Age (model theory)

In mathematics, a Gödel numbering for sequences provides us an effective way to represent each finite sequence of natural numbers as a single natural number. Of course, the embedding is surely possible set theoretically, but the emphasis is on the effectiveness of the functions manipulating such representations of sequences: the operations on sequences (accessing individual members, concatenation) can be "implemented" using total recursive functions, and in fact by primitive recursive functions.

It is usually used to build sequential “data types” in the realm of arithmetic-based formalizations of some fundamental notions of mathematics. It is a specific case of the more general idea of Gödel numbering.

E.g. recursive function theory can be regarded as a formalization of notion “algorithm”, and if we regard it as a programming language, we can mimic arrays, lists by encoding a sequence of natural numbers in a single natural number — to achieve this, we can use various number theoretic ideas. Using the fundamental theorem of arithmetic is a straightforward way, but there are also more economic approaches, e.g. using pairing function combined with Chinese remainder theorem in a sophisticated way.

Gödel numbering

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

Gödel numbering can be used to not only encode unique sequences of symbols into unique natural numbers (i.e. place numbers into mutually exclusive or one-to-one correspondence with the sequences) but also to encode whole “architectures” of sophisticated “machines”. For example we can encode Markov algorithms, or Turing machines into natural numbers and thereby prove that the expressing power of recursive function theory is no less than that of the former machine-like formalizations of algorithms.

Accessing members

We expect from any such representation of sequences that we can get back all the information from it that is contained by the original sequence: most important, to access each individual member. It is not strictly necessary that the length can be also obtained directly: even if we want to handle sequences of different length, we can store length data as a surplus member, or as the other member of an ordered pair by using a pairing function.

Anyway, we expect that this obtaining back information can be done in an effective way, by an appropriate total recursive function.

We want a totally recursive function f that satisfies: For all n and for any n-length sequence of natural numbers $⟨a_0,\dots a_{n-1}⟩$, there exists an appropriate natural number a, called the Gödel number of the sequence such that for all i in the range of 0, …, n - 1,

$f(a,i)=a_i$.

There exist effective functions enabling us to obtain back each member of the original sequence from a Gödel number of the sequence. Moreover, there are ways to define some in a constructive way, thus we are not forced to be satisfied with mere proofs of existence.

Gödel's β-function lemma

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value.

By an ingenious use of Chinese remainder theorem, we can define constructively such a recursive function $\beta$ (using simple number-theoretical functions, all of which can be defined in a total recursive way) fulfilling the "specifications" given above. Also Gödel defined the $\beta$ function using the Chinese remainder theorem in his article written in 1931. This is a primitive recursive function.

Thus, for all n and for any n-length sequence of natural numbers $⟨a_0,\dots a_{n-1}⟩$, there exists an appropriate natural number a, called the Gödel number of the sequence such that

$\beta (a,i)=a_i$

Using a pairing function

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

Our specific solution will depend on a pairing function — there are several ways to implement the latter, let us select one. Now, we can abstract from the details of the “implementation” of the pairing function, we need only to know its “interface”: let $\pi$, K, L denote the pairing function and its two projection functions, respectively, satisying specification

$K\left(\pi (x,y)\right)=x$

$L\left(\pi (x,y)\right)=y$

we shall not discuss and formalize the axiom for excluding alien objects here, it is now not so important.

Remainder for natural numbers

We shall use another auxiliary function: it will compute the remainder for natural numbers. Examples:

• $\mathrm{r}\mathrm{e}\mathrm{m}(5,3)=2$
• $\mathrm{r}\mathrm{e}\mathrm{m}(7,2)=1$

It can be proven that this function can be "implemented" as a recursive function.

Using the Chinese remainder theorem

Implementation of the β function

Using the Chinese remainder theorem, we can prove that implementing $\beta$ as

$\beta (s,i)=\mathrm{r}\mathrm{e}\mathrm{m}(K\left(s\right),(i+1)\cdot L\left(s\right)+1)$

will work, according to the specification we expect $\beta$ to satisfy. We can use a more concise form by an abuse of notation (sort of pattern matching):

$\beta (\pi (x_0,m),i)=\mathrm{r}\mathrm{e}\mathrm{m}(x_0,(i+1)\cdot m+1)$

Let us achieve even more readability by more modularity and reuse (as these notions are used in computer science): defining $\mathrm{\forall }i<n$ the sequence $m_i=(i+1)\cdot m+1$, enables us to write

$\beta (\pi (x_0,m),i)=\mathrm{r}\mathrm{e}\mathrm{m}(x_0,m_i)$

We shall use this $m_i$ notation also in the proof.

Hand-tuned assumptions

For proving the correctness of the above definition of $\beta$ function, we shall use (and prove) several auxiliary theorems, lemmas. These have their own assumptions. Now we try to find out these assumptions, calibrating and tuning their strength carefully: they should not be said in an either superfluously sharp, or unsatisfactorily weak form.

Let $a_0,\dots a_{n-1}$ be a sequence of natural numbers. Let m be chosen to satisfy

$\mathrm{\forall }i\in \bar{n}\setminus \left\{0\right\}(i\mid m)$

$\mathrm{\forall }i<n(a_i<m_i)$

The first assumption is meant as

$1\mid m\wedge \dots \wedge n-1\mid m$

It is needed to meet an assumption of the Chinese remainder theorem (that of being pairwise coprime). In the literature, sometimes this requirement is replaced with a stronger one, e.g. constructively built with the factorial function, but the proof uses just as much strength as formulated here.

The second assumption does not concern the Chinese remainder theorem in any way. It will have importance in proving that the specification for $\beta$ is met eventually. It ensures that an $\tilde{x}$ solution of the simultaneous congruence system

$x\equiv a_i(\mathrm{mod}{m}_i)$ for each i ranging 0,… , n-1

also satisfies

$a_i=\mathrm{r}\mathrm{e}\mathrm{m}(\tilde{x},m_i)$

A stronger assumption for m requiring $\mathrm{\forall }i<n(a_i<m)$ automatically satisfies it (if we define the notation $m_i$ as above).

Proof that (coprimality) assumption for Chinese remainder theorem is met

We shall prove that the (coprimality) assumption for Chinese remainder theorem is met.

As mentioned in section Hand-tuned assumptions, we prescribed that

$\mathrm{\forall }i\in \bar{n}\setminus \left\{0\right\}(i\mid m)$

thus we can use it.

What we want to prove is that we can produce a sequence of pairwise coprime numbers in a way that will turn out to correspond to the Implementation of the β function in a sense.

In details:

$\mathrm{\forall }i<n,j<n(i\ne j\to \mathrm{c}\mathrm{o}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}(m_i,m_j))$

let us remember, $\mathrm{\forall }i<n$ we defined $m_i=(i+1)\cdot m+1$.

Let us use reductio ad absurdum!

Negation of the original statement:

$\mathrm{\exists }i<n,j<n(i\ne j\wedge \mathrm{\neg }\mathrm{c}\mathrm{o}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}(m_i,m_j))$

First steps

We know what “coprime” relation means (in a lucky way, its negation can be formulated in a concise form), thus, let us substitute in the appropriate way:

$\mathrm{\exists }i<n,j<n(i\ne j\wedge \mathrm{\exists }p\in \mathrm{P}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}(p\mid m_i\wedge p\mid m_j))$

Using a “more” prenex normal form (but note allowing a constraint-like notation in quantifiers):

$\mathrm{\exists }i<n,j<n,p\in \mathrm{P}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}(i\ne j\wedge p\mid m_i\wedge p\mid m_j)$

Because of a theorem on divisibility, $p\mid m_i\wedge p\mid m_j$ allows us to say also

$p\mid m_i-m_j$

Substituting the definens of $m_k$-sequence notation, we get $m_i-m_j=(i-j)\cdot m$, thus (as equality axioms postulate identity to be a congruence relation ) we get

$p\mid (i-j)\cdot m$

Using that p is a prime element (note: not the irreducible element property is used!), we get

$p\mid i-j\vee p\mid m$

Resorting to the first hand-tuned assumption

Now this is the point in the proof where we must resort to our assumption

$\mathrm{\forall }i\in \bar{n}\setminus \left\{0\right\}(i\mid m)$

let us remember, we have planned this assumption calibrated carefully to be as weak as possible, but strong enough to enable us to use it now.

The assumed negation of the original statement (let us remember, we use reductio ad absurdum) contains an appropriate existential statement using indices $i<n\wedge j<n\wedge i\ne j$, this entails $i-j\in \bar{n}\setminus \left\{0\right\}$, thus the mentioned assumption can be applied, so $i-j\mid m$ holds.

Using an (object) theorem of the propositional calculus as a lemma

We can prove by several means known in propositional calculus, that

$(A\wedge (A\to B))\to B$

holds.

Because $i-j\mid m$ entails (by the transitivity property of the divisibility relation) that $p\mid i-j\to p\mid m$, thus (as equality axioms postulate identiy to be a congruence relation )

$p\mid m$

can be proven.

Reaching the contradiction

The negation of original statement contained

$p\mid m_i$

and we have just proved

$p\mid m$

thus also

$p\mid m_i-(i+1)\cdot m$

should hold. But, after substituting the definiens for $m_i$, we shall see

$m_i-(i+1)\cdot m=1$

Thus, summarizing the above three statements, by transitivity of the equality, also

$p\mid 1$

should hold. But let us look up the quantification of p in the negation of the original statement: p is existentially quantified and restricted to primes $\mathrm{\exists }p\in \mathrm{P}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}$

The above statement together with the above quantification of p establish the contradiction we wanted to reach.

End of reductio ad absurdum

By reaching contradiction with its negation, we have just proven the original statement:

$\mathrm{\forall }i<n,j<n(i\ne j\to \mathrm{c}\mathrm{o}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}(m_i,m_j))$

The system of simultaneous congruences

We build a system of simultaneous congruences

$x\equiv a_0(\mathrm{mod}{m}_0)$

$⋮$

$x\equiv a_{n-1}(\mathrm{mod}{m}_{n-1})$

We can write it in a more concise way:

$\mathrm{\forall }i<n(x\equiv a_i(\mathrm{mod}{m}_i))$

In the followings, many statements will be said, all beginning with $\mathrm{\forall }i<n(\dots )$. To achieve a more ergonomic treatment, from now on all statements will be regarded in the scope of an $\mathrm{\forall }i<n(\dots )$ qantification. Thus: $\mathrm{\forall }i<n($ begins!

Let us chose a solution $x_0$ for the system of simultaneous congruences. At least one solution must exist, because $m_0,\dots m_{n-1}$ are pairwise comprime (that's what we have been proving so long in the previous sections!), thus we can refer to the Chinese remainder theorem's ensuring solution. Thus, from now on, we can regard $x_0$ satisfying

$x_0\equiv a_i(\mathrm{mod}{m}_i)$

it means (by definition of modular arithmetic) that

$\mathrm{r}\mathrm{e}\mathrm{m}(x_0,m_i)=\mathrm{r}\mathrm{e}\mathrm{m}(a_i,m_i)$

Resorting to the second hand-tuned assumption

Again, we must resort to the assumptions whose strength we specifically “tuned” for using in the proof. But now, it is the second assumption (which does not concern the Chinese remainder theorem in any way) that we need: “$\mathrm{\forall }i<n(a_i<m_i)$”. Let us remember: we are now in the scope of a “big” quantification for i, thus we don't repeat its quantification for each statement.

The second hand-tuned assumption $a_i<m_i$ will join in at this point, because it entails that

$\mathrm{r}\mathrm{e}\mathrm{m}(a_i,m_i)=a_i$

Now by transitivity of equality we get

$\mathrm{r}\mathrm{e}\mathrm{m}(x_0,m_i)=a_i$

QED

Our original goal was to prove that the definition

$\beta (\pi (x_0,m),i)=\mathrm{r}\mathrm{e}\mathrm{m}(x_0,m_i)$

is good for achieving what we declared in the specification of $\beta$: we want $\beta (\pi (x_0,m),i)=a_i$ to hold.

That's it, it can be seen now by transitivity of equality, looking at the above three equations.

Scope of i ends here.

Existence and uniqueness

We have just proven the correctness of the definition of $\beta$: its specification requiring

$\mathrm{\forall }{a}_0,\dots ,a_{n-1}\mathrm{\exists }s\mathrm{\forall }i<n\beta (s,i)=a_i$

is met. Although proving this was the most important, if we want to establish an encoding scheme for sequences, but we have to fill in some gaps yet. These are related notions similar to existence and uniqueness (although on uniqueness, “at most one” should be meant here, and the conjunction of both is delayed as a final result).

Uniqueness of encoding, achieved by minimalization

Because let us remember, our ultimate question is: what number should stand for the encoding of sequence $⟨a_0,\dots ,a_{n-1}⟩$? The specification declares only an existential quantification, not yet a functional connection. We want a constructive and algorithmic way, even more, a (total) recursive function for the encoding.

Totality, because minimalization is restricted to special functions

This gap can be filled in in a straightforward way: we shall use minimalization, and the totality of the resulting function is ensured by everything we have proven till now (i.e. the correctness of the definition of $\beta$ by meeting its specification). In fact, the specification

$\mathrm{\forall }{a}_0,\dots ,a_{n-1}\mathrm{\exists }s\mathrm{\forall }i<n\beta (s,i)=a_i$

plays a role here of a more general notion (“special function”). The importance of this notion is that it enables us to split off the (sub)class of (total) recursive functions from the (super)class of partial recursive functions. In brief, the specification says exactly: a function f satisfying specification

$f(a_0,\dots ,a_{n-1},s)=0\leftrightarrow \mathrm{\forall }i<n(\beta (s,i)=a_i)$

is a special function, i.e. for each fixed combination of all-but-last arguments, the function f has root in its last argument:

$\mathrm{\forall }{a}_0,\dots ,a_{n-1}\mathrm{\exists }s(f(a_0,\dots ,a_{n-1},s)=0)$

The Gödel numbering function g can be chosen to be total recursive

Thus, let us choose the minimal possible number that fits well in the specification of the $\beta$ function:

$g:{\mathbb{\mathbb{N}}}^n\to \mathbb{\mathbb{N}}$

$⟨a_0,\dots ,a_{n-1}⟩⟼\mu a.[\mathrm{\forall }i<n(\beta (a,i)=a_i)]$

and it can be proven (using the notions of the previous section ) that g is (total) recursive.

Access of length

If we use the above scheme for encoding sequences only in contexts where the length of the sequences is fixed, then no problem arises. In other words, we can use them in an analogous way as arrays are used in programming.

But sometimes we need dynamically stretching sequences, or we need to deal with sequences whose length cannot be typed in a static way. In other words, we may encode sequences in an analogous way as we use lists in programming.

An example for both cases: if we make the Gödel numbering of a Turing machine, then the each row in matrix of the “program” can be represented with tuples, sequences of fixed length (thus, without storing the length), because the number of the columns is fixed. But if we want to reason about configuration-like things (of Turing-machines), and specially, we want to encode the significant part of the tape of a running Turing machine, then we have to represent sequences together with their length. Moreover, we can mimic dynamically stretching sequences by representing sequence concatenation (or at least, augmenting a sequence with one more element) with a [totally] recursive function.

Length can be stored stored simply as a surplus member:

$g:{\mathbb{\mathbb{N}}}^{*}\to \mathbb{\mathbb{N}}$

$⟨a_0,\dots ,a_{n-1},a_n⟩⟼\mu a.[a_0=n\wedge \mathrm{\forall }i<n(\beta (a,i+1)=a_i)]$

The corresponding modification of the proof is straightforward, by adding a surplus

$x\equiv n(\mathrm{mod}{m}_0)$

to the system of simultaneous congruences (provided that the surplus member index is chosen to be 0). Also the assumptions etc. have to be modified accordingly.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Each chapter is downloadable verbatim on author's page.
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  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 Template:Dead link
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Translation of Smullyan 1992.

External links

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534