Group representation

The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a rewritten version of the dissertation, published as an article in 1930) is not easy to read today; it uses concepts and formalism that are outdated and terminology that is often obscure. The version given below attempts to represent all the steps in the proof and all the important ideas faithfully, while restating the proof in the modern language of mathematical logic. This outline should not be considered a rigorous proof of the theorem.

Definitions and assumptions

We work with first-order predicate calculus. Our languages allow constant, function and relation symbols. Structures consist of (non-empty) domains and interpretations of the relevant symbols as constant members, functions or relations over that domain.

We fix some axiomatization of the predicate calculus: logical axioms and rules of inference. Any of the several well-known axiomatisations will do; we assume without proof all the basic well-known results about our formalism (such as the normal form theorem or the soundness theorem) that we need.

We axiomatize predicate calculus without equality, i.e. there are no special axioms expressing the properties of equality as a special relation symbol. After the basic form of the theorem is proved, it will be easy to extend it to the case of predicate calculus with equality.

Statement of the theorem and its proof

In the following, we state two equivalent forms of the theorem, and show their equivalence.

Later, we prove the theorem. This is done in the following steps:

1. Reducing the theorem to sentences (formulas with no free variables) in prenex form, i.e. with all quantifiers (Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
   
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   There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
   
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   A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
   
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   A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
   
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   This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
   
   A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
   
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2. Reducing the theorem to sentences of the form Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
   
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   There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
   
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   A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
   
   If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
   
   Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.. While we cannot do this by simply rearranging the quantifiers, we show that it is yet enough to prove the theorem for sentences of that form.
3. Finally we prove the theorem for sentences of that form.
   • This is done by first noting that a sentence such as Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
     
     Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
     
     There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
     
     This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
     
     A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
     
     If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
     
     Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is either refutable or has some model in which it holds; this model is simply assigning truth values to the subpropositions from which B is built. The reason for that is the completeness of propositional logic, with the existential quantifiers playing no role.
   • We extend this result to more and more complex and lengthy sentences, D_n (n=1,2...), built out from B, so that either any of them is refutable and therefore so is φ, or all of them are not refutable and therefore each holds in some model.
   • We finally use the models in which the D_n hold (in case all are not refutable) in order to build a model in which φ holds.

Theorem 1. Every formula valid in all structures is provable.

This is the most basic form of the completeness theorem. We immediately restate it in a form more convenient for our purposes:

Theorem 2. Every formula φ is either refutable or satisfiable in some structure.

"φ is refutable" means by definition "¬φ is provable".

Equivalence of both theorems

To see the equivalence, note first that if Theorem 1 holds, and φ is not satisfiable in any structure, then ¬φ is valid in all structures and therefore provable, thus φ is refutable and Theorem 2 holds. If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬φ is not satisfiable in any structure and therefore refutable; then ¬¬φ is provable and then so is φ, thus Theorem 1 holds.

Proof of theorem 2: first step

We approach the proof of Theorem 2 by successively restricting the class of all formulas φ for which we need to prove "φ is either refutable or satisfiable". At the beginning we need to prove this for all possible formulas φ in our language. However, suppose that for every formula φ there is some formula ψ taken from a more restricted class of formulas C, such that "ψ is either refutable or satisfiable" → "φ is either refutable or satisfiable". Then, once this claim (expressed in the previous sentence) is proved, it will suffice to prove "φ is either refutable or satisfiable" only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ (i.e., (φ≡ψ) is provable), then it is indeed the case that "ψ is either refutable or satisfiable" → "φ is either refutable or satisfiable" (the soundness theorem is needed to show this).

There are standard techniques for rewriting an arbitrary formula into one which does not use function or constant symbols, at the cost of introducing additional quantifiers; we will therefore assume that all formulas are free of such symbols. In Gödel's paper, he uses a version of first-order predicate calculus which has no function or constant symbols to begin with.

Next we consider a generic formula φ (which no longer uses function or constant symbols) and apply the prenex form theorem to find a formula ψ in normal form such that φ≡ψ (ψ being in normal form means that all the quantifiers in ψ, if there are any, are found at the very beginning of ψ). It follows now that we need only prove Theorem 2 for formulas φ in normal form.

Next, we eliminate all free variables from φ by quantifying them existentially: if, say, x₁...x_n are free in φ, we form $\psi =\mathrm{\exists }{x}_1...\mathrm{\exists }{x}_n\varphi$. If ψ is satisfiable in a structure M, then certainly so is φ and if ψ is refutable, then $\mathrm{\neg }\psi =\mathrm{\forall }{x}_1...\mathrm{\forall }{x}_n\mathrm{\neg }\varphi$ is provable, and then so is ¬φ, thus φ is refutable. We see that we can restrict φ to be a sentence, that is, a formula with no free variables.

Finally, we would like, for reasons of technical convenience, that the prefix of φ (that is, the string of quantifiers at the beginning of φ, which is in normal form) begin with a universal quantifier and end with an existential quantifier. To achieve this for a generic φ (subject to restrictions we have already proved), we take some one-place relation symbol F unused in φ, and two new variables y and z.. If φ = (P)Φ, where (P) stands for the prefix of φ and Φ for the matrix (the remaining, quantifier-free part of φ) we form $\psi =\mathrm{\forall }y(P)\mathrm{\exists }z(\mathrm{\Phi }\wedge [F(y)\vee \mathrm{\neg }F(z)])$. Since $\mathrm{\forall }y\mathrm{\exists }z(F(y)\vee \mathrm{\neg }F(z))$ is clearly provable, it is easy to see that $\varphi =\psi$ is provable.

Reducing the theorem to formulas of degree 1

Our generic formula φ now is a sentence, in normal form, and its prefix starts with a universal quantifier and ends with an existential quantifier. Let us call the class of all such formulas R. We are faced with proving that every formula in R is either refutable or satisfiable. Given our formula φ, we group strings of quantifiers of one kind together in blocks:

$\varphi =(\mathrm{\forall }{x}_1...\mathrm{\forall }{x}_{k_1})(\mathrm{\exists }{x}_{k_1+1}...\mathrm{\exists }{x}_{k_2}).......(\mathrm{\forall }{x}_{k_{n-2}+1}...\mathrm{\forall }{x}_{k_{n-1}})(\mathrm{\exists }{x}_{k_{n-1}+1}...\mathrm{\exists }{x}_{k_n})(\mathrm{\Phi })$

We define the degree of $\varphi$ to be the number of universal quantifier blocks, separated by existential quantifier blocks as shown above, in the prefix of $\varphi$. The following lemma, which Gödel adapted from Skolem's proof of the Löwenheim-Skolem theorem, lets us sharply reduce the complexity of the generic formula $\varphi$ for which we need to prove the theorem:

Lemma. Let k>=1. If every formula in R of degree k is either refutable or satisfiable, then so is every formula in R of degree k+1.

Comment: Take a formula φ of degree k+1 of the form $\varphi =(\mathrm{\forall }x)(\mathrm{\exists }y)(\mathrm{\forall }u)(\mathrm{\exists }v)(P)\psi$, where $(P)\psi$ is the remainder of $\varphi$ (it is thus of degree k-1). φ states that for every x there is a y such that... (something). It would have been nice to have a predicate Q' so that for every x, Q'(x,y) would be true if and only if y is the required one to make (something) true. Then we could have written a formula of degree k which is equivalent to φ, namely $(\mathrm{\forall }{x}^{\prime })(\mathrm{\forall }x)(\mathrm{\forall }y)(\mathrm{\forall }u)(\mathrm{\exists }v)(\mathrm{\exists }{y}^{\prime })(P)Q^{\prime }(x^{\prime },y^{\prime })\wedge (Q^{\prime }(x,y)\to \psi )$. This formula is indeed equivalent to φ because it states that for every x, if there is a y which satisfies Q'(x,y), then (something) holds, and furthermore, we know that there is such a y, because for every x', there is a y' which satisfies Q'(x',y'). Therefore φ follows from this formula. It is also easy to show that if the formula is false, then so is φ. Unfortunately, in general there is no such predicate Q'. However, this idea can be understood as a basis for the following proof of the Lemma.

Proof. Let φ be a formula of degree k+1; then we can write it as

$\varphi =(\mathrm{\forall }x)(\mathrm{\exists }y)(\mathrm{\forall }u)(\mathrm{\exists }v)(P)\psi$

where (P) is the remainder of the prefix of $\varphi$ (it is thus of degree k-1) and $\psi$ is the quantifier-free matrix of $\varphi$. x, y, u and v denote here tuples of variables rather than single variables; e.g. $(\mathrm{\forall }x)$ really stands for $\mathrm{\forall }{x}_1\mathrm{\forall }{x}_2...\mathrm{\forall }{x}_n$ where $x_1...x_n$ are some distinct variables.

Let now x' and y' be tuples of previously unused variables of the same length as x and y respectively, and let Q be a previously unused relation symbol which takes as many arguments as the sum of lengths of x and y; we consider the formula

$\mathrm{\Phi }=(\mathrm{\forall }{x}^{\prime })(\mathrm{\exists }{y}^{\prime })Q(x^{\prime },y^{\prime })\wedge (\mathrm{\forall }x)(\mathrm{\forall }y)(Q(x,y)\to (\mathrm{\forall }u)(\mathrm{\exists }v)(P)\psi )$

Clearly, $\mathrm{\Phi }\to \varphi$ is provable.

Now since the string of quantifiers $(\mathrm{\forall }u)(\mathrm{\exists }v)(P)$ does not contain variables from x or y, the following equivalence is easily provable with the help of whatever formalism we're using:

$(Q(x,y)\to (\mathrm{\forall }u)(\mathrm{\exists }v)(P)\psi )\equiv (\mathrm{\forall }u)(\mathrm{\exists }v)(P)(Q(x,y)\to \psi )$

And since these two formulas are equivalent, if we replace the first with the second inside Φ, we obtain the formula Φ' such that Φ≡Φ':

${\mathrm{\Phi }}^{\prime }=(\mathrm{\forall }{x}^{\prime })(\mathrm{\exists }{y}^{\prime })Q(x^{\prime },y^{\prime })\wedge (\mathrm{\forall }x)(\mathrm{\forall }y)(\mathrm{\forall }u)(\mathrm{\exists }v)(P)(Q(x,y)\to \psi )$

Now Φ' has the form $(S)\rho \wedge (S^{\prime }){\rho }^{\prime }$, where (S) and (S') are some quantifier strings, ρ and ρ' are quantifier-free, and, furthermore, no variable of (S) occurs in ρ' and no variable of (S') occurs in ρ. Under such conditions every formula of the form $(T)(\rho \wedge {\rho }^{\prime })$, where (T) is a string of quantifiers containing all quantifiers in (S) and (S') interleaved among themselves in any fashion, but maintaining the relative order inside (S) and (S'), will be equivalent to the original formula Φ'(this is yet another basic result in first-order predicate calculus that we rely on). To wit, we form Ψ as follows:

$\mathrm{\Psi }=(\mathrm{\forall }{x}^{\prime })(\mathrm{\forall }x)(\mathrm{\forall }y)(\mathrm{\forall }u)(\mathrm{\exists }{y}^{\prime })(\mathrm{\exists }v)(P)Q(x^{\prime },y^{\prime })\wedge (Q(x,y)\to \psi )$

and we have ${\mathrm{\Phi }}^{\prime }\equiv \mathrm{\Psi }$.

Now $\mathrm{\Psi }$ is a formula of degree k and therefore by assumption either refutable or satisfiable. If $\mathrm{\Psi }$ is satisfiable in a structure M, then, considering $\mathrm{\Psi }\equiv {\mathrm{\Phi }}^{\prime }\equiv \mathrm{\Phi }\wedge \mathrm{\Phi }\to \varphi$, we see that $\varphi$ is satisfiable as well. If $\mathrm{\Psi }$ is refutable, then so is $\mathrm{\Phi }$ which is equivalent to it; thus $\mathrm{\neg }\mathrm{\Phi }$ is provable. Now we can replace all occurrences of Q inside the provable formula $\mathrm{\neg }\mathrm{\Phi }$ by some other formula dependent on the same variables, and we will still get a provable formula. (This is yet another basic result of first-order predicate calculus. Depending on the particular formalism adopted for the calculus, it may be seen as a simple application of a "functional substitution" rule of inference, as in Gödel's paper, or it may be proved by considering the formal proof of $\mathrm{\neg }\mathrm{\Phi }$, replacing in it all occurrences of Q by some other formula with the same free variables, and noting that all logical axioms in the formal proof remain logical axioms after the substitution, and all rules of inference still apply in the same way.)

In this particular case, we replace Q(x',y') in $\mathrm{\neg }\mathrm{\Phi }$ with the formula $(\mathrm{\forall }u)(\mathrm{\exists }v)(P)\psi (x,y|x^{\prime },y^{\prime })$. Here (x,y|x',y') means that instead of ψ we are writing a different formula, in which x and y are replaced with x' and y'. Note that Q(x,y) is simply replaced by $(\mathrm{\forall }u)(\mathrm{\exists }v)(P)\psi$.

$\mathrm{\neg }\mathrm{\Phi }$ then becomes

$\mathrm{\neg }((\mathrm{\forall }{x}^{\prime })(\mathrm{\exists }{y}^{\prime })(\mathrm{\forall }u)(\mathrm{\exists }v)(P)\psi (x,y|x^{\prime },y^{\prime })\wedge (\mathrm{\forall }x)(\mathrm{\forall }y)((\mathrm{\forall }u)(\mathrm{\exists }v)(P)\psi \to (\mathrm{\forall }u)(\mathrm{\exists }v)(P)\psi ))$

and this formula is provable; since the part under negation and after the $\wedge$ sign is obviously provable, and the part under negation and before the $\wedge$ sign is obviously φ, just with x and y replaced by x' and y', we see that $\mathrm{\neg }\varphi$ is provable, and φ is refutable. We have proved that φ is either satisfiable or refutable, and this concludes the proof of the Lemma.

Notice that we could not have used $(\mathrm{\forall }u)(\mathrm{\exists }v)(P)\psi (x,y|x^{\prime },y^{\prime })$ instead of Q(x',y') from the beginning, because $\mathrm{\Psi }$ would not have been a well-formed formula in that case. This is why we cannot naively use the argument appearing at the comment which precedes the proof.

Proving the theorem for formulas of degree 1

As shown by the Lemma above, we only need to prove our theorem for formulas φ in R of degree 1. φ cannot be of degree 0, since formulas in R have no free variables and don't use constant symbols. So the formula φ has the general form:

$(\mathrm{\forall }{x}_1...x_k)(\mathrm{\exists }{y}_1...y_m)\varphi (x_1...x_k,y_1...y_m).$

Now we define an ordering of the k-tuples of natural numbers as follows: $(x_1...x_k)<(y_1...y_k)$ should hold if either ${\mathrm{\Sigma }}_k(x_1...x_k)<{\mathrm{\Sigma }}_k(y_1...y_k)$, or ${\mathrm{\Sigma }}_k(x_1...x_k)={\mathrm{\Sigma }}_k(y_1...y_k)$, and $(x_1...x_k)$ precedes $(y_1...y_k)$ in lexicographic order. [Here ${\mathrm{\Sigma }}_k(x_1...x_k)$ denotes the sum of the terms of the tuple.] Denote the nth tuple in this order by $(a_1^n...a_k^n)$.

Set the formula $B_n$ as $\varphi (z_{a_1^n}...z_{a_k^n},z_{(n-1)m+2},z_{(n-1)m+3}...z_{nm+1})$. Then put $D_n$ as

$(\mathrm{\exists }{z}_1...z_{nm+1})(B_1\wedge B_2...\wedge B_n).$

Lemma: For every n, φ$\to D_n$.

Proof: By induction on n; we have $D_k\Leftarrow D_{k-1}\wedge (\mathrm{\forall }{z}_1...z_{(n-1)m+1})(\mathrm{\exists }{z}_{(n-1)m+2}...z_{nm+1})B_n\Leftarrow D_{k-1}\wedge (\mathrm{\forall }{z}_{a_1^n}...z_{a_k^n})(\mathrm{\exists }{y}_1...y_m)\varphi (z_{a_1^n}...z_{a_k^n},y_1...y_m)$, where the latter implication holds by variable substitution, since the ordering of the tuples is such that $(\mathrm{\forall }k)(a_1^n...a_k^n)<(n-1)m+2$. But the last formula is equivalent to $D_{k-1}\wedge$φ.

For the base case, $D_1\equiv (\mathrm{\exists }{z}_1...z_{m+1})\varphi (z_{a_1^1}...z_{a_k^1},z_2,z_3...z_{m+1})\equiv (\mathrm{\exists }{z}_1...z_{m+1})\varphi (z_1...z_1,z_2,z_3...z_{m+1})$ is obviously a corollary of φ as well. So the Lemma is proven.

Now if $D_n$ is refutable for some n, it follows that φ is refutable. On the other hand, suppose that $D_n$ is not refutable for any n. Then for each n there is some way of assigning truth values to the distinct subpropositions $E_h$ (ordered by their first appearance in $D_n$; "distinct" here means either distinct predicates, or distinct bound variables) in $B_k$, such that $D_n$ will be true when each proposition is evaluated in this fashion. This follows from the completeness of the underlying propositional logic.

We will now show that there is such an assignment of truth values to $E_h$, so that all $D_n$ will be true: The $E_h$ appear in the same order in every $D_n$; we will inductively define a general assignment to them by a sort of "majority vote": Since there are infinitely many assignments (one for each $D_n$) affecting $E_1$, either infinitely many make $E_1$ true, or infinitely many make it false and only finitely many make it true. In the former case, we choose $E_1$ to be true in general; in the latter we take it to be false in general. Then from the infinitely many n for which $E_1$ through $E_{h-1}$ are assigned the same truth value as in the general assignment, we pick a general assignment to $E_h$ in the same fashion.

This general assignment must lead to every one of the $B_k$ and $D_k$ being true, since if one of the $B_k$ were false under the general assignment, $D_n$ would also be false for every n > k. But this contradicts the fact that for the finite collection of general $E_h$ assignments appearing in $D_k$, there are infinitely many n where the assignment making $D_n$ true matches the general assignment.

Now from this general assignment which makes all of the $D_k$ true, we construct an interpretation of the language's predicates which makes φ true. The universe of the model will be the natural numbers. Each i-ary predicate $\mathrm{\Psi }$ should be true of the naturals $(u_1...u_i)$ precisely when the proposition $\mathrm{\Psi }(z_{u_1}...z_{u_i})$ is either true in the general assignment, or not assigned by it (because it never appears in any of the $D_k$).

In this model, each of the formulas $(\mathrm{\exists }{y}_1...y_m)\varphi (a_1^n...a_k^n,y_1...y_m)$ is true by construction. But this implies that φ itself is true in the model, since the $a^n$ range over all possible k-tuples of natural numbers. So φ is satisfiable, and we are done.

Intuitive explanation

We may write each B_i as Φ(x₁...x_k,y₁...y_m) for some x-s which we may call "first arguments" and y-s which we may call "last arguments".

Take B₁ for example. Its "last arguments" are z₂,z₃...z_m+1, and for every possible combination of k of these variables there is some j so that they appear as "first arguments" in B_j. Thus for large enough n₁, D_n1 has the property that the "last arguments" of B₁ appear, in every possible combinations of k of them, as "first arguments" in other B_j-s within D_n. For every B_i there is a D_ni with the corresponding property.

Therefore in a model which satisfies all the D_n-s, there are objects corresponding to z₁, z₂... and each combination of k of these appear as "first arguments" in some B_j, meaning that for every k of these objects z_p1...z_pk there are z_q1...z_qm which makes Φ(z_p1...z_pk,z_q1...z_qm) satisfied. By taking a submodel which contains only these z₁, z₂... objects, we have a model satisfying φ.

Extensions

Extension to first-order predicate calculus with equality

Gödel reduced a formula containing instances of the equality predicate to ones without it in an extended language. His method involves replacing a formula φ containing some instances of equality with the formula

$(\mathrm{\forall }x)Eq(x,x)\wedge (\mathrm{\forall }x,y,z)[Eq(x,y)\to (Eq(x,z)\to Eq(y,z))]$ $\wedge (\mathrm{\forall }x,y,z)[Eq(x,y)\to (Eq(z,x)\to Eq(z,y))]$ $\wedge (\mathrm{\forall }{x}_1...x_k,y_1...x_k)[(Eq(x_1,y_1)\wedge ...\wedge Eq(x_k,y_k))\to (A(x_1...x_k)\equiv A(y_1...y_k))]$ $\wedge ...\wedge (\mathrm{\forall }{x}_1...x_m,y_1...x_m)[(Eq(x_1,y_1)\wedge ...\wedge Eq(x_m,y_m))\to (Z(x_1...x_m)\equiv Z(y_1...y_m))]$ $\wedge {\phi }^{\prime }.$

Here $A...Z$ denote the predicates appearing in φ (with $k...m$ their respective arities), and φ' is the formula φ with all occurrences of equality replaced with the new predicate Eq. If this new formula is refutable, the original φ was as well; the same is true of satisfiability, since we may take a quotient of satisfying model of the new formula by the equivalence relation representing Eq. This quotient is well-defined with respect to the other predicates, and therefore will satisfy the original formula φ.

Extension to countable sets of formulas

Gödel also considered the case where there are a countably infinite collection of formulas. Using the same reductions as above, he was able to consider only those cases where each formula is of degree 1 and contains no uses of equality. For a countable collection of formulas ${\varphi }^i$ of degree 1, we may define $B_k^i$ as above; then define $D_k$ to be the closure of $B_1^1...B_k^1,...,B_1^k...B_k^k$. The remainder of the proof then went through as before.

Extension to arbitrary sets of formulas

When there is an uncountably infinite collection of formulas, the Axiom of Choice (or at least some weak form of it) is needed. Using the full AC, one can well-order the formulas, and prove the uncountable case with the same argument as the countable one, except with transfinite induction. Other approaches can be used to prove that the completeness theorem in this case is equivalent to the Boolean prime ideal theorem, a weak form of AC.

References

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  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang The same material as the dissertation, except with briefer proofs, more succinct explanations, and omitting the lengthy introduction.

External links

• Stanford Encyclopedia of Philosophy: "Kurt Gödel"—by Juliette Kennedy.
• MacTutor biography: Kurt Gödel.