direct proof of the unprovability of ε₀-induction in Peano arithmetic and the Paris–Harrington theorem). Earlier statements of this type had either been, except for Gentzen, extremely complicated, ad-hoc constructions (such as the statements generated by the construction given in Gödel's incompleteness theorem) or concerned metamathematics or combinatorial results Template:Harv.

Laurie Kirby and Jeff Paris gave an interpretation of the Goodstein's theorem as a hydra game: the "Hydra" is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. The Kirby–Paris interpretation of the theorem says that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very, very long time.

Hereditary base-n notation

Goodstein sequences are defined in terms of a concept called "hereditary base-n notation". This notation is very similar to usual base-n positional notation, but the usual notation does not suffice for the purposes of Goodstein's theorem.

In ordinary base-n notation, where n is a natural number greater than 1, an arbitrary natural number m is written as a sum of multiples of powers of n:

$m=a_k{n}^k+a_{k-1}{n}^{k-1}+\cdots +a_0,$

where each coefficient $a_i$ satisfies $0\le a_i<n$, and $a_k=0$. For example, in base 2,

$35=32+2+1=2^5+2^1+2^0.$

Thus the base 2 representation of 35 is $2^5+2^1+2^0$. (This expression could be written in binary notation as 100011.) Similarly, one can write 100 in base 3:

$100=81+18+1=3^4+2\cdot 3^2+1.$

Note that the exponents themselves are not written in base-n notation. For example, the expressions above include $2^5$ and $3^4$.

To convert a base-n representation to hereditary base n notation, first rewrite all of the exponents in base-n notation. Then rewrite any exponents inside the exponents, and continue in this way until every digit appearing in the expression is n or less.

For example, while 35 in ordinary base-2 notation is $2^5+2+1$, it is written in hereditary base-2 notation as

$35=2^{2^2+1}+2+1,$

using the fact that $5=2^2+1.$ Similarly, 100 in hereditary base 3 notation is

$100=3^{3+1}+2\cdot 3^2+1.$

Goodstein sequences

The Goodstein sequence G(m) of a number m is a sequence of natural numbers. The first element in the sequence G(m) is m itself. To get the next element Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., write m in hereditary base 2 notation, change all the 2s to 3s, and then subtract 1 from the result; this is the second element of G(m). To get the third element of G(m) Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland., write the second element in hereditary base 3 notation, change all 3s to 4s, and subtract 1 again. Continue until the result is zero, at which point the sequence terminates.

Early Goodstein sequences terminate quickly. For example, G(3) Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. terminates at the sixth step:

Base	Hereditary notation	Value	Notes
2	$2^1+1$	3	Write 3 in base 2 notation
3	$3^1+1-1=3^1$	3	Switch the 2 to a 3, then subtract 1
4	$4^1-1=3$	3	Switch the 3 to a 4, then subtract 1. Now there are no more 4s left
5	$3-1=2$	2	No 4s left to switch to 5s. Just subtract 1
6	$2-1=1$	1
7	$1-1=0$	0

Later Goodstein sequences increase for a very large number of steps. For example, G(4) Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. starts as follows:

Hereditary notation	Value
$2^2$	4
$3^3-1=2\cdot 3^2+2\cdot 3+2$	26
$2\cdot 4^2+2\cdot 4+1$	41
$2\cdot 5^2+2\cdot 5$	60
$2\cdot 6^2+2\cdot 6-1=2\cdot 6^2+6+5$	83
$2\cdot 7^2+7+4$	109
$⋮$	$⋮$
$2\cdot 11^2+11$	253
$2\cdot 12^2+12-1=2\cdot 12^2+11$	299
$⋮$	$⋮$

Elements of G(4) continue to increase for a while, but at base $3\cdot 2^{402653209}$, they reach the maximum of $3\cdot 2^{402653210}-1$, stay there for the next $3\cdot 2^{402653209}$ steps, and then begin their first and final descent.

The value 0 is reached at base $3\cdot 2^{402653211}-1$ (curiously, this is a generalized Woodall number: $3\cdot 2^{402653211}-1=402653184\cdot 2^{402653184}-1$. This is also the case with all other final bases for starting values greater than 4Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.).

However, even G(4) doesn't give a good idea of just how quickly the elements of a Goodstein sequence can increase. G(19) increases much more rapidly, and starts as follows:

In spite of this rapid growth, Goodstein's theorem states that every Goodstein sequence eventually terminates at 0, no matter what the starting value is.

Proof of Goodstein's theorem

Goodstein's theorem can be proved (using techniques outside Peano arithmetic, see below) as follows: Given a Goodstein sequence G(m), we will construct a parallel sequence of ordinal numbers whose elements are no smaller than those in the given sequence. If the elements of the parallel sequence go to 0, the elements of the Goodstein sequence must also go to 0.

To construct the parallel sequence, take the hereditary base n representation of the (n − 1)-th element of the Goodstein sequence, and replace every instance of n with the first infinite ordinal number ω. Addition, multiplication and exponentiation of ordinal numbers is well defined, and the resulting ordinal number clearly cannot be smaller than the original element.

The 'base-changing' operation of the Goodstein sequence does not change the element of the parallel sequence: replacing all the 4s in $4^{4^4}+4$ with ω is the same as replacing all the 4s with 5s and then replacing all the 5s with ω. The 'subtracting 1' operation, however, corresponds to decreasing the infinite ordinal number in the parallel sequence; for example, ${\omega }^{{\omega }^{\omega }}+\omega$ decreases to ${\omega }^{{\omega }^{\omega }}+4$ if the step above is performed. Because the ordinals are well-ordered, there are no infinite strictly decreasing sequences of ordinals. Thus the parallel sequence must terminate at 0 after a finite number of steps. The Goodstein sequence, which is bounded above by the parallel sequence, must terminate at 0 also.

While this proof of Goodstein's theorem is fairly easy, the Kirby–Paris theorem which says that Goodstein's theorem is not a theorem of Peano arithmetic, is technical and considerably more difficult. It makes use of countable nonstandard models of Peano arithmetic. What Kirby showed is that Goodstein's theorem leads to Gentzen's theorem, i.e. it can substitute for induction up to ε₀.

Sequence length as a function of the starting value

The Goodstein function, $\mathcal{𝒢}:\mathbb{\mathbb{N}}\to \mathbb{\mathbb{N}}$, is defined such that $\mathcal{𝒢}(n)$ is the length of the Goodstein sequence that starts with n. (This is a total function since every Goodstein sequence terminates.) The extreme growth-rate of $\mathcal{𝒢}$ can be calibrated by relating it to various standard ordinal-indexed hierarchies of functions, such as the functions $H_{\alpha }$ in the Hardy hierarchy, and the functions $f_{\alpha }$ in the fast-growing hierarchy of Löb and Wainer:

• Kirby and Paris (1982) proved that

$\mathcal{𝒢}$ has approximately the same growth-rate as $H_{{ϵ}_0}$ (which is the same as that of $f_{{ϵ}_0}$); more precisely, $\mathcal{𝒢}$ dominates $H_{\alpha }$ for every $\alpha <{ϵ}_0$, and $H_{{ϵ}_0}$ dominates $\mathcal{𝒢}.$

(For any two functions $f,g:\mathbb{\mathbb{N}}\to \mathbb{\mathbb{N}}$, $f$ is said to dominate $g$ if $f(n)>g(n)$ for all sufficiently large $n$.)

• Cichon (1983) showed that

$\mathcal{𝒢}(n)=H_{R_2^{\omega }(n+1)}(1)-1,$

where $R_2^{\omega }(n)$ is the result of putting n in hereditary base-2 notation and then replacing all 2s with ω (as was done in the proof of Goodstein's theorem).

• Caicedo (2007) showed that if $n=2^{m_1}+2^{m_2}+\cdots +2^{m_k}$ with $m_1>m_2>\cdots >m_k,$ then

$\mathcal{𝒢}(n)=f_{R_2^{\omega }(m_1)}(f_{R_2^{\omega }(m_2)}(\cdots (f_{R_2^{\omega }(m_k)}(3))\cdots ))-2$.

Some examples:

n			$\mathcal{𝒢}(n)$
1	$2^0$	$2-1$	$H_{\omega }(1)-1$	$f_0(3)-2$	2
2	$2^1$	$2^1+1-1$	$H_{\omega +1}(1)-1$	$f_1(3)-2$	4
3	$2^1+2^0$	$2^2-1$	$H_{{\omega }^{\omega }}(1)-1$	$f_1(f_0(3))-2$	6
4	$2^2$	$2^2+1-1$	$H_{{\omega }^{\omega }+1}(1)-1$	$f_{\omega }(3)-2$	3·2402653211 − 2
5	$2^2+2^0$	$2^2+2-1$	$H_{{\omega }^{\omega }+\omega }(1)-1$	$f_{\omega }(f_0(3))-2$	> A(4,4)
6	$2^2+2^1$	$2^2+2+1-1$	$H_{{\omega }^{\omega }+\omega +1}(1)-1$	$f_{\omega }(f_1(3))-2$	> A(6,6)
7	$2^2+2^1+2^0$	$2^{2+1}-1$	$H_{{\omega }^{\omega +1}}(1)-1$	$f_{\omega }(f_1(f_0(3)))-2$	> A(8,8)
8	$2^{2+1}$	$2^{2+1}+1-1$	$H_{{\omega }^{\omega +1}+1}(1)-1$	$f_{\omega +1}(3)-2$	> A3(3,3) = A(A(61, 61), A(61, 61))
$⋮$
12	$2^{2+1}+2^2$	$2^{2+1}+2^2+1-1$	$H_{{\omega }^{\omega +1}+{\omega }^{\omega }+1}(1)-1$	$f_{\omega +1}(f_{\omega }(3))-2$	> fω+1(64) > Graham's number
$⋮$
19	$2^{2^2}+2^1+2^0$	$2^{2^2}+2^2-1$	$H_{{\omega }^{{\omega }^{\omega }}+{\omega }^{\omega }}(1)-1$	$f_{{\omega }^{\omega }}(f_1(f_0(3)))-2$

(For Ackermann function and Graham's number bounds see fast-growing hierarchy#Functions in fast-growing hierarchies.)

Application to computable functions

Goodstein's theorem can be used to construct a total computable function that Peano arithmetic cannot prove to be total. The Goodstein sequence of a number can be effectively enumerated by a Turing machine; thus the function which maps n to the number of steps required for the Goodstein sequence of n to terminate is computable by a particular Turing machine. This machine merely enumerates the Goodstein sequence of n and, when the sequence reaches 0, returns the length of the sequence. Because every Goodstein sequence eventually terminates, this function is total. But because Peano arithmetic does not prove that every Goodstein sequence terminates, Peano arithmetic does not prove that this Turing machine computes a total function.

See also

• Non-standard model of arithmetic
• Fast-growing hierarchy
• Paris–Harrington theorem
• Kanamori–McAloon theorem
• Kruskal's tree theorem

References

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  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.
• 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.

External links

• 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
  
  Here is my web site - cottagehillchurch.com
• Some elements of a proof that Goodstein's theorem is not a theorem of PA, from an undergraduate thesis by Justin T Miller
• A Classification of non standard models of Peano Arithmetic by Goodstein's theorem - Thesis by Dan Kaplan, Franklan and Marshall College Library
• Definitions of Goodstein sequences in the programming languages Ruby and Haskell, as well as a large-scale plot
• The Hydra game implemented as a Java applet
• Goodstein Sequences: The Power of a Detour via Infinity - good exposition with illustrations of Goodstein Sequences and the hydra game.