Mahler's compactness theorem

Template:More footnotes In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, stating that any sentence in the first-order language of fields true for the complex numbers is also true for any algebraically closed field of characteristic 0.

History

An incipient form of a transfer principle was described by Leibniz under the name of "the Law of Continuity". Here infinitesimals are expected to have the "same" properties as appreciable numbers. Similar tendencies are found in Cauchy, who used infinitesimals to define both the continuity of functions (in Cours d'Analyse) and a form of the Dirac delta function.

In 1955, Jerzy Łoś proved the transfer principle for any hyperreal number system. Its most common use is in Abraham Robinson's non-standard analysis of the hyperreal numbers, where the transfer principle states that any sentence expressible in a certain formal language that is true of real numbers is also true of hyperreal numbers.

Transfer principle for the hyperreals

The transfer principle concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the hyperreals. The field *R includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical realisation of a project initiated by Leibniz.

The idea is to express analysis over R in a suitable language of mathematical logic, and then point out that this language applies equally well to *R. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to internal sets rather than to all sets. As Robinson put it, the sentences of [the theory] are interpreted in *R in Henkin's sense.

The theorem to the effect that each proposition valid over R, is also valid over *R, is called the transfer principle.

There are several different versions of the transfer principle, depending on what model of non-standard mathematics is being used. In terms of model theory, the transfer principle states that a map from a standard model to a non-standard model is an elementary embedding (an embedding preserving the truth values of all statements in a language), or sometimes a bounded elementary embedding (similar, but only for statements with bounded quantifiers).

The transfer principle appears to lead to contradictions if it is not handled correctly. For example, since the hyperreal numbers form a non-Archimedean ordered field and the reals form an Archimedean ordered field, the property of being Archimedean ("every positive real is larger than 1/n for some positive integer n") seems at first sight not to satisfy the transfer principle. The statement "every positive hyperreal is larger than 1/n for some positive integer n" is false; however the correct interpretation is "every positive hyperreal is larger than 1/n for some positive hyperinteger n". In other words, the hyperreals appear to be archimedean to an internal observer living in the non-standard universe, but appear to be non-archimedean to an external observer outside the universe.

A freshman-level accessible formulation of the transfer principle is Keisler's book Elementary Calculus: An Infinitesimal Approach.

Example

Every real x satisfies the inequality

$x\geq [x]$

where [x] is the integer part function. By a typical application of the transfer principle, every hyperreal x satisfies the inequality

$x\geq {}^{*}\![x]$,

where *[.] is the natural extension of the integer part function. If x is infinite, then the hyperinteger *[x] is infinite, as well.

Generalizations of the concept of number

Historically, the concept of number has been repeatedly generalized. The addition of 0 to the natural numbers $\mathbb {N}$ was a major intellectual accomplishment in its time. The addition of negative integers to form $\mathbb {Z}$ already constituted a departure from the realm of immediate experience to the realm of mathematical models. The further extension, the rational numbers $\mathbb {Q}$, is more familiar to a layperson than their completion $\mathbb {R}$, partly because the reals do not correspond to any physical reality (in the sense of measurement and computation) different from that represented by $\mathbb {Q}$. Thus, the notion of an irrational number is meaningless to even the most powerful floating-point computer. The necessity for such an extension stems not from physical observation but rather from the internal requirements of mathematical coherence. The infinitesimals entered mathematical discourse at a time when such a notion was required by mathematical developments at the time, namely the emergence of what became known as the infinitesimal calculus. As already mentioned above, the mathematical justification for this latest extension was delayed by three centuries. Keisler wrote:

"In discussing the real line we remarked that we have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line."

The self-consistent development of the hyperreals turned out to be possible if every true first-order logic statement that uses basic arithmetic (the natural numbers, plus, times, comparison) and quantifies only over the real numbers was assumed to be true in a reinterpreted form if we presume that it quantifies over hyperreal numbers. For example, we can state that for every real number there is another number greater than it:

$\forall x\in \mathbb {R} \quad \exists y\in \mathbb {R} \quad x<y.$

The same will then also hold for hyperreals:

$\forall x\in {}^{\star }\mathbb {R} \quad \exists y\in {}^{\star }\mathbb {R} \quad x<y.$

Another example is the statement that if you add 1 to a number you get a bigger number:

$\forall x\in \mathbb {R} \quad x<x+1$

which will also hold for hyperreals:

$\forall x\in {}^{\star }\mathbb {R} \quad x<x+1.$

The correct general statement that formulates these equivalences is called the transfer principle. Note that in many formulas in analysis quantification is over higher order objects such as functions and sets which makes the transfer principle somewhat more subtle than the above examples suggest.

Differences between R and ^*R

The transfer principle however doesn't mean that R and *R have identical behavior. For instance, in *R there exists an element ω such that

$1<\omega ,\quad 1+1<\omega ,\quad 1+1+1<\omega ,\quad 1+1+1+1<\omega ,\ldots$

but there is no such number in R. This is possible because the nonexistence of this number cannot be expressed as a first order statement of the above type. A hyperreal number like ω is called infinitely large; the reciprocals of the infinitely large numbers are the infinitesimals.

The hyperreals *R form an ordered field containing the reals R as a subfield. Unlike the reals, the hyperreals do not form a standard metric space, but by virtue of their order they carry an order topology.

Constructions of the hyperreals

The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed. Vladimir Kanovei and Shelah give a construction of a definable, countably saturated elementary extension of the structure consisting of the reals and all finitary relations on it.

In its most general form, transfer is a bounded elementary embedding between structures.

Statement

The ordered field ^*R of nonstandard real numbers properly includes the real field R. Like all ordered fields that properly include R, this field is non-Archimedean. It means that some members x ≠ 0 of ^*R are infinitesimal, i.e.,

$\underbrace {\left|x\right|+\cdots +\left|x\right|} _{n{\text{ terms}}}<1{\text{ for every finite cardinal number }}n.\,$

The only infinitesimal in R is 0. Some other members of ^*R, the reciprocals y of the nonzero infinitesimals, are infinite, i.e.,

$\underbrace {1+\cdots +1} _{n{\text{ terms}}}<\left|y\right|{\text{ for every finite cardinal number }}n.\,$

The underlying set of the field ^*R is the image of R under a mapping A Template:Mapsto ^*A from subsets A of R to subsets of ^*R. In every case

$A\subseteq {^{*}\!A},\,$

with equality if and only if A is finite. Sets of the form ^*A for some $\scriptstyle A\,\subseteq \,\mathbb {R}$ are called standard subsets of ^*R. The standard sets belong to a much larger class of subsets of ^*R called internal sets. Similarly each function

$f:A\rightarrow \mathbb {R} \,$

extends to a function

${^{*}\!f}:{^{*}\!A}\rightarrow {^{*}\mathbb {R} };\,$

these are called standard functions, and belong to the much larger class of internal functions. Sets and functions that are not internal are external.

The importance of these concepts stems from their role in the following proposition and is illustrated by the examples that follow it.

The transfer principle:

• Suppose a proposition that is true of ^*R can be expressed via functions of finitely many variables (e.g. (x, y) Template:Mapsto x + y), relations among finitely many variables (e.g. x ≤ y), finitary logical connectives such as and, or, not, if...then..., and the quantifiers

$\forall x\in \mathbb {R} {\text{ and }}\exists x\in \mathbb {R} .\,$

For example, one such proposition is

$\forall x\in \mathbb {R} \ \exists y\in \mathbb {R} \ x+y=0.\,$

Such a proposition is true in R if and only if it is true in ^*R when the quantifier

$\forall x\in {^{*}\!\mathbb {R} }\,$

replaces

$\forall x\in \mathbb {R} ,\,$

and similarly for $\exists$.

• Suppose a proposition otherwise expressible as simply as those considered above mentions some particular sets $\scriptstyle A\,\subseteq \,\mathbb {R}$. Such a proposition is true in R if and only if it is true in ^*R with each such "A" replaced by the corresponding ^*A. Here are two examples:

• The set

$[0,1]^{\ast }=\{\,x\in \mathbb {R} :0\leq x\leq 1\,\}^{\ast }$

must be

$\{\,x\in {^{*}\mathbb {R} }:0\leq x\leq 1\,\},$

including not only members of R between 0 and 1 inclusive, but also members of ^*R between 0 and 1 that differ from those by infinitesimals. To see this, observe that the sentence

$\forall x\in \mathbb {R} \ (x\in [0,1]{\text{ if and only if }}0\leq x\leq 1)$

is true in R, and apply the transfer principle.

• The set ^*N must have no upper bound in ^*R (since the sentence expressing the non-existence of an upper bound of N in R is simple enough for the transfer principle to apply to it) and must contain n + 1 if it contains n, but must not contain anything between n and n + 1. Members of

${^{*}\mathbb {N} }\setminus \mathbb {N} \,$

are "infinite integers".)

• Suppose a proposition otherwise expressible as simply as those considered above contains the quantifier

$\forall A\subseteq \mathbb {R} \dots {\text{ or }}\exists A\subseteq \mathbb {R} \dots \ .$

Such a proposition is true in R if and only if it is true in ^*R after the changes specified above and the replacement of the quantifiers with

$[\forall {\text{ internal }}A\subseteq {^{*}\mathbb {R} }\dots ]\,$

and

$[\exists {\text{ internal }}A\subseteq {^{*}\mathbb {R} }\dots ]\ .$

Three examples

• Every nonempty internal subset of ^*R that has an upper bound in ^*R has a least upper bound in ^*R. Consequently the set of all infinitesimals is external.
  • The well-ordering principle implies every nonempty internal subset of ^*N has a smallest member. Consequently the set

${^{*}\mathbb {N} }\setminus \mathbb {N} \,$

of all infinite integers is external.

• If n is an infinite integer, then the set {1, ..., n} (which is not standard) must be internal. To prove this, first observe that the following is trivially true:

$\forall n\in \mathbb {N} \ \exists A\subseteq \mathbb {N} \ \forall x\in \mathbb {N} \ [x\in A{\text{ iff }}x\leq n].$

Consequently

$\forall n\in {^{*}\mathbb {N} }\ \exists {\text{ internal }}A\subseteq {^{*}\mathbb {N} }\ \forall x\in {^{*}\mathbb {N} }\ [x\in A{\text{ iff }}x\leq n].$

• As with internal sets, so with internal functions: Replace

$\forall f:A\rightarrow \mathbb {R} \dots \,$

with

$\forall {\text{ internal }}f:{^{*}\!A}\rightarrow {^{*}\mathbb {R} }\dots$

and similarly with $\exists$ in place of $\forall$.

For example: If n is an infinite integer, then the complement of the image of any internal one-to-one function ƒ from the infinite set {1, ..., n} into {1, ..., n, n + 1, n + 2, n + 3} has exactly three members. Because of the infiniteness of the domain, the complements of the images of one-to-one functions from the former set to the latter come in many sizes, but most of these functions are external.

This last example motivates an important definition: A *-finite (pronounced star-finite) subset of ^*R is one that can be placed in internal one-to-one correspondence with {1, ..., n} for some n ∈ ^*N.

See also

• Elementary Calculus: An Infinitesimal Approach

Notes

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References

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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Hardy, Michael: "Scaled Boolean algebras". Adv. in Appl. Math. 29 (2002), no. 2, 243–292.
• 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
• Template:Eom
• Łoś, Jerzy (1955) Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres. Mathematical interpretation of formal systems, pp. 98–113. North-Holland Publishing Co., Amsterdam.
• 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

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