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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 | |||
|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) – (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 ω<sub>1</sub> either has a branch of length ω<sub>1</sub> or an [[antichain]] of cardinality <math>\aleph_1.</math> | |||
The '''generalized Suslin hypothesis''' says that for every infinite [[regular cardinal]] κ every tree of height κ either has a branch of length κ or an antichain of cardinality κ. | |||
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 00:52, 4 February 2014
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin in a work published posthumously in 1920.[1] It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: the statement can neither be proven nor disproven from those axioms.[2]
(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 least nor a greatest element;
- the order on R is dense (between any two elements there is another);
- the order on R is complete, in the sense that every non-empty bounded subset has a supremum and an infimum;
- every collection of mutually disjoint non-empty open intervals in R is countable (this is the countable chain condition).
Is R necessarily 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) – (4) is known as a Suslin line. The existence of Suslin lines has been proven to be equivalent to the existence of Suslin trees. Suslin lines exist if the additional constructibility axiom 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 of height ω1 either has a branch of length ω1 or an antichain of cardinality
The generalized Suslin hypothesis says that for every infinite regular cardinal κ every tree of height κ either has a branch of length κ or an antichain of cardinality κ.
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
Notes
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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
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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
References
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