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In [[mathematics]], '''Suslin's problem''' is a question about [[totally ordered set]]s posed by [[Mikhail Yakovlevich Suslin]] in a work published posthumously in 1920.<ref>{{cite journal
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|title=Problème 3
|last= Souslin
|first=M.
|journal=Fundamenta Mathematicae
|volume=1
|year=1920
|page=223
}}</ref>
It has been shown to be [[independence (mathematical logic)|independent]] of the standard axiomatic system of [[set theory]] known as [[ZFC]]: the statement can neither be proven nor disproven from those axioms.<ref>{{cite journal
|title=Iterated Cohen extensions and Souslin's problem
|last=Solovay
|first=R. M.
|coauthors=Tennenbaum, S.
|journal=Ann. Of Math. (2)
|volume=94
|year=1971
|pages=201–245
|doi=10.2307/1970860
|issue=2
|publisher=Annals of Mathematics
|jstor=1970860
}}</ref>
 
(Suslin is also sometimes written with the French transliteration as Souslin, from the Cyrillic Суслин.)
 
==Formulation==
Given a [[non-empty]] [[totally ordered set]] ''R'' with the following four properties:
# ''R'' does not have a [[greatest element|least nor a greatest element]];
# the order on ''R'' is [[dense order|dense]] (between any two elements there is another);
# the order on ''R'' is [[completeness (order theory)|complete]], in the sense that every non-empty bounded subset has a [[supremum]] and an [[infimum]];
# every collection of mutually [[disjoint sets|disjoint]] non-empty open [[interval (mathematics)|interval]]s in ''R'' is [[countable]] (this is the [[countable chain condition]]).
Is ''R'' necessarily [[order isomorphism|order-isomorphic]] to the [[real line]] '''R'''?
 
If the requirement for the countable chain condition is replaced with the requirement that ''R'' contains a countable dense subset (i.e., ''R'' is a [[separable space]]) then the answer is indeed yes: any such set ''R'' is necessarily isomorphic to '''R'''.
 
==Implications==
Any totally ordered set that is ''not'' isomorphic to '''R''' but satisfies (1)&nbsp;–&nbsp;(4) is known as a '''Suslin line'''. The existence of Suslin lines has been proven to be equivalent to the existence of [[Suslin tree]]s. Suslin lines exist if the additional constructibility axiom [[Axiom of constructibility|V equals L]] is assumed.
 
The '''Suslin hypothesis''' says that there are no Suslin lines: that every countable-chain-condition dense complete linear order without endpoints is isomorphic to the real line. Equivalently, that every [[tree (set theory)|tree]] of height &omega;<sub>1</sub> either has a branch of length &omega;<sub>1</sub> or an [[antichain]] of cardinality <math>\aleph_1.</math>
 
The '''generalized Suslin hypothesis''' says that for every infinite [[regular cardinal]] &kappa; every tree of height &kappa; either has a branch of length &kappa; or an antichain of cardinality &kappa;.
 
The Suslin hypothesis is independent of ZFC, and is independent of both the [[generalized continuum hypothesis]] and of the negation of the [[continuum hypothesis]]. However, [[Martin's axiom]] plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis. It is not known whether the Generalized Suslin Hypothesis is consistent with the Generalized Continuum Hypothesis; however, since the combination implies the negation of the [[square principle]] at a singular strong [[limit cardinal]]—in fact, at all singular cardinals and all regular successor cardinals—it implies that the [[axiom of determinacy]] holds in L(R) and is believed to imply the existence of an [[inner model]] with a [[superstrong cardinal]].
 
== See also ==
* [[List of statements undecidable in ZFC]]
* [[AD+]]
 
==Notes==
<references/>
 
==References==
*{{springer|id=S/s091460
|first=V.N. |last=Grishin|title=Suslin hypothesis}}
 
{{Set theory}}
 
[[Category:Independence results]]
[[Category:Order theory]]

Revision as of 19:22, 3 March 2014

There have been numerous recent research on the amount of obese folks inside society today. Both adults plus youngsters, young individuals plus seniors are usually too heavy. If you too are overweight and desire to make a change inside your appearance, or if a doctor has told we which fat loss is mandatory, then the following post will provide several background info, tips and tricks on how to lose fat swiftly.

Frequently you think we are hungry whenever you are really thirsty. This means you eat over we want. So we gain fat. If you wish To lose weight fast, you have to train ourselves to drink more fluids, throughout the day, particularly water. We must even teach ourselves to carry water with us as we are running our errands or keeping it close at hand at the desks, whilst you function. This usually keep us sipping throughout the day.

The right time to do any cardio exercise is initially thing inside the morning before you eat anything. What happens is that considering you haven't eaten anything throughout the evening, as we have been asleep, a body has to employ calories that are stored inside a body plus burning calories is what you require if you would like to lose weight.

One important tip for losing fat naturally is to exercise everyday. Cardiovascular exercises such as aerobics, swimming, jogging plus cycling ought to be done everyday as they assist in losing weight from all over the body. These ought to be combined with exercises aimed at a specific body part, that strengthen plus tone up muscles of the particular body area. Weight training, flexibility exercises and resistance training could be a part of your exercise regimen. But, these ought to be undertaken just beneath the guidance plus supervision of the trained expert. So, when you may afford, hire a trainer whom would come plus teach you effective weight loss exercises. Otherwise, receive some exercise videos within the marketplace plus undertake exercises at house, watching them.

After a healthy lifestyle is important for keeping a healthy body weight. Our body requirements rest to restore so does the mind. Be sure you receive enough rest, follow the older adage 'early to bed and early to rise makes a man healthy, rich plus lose weight wise'. If your schedule permits you, take a energy nap of 10 - 15 minutes inside the afternoon, this usually keep the stamina levels significant. A word of caution here is to not surpass a energy naps, otherwise you'll feel sleepy. Supplement a habit of drinking soda with fruit juices or greater yet with water. Avoid processed plus sugary snacks like candies, cookies, rather choose snacks that are significant inside fiber and important nutrients.

These meals arrive because single servings so you eat simply the appropriate amount to stay inside a calorie count. All you must do is warm the meal inside the microwave and enjoy. Along with removing temptation, diet plans moreover remove the difficulty associated with meal planning and buying whenever you go on a diet. Forget all the complicated stuff. Just order the food that appeal to you plus they will arrive at a door. If you don't wish To pick plus choose a food you are able to even allow the company choose the meals for you so we have a balanced diet.

Losing fat could be rather difficult, however with this you need to be able to get to a weight objective. If you have any query, comments or concerns please feel free to comment. For an expert opinion please contact your local physician or dietician.