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A hypothesis that states that every linearly ordered set without first and last elements that is moreover complete, dense and satisfies the Suslin condition, is isomorphic to the real line. Here, completeness signifies the existence of a least upper bound for every non-empty bounded subset, density denotes the non-emptiness of any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s0914601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s0914602.png" />, and the [[Suslin condition|Suslin condition]] requires that every family of pairwise-disjoint intervals is countable. The real line possesses all the properties that figure in the formulation of the Suslin hypothesis. The Suslin hypothesis thus states that the above properties of an ordered set define it completely. This hypothesis was formulated in 1920 by M.Ya. Suslin [[#References|[1]]].
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A hypothesis that states that every linearly ordered set without first and last elements that is moreover complete, dense and satisfies the Suslin condition, is order-isomorphic to the real line $ \mathbb{R} $. Here, completeness signifies the existence of a least upper bound for every non-empty bounded subset, density denotes the non-emptiness of any interval $ (a,b) $, where $ a < b $, and the [[Suslin condition|Suslin condition]] requires that every family of pairwise-disjoint intervals is countable. The real line possesses all the properties that appear in the formulation of the Suslin hypothesis. The Suslin hypothesis thus states that the above properties of an ordered set define it completely. This hypothesis was formulated in 1920 by M.Ya. Suslin [[#References|[1]]].
  
Within the framework of the system ZFC (the system ZF with the [[Axiom of choice|axiom of choice]]) it is impossible to prove or disprove the Suslin hypothesis, assuming that ZF is consistent. It follows from Gödel's [[axiom of constructibility]] (see [[Gödel constructive set|Gödel constructive set]]) that the negation of the Suslin hypothesis holds. The consistency of the Suslin hypothesis with the axioms of ZFC is proved by the construction of a corresponding model, using a variant of the [[Forcing method|forcing method]] (iterated forcing). The addition of the [[Continuum hypothesis|continuum hypothesis]] to ZFC does not give either a positive or negative solution of the Suslin hypothesis.
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Within the framework of the system $ \mathsf{ZFC} $ (the system $ \mathsf{ZF} $ with the [[Axiom of choice|axiom of choice]]), it is impossible to prove or disprove the Suslin hypothesis, assuming that $ \mathsf{ZF} $ is consistent. It follows from Gödel’s [[axiom of constructibility]], $ V = L $ (see [[Gödel constructive set|Gödel constructive set]]), that the negation of the Suslin hypothesis holds. The consistency of the Suslin hypothesis with the axioms of $ \mathsf{ZFC} $ is proved by the construction of a corresponding model, using a variant of the [[Forcing method|forcing method]], namely, iterated forcing. The addition of the [[Continuum hypothesis|continuum hypothesis]] ($ \mathsf{CH} $) to $ \mathsf{ZFC} $ does not give either a positive or negative solution to the Suslin hypothesis.
  
The Suslin hypothesis and its generalizations have had a great influence on the development of [[Axiomatic set theory|axiomatic set theory]]. Many ideas and methods have been worked out in conjunction with it. These include Jensen's combinatorial principles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s0914603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s0914604.png" /> (see [[#References|[4]]]), the theory of the fine structure of the constructible hierarchy (see [[#References|[5]]]), Martin's axiom [[#References|[7]]], and the iterated forcing method [[#References|[2]]].
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The Suslin hypothesis and its generalizations have had a great influence on the development of [[Axiomatic set theory|axiomatic set theory]]. Many ideas and methods have been worked out in conjunction with it. These include Jensen’s combinatorial principles $ \diamondsuit_{\kappa} $ and $ \Box_{\kappa} $ ([[#References|[4]]]), the theory of the fine structure of the constructible hierarchy ([[#References|[5]]]), Martin’s axiom ([[#References|[7]]]), and the iterated forcing method ([[#References|[2]]]).
  
Jensen's principle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s0914605.png" />: A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s0914606.png" /> of a cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s0914607.png" /> is said to be closed unbounded if it contains all its limit points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s0914608.png" /> and if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s0914609.png" />, there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146011.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146012.png" /> is said to be stationary if its intersection with every closed unbounded subset of a cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146013.png" /> is non-empty.
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'''Jensen’s principle $ \diamondsuit_{\kappa} $:''' A subset $ A \subseteq \kappa $ of a cardinal $ \kappa = \{ \alpha \mid \alpha < \kappa \} $ is said to be '''closed unbounded''' (or '''club''' for short) if and only if it contains all of its limit points $ < \kappa $ and if, for any $ \alpha < \kappa $, there is a $ \beta \in A $ such that $ \alpha < \beta $. A subset $ A \subseteq \kappa $ is said to be '''stationary''' if and only if its intersection with every club subset of $ \kappa $ is non-empty.
  
Jensen's principle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146014.png" />: There exists a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146016.png" />, such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146017.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146018.png" /> is stationary. For every regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146019.png" />, the principle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146020.png" /> follows from the axiom of constructibility, while the negation of the Suslin hypothesis follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146021.png" />. Jensen's combinatorial principles, as well as Martin's axiom (see below), have been successfully used in topology (see [[#References|[4]]], [[#References|[6]]], [[#References|[8]]]).
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'''Jensen’s principle $ \Box_{\kappa} $:''' There exists a sequence $ \langle S_{\alpha} \mid \alpha < \kappa \rangle $, where $ S_{\alpha} \subseteq \alpha $, such that for every $ X \subseteq \kappa $, the subset $ \{ \alpha < \kappa \mid S_{\alpha} = X \cap \alpha \} $ of $ \kappa $ is stationary. For every regular cardinal $ \kappa $, the principle $ \diamondsuit_{\kappa} $ follows from the axiom of constructibility, while the negation of the Suslin hypothesis follows from $ \diamondsuit_{\omega_{1}} $. Jensen’s combinatorial principles, as well as Martin’s axiom (see below), have been successfully used in topology ([[#References|[4]]], [[#References|[6]]], [[#References|[8]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146022.png" /> be a partially ordered set. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146023.png" /> is called dense if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146024.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146026.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146027.png" /> is said to be compatible if for any finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146028.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146030.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146031.png" />. Two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146033.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146034.png" /> are said to be incompatible if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146035.png" /> is not compatible. It is said that a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146036.png" /> satisfies the countable anti-chain condition if every set that consists of pairwise-incompatible elements is countable. Martin's axiom (MA) states the following: If a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146037.png" /> satisfies the countable anti-chain condition and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146038.png" /> is a family of dense subsets of cardinality less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146039.png" />, then there is a compatible set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146040.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146041.png" /> the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146042.png" /> is non-empty.
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Let $ \langle \mathbb{P},\leq \rangle $ be a partially ordered set. A set $ D \subseteq \mathbb{P} $ is called '''dense''' if and only if for every $ p \in \mathbb{P} $, there is a $ d \in D $ such that $ d \leq p $. A set $ Q \subseteq \mathbb{P} $ is said to be '''compatible''' if and only if for any finite subset $ F \subseteq Q $, there is a $ p \in \mathbb{P} $ such that $ p \leq r $ for every $ r \in F $. Two elements, $ p_{1} $ and $ p_{2} $, of $ \mathbb{P} $ are said to be '''incompatible''' if and only if the set $ \{ p_{1},p_{2} \} $ is not compatible. It is said that a partially ordered set $ \langle \mathbb{P},\leq \rangle $ satisfies the '''countable anti-chain condition''' if and only if every set that consists of pairwise-incompatible elements is countable. Martin's axiom ($ \mathsf{MA} $) states the following: If a partially ordered set $ \langle \mathbb{P},\leq \rangle $ satisfies the countable anti-chain condition and if $ \mathcal{P} $ is a family of dense subsets of cardinality less than $ 2^{\aleph_{0}} $, then there is a compatible set $ Q \subseteq \mathbb{P} $ such that for every $ D \in \mathcal{P} $, the intersection $ D \cap Q $ is non-empty.
  
In the presence of the [[Continuum hypothesis|continuum hypothesis]] (CH), Martin's axiom can be proved. The most interesting results are obtained by a combination of Martin's axiom (MA) and the negation of the continuum hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146043.png" />. The principle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146044.png" /> contradicts the combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146045.png" />, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146046.png" /> implies CH. It often turns out that a result that can be inferred from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146047.png" /> will be disproved under the assumption <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146048.png" />. This is, e.g., the case with the Suslin hypothesis. Indeed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146049.png" /> implies the Suslin hypothesis, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146050.png" /> implies the negation of the Suslin hypothesis.
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In the presence of the [[Continuum hypothesis|continuum hypothesis]], Martin’s axiom can be proved. The most interesting results are obtained by a combination of Martin’s axiom and the negation of the continuum hypothesis ($ \neg \mathsf{CH} $). The principle $ \diamondsuit_{\omega_{1}} $ contradicts the combination $ \mathsf{MA} + \neg \mathsf{CH} $, since $ \diamondsuit_{\omega_{1}} $ implies $ \mathsf{CH} $. It often turns out that a result that can be inferred from $ \diamondsuit_{\omega_{1}} $ will be disproved under the assumption $ \mathsf{MA} + \neg \mathsf{CH} $. This is, for example, the case with the Suslin hypothesis. Indeed, $ \mathsf{MA} + \neg \mathsf{CH} $ implies the Suslin hypothesis, while $ \diamondsuit_{\omega_{1}} $ implies the negation of the Suslin hypothesis.
  
The combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146051.png" /> is compatible with ZFC if ZF is consistent.
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The combination $ \mathsf{MA} + \neg \mathsf{CH} $ is compatible with $ \mathsf{ZFC} $ if $ \mathsf{ZF} $ is consistent.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. [M.Ya. Suslin] Souslin,  "Problème 3"  ''Fundam. Math.'' , '''1'''  (1920)  pp. 223</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K.J. Devlin,  H. Johnsbråten,  "The Souslin problem" , ''Lect. notes in math.'' , '''405''' , Springer  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T.J. Jech,  "Lectures in set theory: with particular emphasis on the method of forcing" , ''Lect. notes in math.'' , '''217''' , Springer  (1971)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Barwise (ed.) , ''Handbook of mathematical logic'' , North-Holland  (1977)  pp. Chapts. B4-B7</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K.J. Devlin,  "Aspects of constructibility" , ''Lect. notes in math.'' , '''354''' , Springer  (1973)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.V. Fedorchuk,  "Completely closed mappings and the compatibility of certain general topology theorems with the axioms of set theory"  ''Mat. Sb.'' , '''99''' :  1  (1976)  pp. 3–33  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D.A. Martin,  R. Solovay,  "Internal Cohen extensions"  ''Ann. Math. Logic'' , '''2'''  (1970)  pp. 143–178</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.I. Malykhin,  "Topology and forcing"  ''Russian Math. Surveys'' , '''38''' :  1  (1983)  pp. 77–136  ''Uspekhi Mat. Nauk'' , '''38''' :  1  (1983)  pp. 69–118</TD></TR></table>
 
  
 +
<table>
 +
<TR><TD valign="top">[1]</TD><TD valign="top"> M. [M.Ya. Suslin] Souslin, “Problème 3”, ''Fundam. Math.'', '''1''' (1920), pp. 223.</TD></TR>
 +
<TR><TD valign="top">[2]</TD><TD valign="top"> K.J. Devlin, H. Johnsbråten, “The Souslin problem”, ''Lect. notes in math.'', '''405''', Springer (1974).</TD></TR>
 +
<TR><TD valign="top">[3]</TD><TD valign="top"> T.J. Jech, “Lectures in set theory: with particular emphasis on the method of forcing”, ''Lect. notes in math.'', '''217''', Springer (1971).</TD></TR>
 +
<TR><TD valign="top">[4]</TD><TD valign="top"> J. Barwise (ed.), ''Handbook of mathematical logic'', North-Holland (1977), Chapts. B4-B7.</TD></TR>
 +
<TR><TD valign="top">[5]</TD><TD valign="top"> K.J. Devlin, “Aspects of constructibility”, ''Lect. notes in math.'', '''354''', Springer (1973).</TD></TR>
 +
<TR><TD valign="top">[6]</TD><TD valign="top"> V.V. Fedorchuk, “Completely closed mappings and the compatibility of certain general topology theorems with the axioms of set theory”, ''Mat. Sb.'', '''99''': 1 (1976), pp. 3–33. (In Russian)</TD></TR>
 +
<TR><TD valign="top">[7]</TD><TD valign="top"> D.A. Martin, R. Solovay, “Internal Cohen extensions”, ''Ann. Math. Logic'', '''2''' (1970), pp. 143–178.</TD></TR>
 +
<TR><TD valign="top">[8]</TD><TD valign="top"> V.I. Malykhin, “Topology and forcing”, ''Russian Math. Surveys'', '''38''': 1 (1983), pp. 77–136; ''Uspekhi Mat. Nauk'', '''38''': 1 (1983), pp. 69–118.</TD></TR>
 +
</table>
  
 +
====Comments====
  
====Comments====
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Jensen’s combinatorial principles $ \diamondsuit $ and $ \Box $ are called '''diamond''' and '''square''' respectively. The countable anti-chain condition is sometimes called the '''countable chain condition''', and is then also abbreviated to '''c.c.c.'''
Jensen's combinatorial principles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091460/s09146053.png" /> are called diamond and square, respectively. The countable anti-chain condition is sometimes called the countable chain condition, and is then also abbreviated to ccc.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.J. Devlin,   "Constructibility" , Springer (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D.H. Fremlin,   "Consequences of Martin's axiom" , Cambridge Univ. Press (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> T.J. Jech,   "Multiple forcing" , Cambridge Univ. Press (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> T.J. Jech,   "Set theory" , Acad. Press (1978) (Translated from German)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> K. Kunen,   "Set theory, an introduction to independence proofs" , North-Holland (1980)</TD></TR></table>
+
 
 +
<table>
 +
<TR><TD valign="top">[a1]</TD><TD valign="top"> K.J. Devlin, “Constructibility”, Springer (1984).</TD></TR>
 +
<TR><TD valign="top">[a2]</TD><TD valign="top"> D.H. Fremlin, “Consequences of Martin’s axiom”, Cambridge Univ. Press (1984).</TD></TR>
 +
<TR><TD valign="top">[a3]</TD><TD valign="top"> T.J. Jech, “Multiple forcing”, Cambridge Univ. Press (1986).</TD></TR>
 +
<TR><TD valign="top">[a4]</TD><TD valign="top"> T.J. Jech, “Set theory”, Acad. Press (1978). (Translated from German)</TD></TR>
 +
<TR><TD valign="top">[a5]</TD><TD valign="top"> K. Kunen, “Set theory, an introduction to independence proofs”, North-Holland (1980).</TD></TR>
 +
</table>

Revision as of 21:51, 3 December 2016

A hypothesis that states that every linearly ordered set without first and last elements that is moreover complete, dense and satisfies the Suslin condition, is order-isomorphic to the real line $ \mathbb{R} $. Here, completeness signifies the existence of a least upper bound for every non-empty bounded subset, density denotes the non-emptiness of any interval $ (a,b) $, where $ a < b $, and the Suslin condition requires that every family of pairwise-disjoint intervals is countable. The real line possesses all the properties that appear in the formulation of the Suslin hypothesis. The Suslin hypothesis thus states that the above properties of an ordered set define it completely. This hypothesis was formulated in 1920 by M.Ya. Suslin [1].

Within the framework of the system $ \mathsf{ZFC} $ (the system $ \mathsf{ZF} $ with the axiom of choice), it is impossible to prove or disprove the Suslin hypothesis, assuming that $ \mathsf{ZF} $ is consistent. It follows from Gödel’s axiom of constructibility, $ V = L $ (see Gödel constructive set), that the negation of the Suslin hypothesis holds. The consistency of the Suslin hypothesis with the axioms of $ \mathsf{ZFC} $ is proved by the construction of a corresponding model, using a variant of the forcing method, namely, iterated forcing. The addition of the continuum hypothesis ($ \mathsf{CH} $) to $ \mathsf{ZFC} $ does not give either a positive or negative solution to the Suslin hypothesis.

The Suslin hypothesis and its generalizations have had a great influence on the development of axiomatic set theory. Many ideas and methods have been worked out in conjunction with it. These include Jensen’s combinatorial principles $ \diamondsuit_{\kappa} $ and $ \Box_{\kappa} $ ([4]), the theory of the fine structure of the constructible hierarchy ([5]), Martin’s axiom ([7]), and the iterated forcing method ([2]).

Jensen’s principle $ \diamondsuit_{\kappa} $: A subset $ A \subseteq \kappa $ of a cardinal $ \kappa = \{ \alpha \mid \alpha < \kappa \} $ is said to be closed unbounded (or club for short) if and only if it contains all of its limit points $ < \kappa $ and if, for any $ \alpha < \kappa $, there is a $ \beta \in A $ such that $ \alpha < \beta $. A subset $ A \subseteq \kappa $ is said to be stationary if and only if its intersection with every club subset of $ \kappa $ is non-empty.

Jensen’s principle $ \Box_{\kappa} $: There exists a sequence $ \langle S_{\alpha} \mid \alpha < \kappa \rangle $, where $ S_{\alpha} \subseteq \alpha $, such that for every $ X \subseteq \kappa $, the subset $ \{ \alpha < \kappa \mid S_{\alpha} = X \cap \alpha \} $ of $ \kappa $ is stationary. For every regular cardinal $ \kappa $, the principle $ \diamondsuit_{\kappa} $ follows from the axiom of constructibility, while the negation of the Suslin hypothesis follows from $ \diamondsuit_{\omega_{1}} $. Jensen’s combinatorial principles, as well as Martin’s axiom (see below), have been successfully used in topology ([4], [6], [8]).

Let $ \langle \mathbb{P},\leq \rangle $ be a partially ordered set. A set $ D \subseteq \mathbb{P} $ is called dense if and only if for every $ p \in \mathbb{P} $, there is a $ d \in D $ such that $ d \leq p $. A set $ Q \subseteq \mathbb{P} $ is said to be compatible if and only if for any finite subset $ F \subseteq Q $, there is a $ p \in \mathbb{P} $ such that $ p \leq r $ for every $ r \in F $. Two elements, $ p_{1} $ and $ p_{2} $, of $ \mathbb{P} $ are said to be incompatible if and only if the set $ \{ p_{1},p_{2} \} $ is not compatible. It is said that a partially ordered set $ \langle \mathbb{P},\leq \rangle $ satisfies the countable anti-chain condition if and only if every set that consists of pairwise-incompatible elements is countable. Martin's axiom ($ \mathsf{MA} $) states the following: If a partially ordered set $ \langle \mathbb{P},\leq \rangle $ satisfies the countable anti-chain condition and if $ \mathcal{P} $ is a family of dense subsets of cardinality less than $ 2^{\aleph_{0}} $, then there is a compatible set $ Q \subseteq \mathbb{P} $ such that for every $ D \in \mathcal{P} $, the intersection $ D \cap Q $ is non-empty.

In the presence of the continuum hypothesis, Martin’s axiom can be proved. The most interesting results are obtained by a combination of Martin’s axiom and the negation of the continuum hypothesis ($ \neg \mathsf{CH} $). The principle $ \diamondsuit_{\omega_{1}} $ contradicts the combination $ \mathsf{MA} + \neg \mathsf{CH} $, since $ \diamondsuit_{\omega_{1}} $ implies $ \mathsf{CH} $. It often turns out that a result that can be inferred from $ \diamondsuit_{\omega_{1}} $ will be disproved under the assumption $ \mathsf{MA} + \neg \mathsf{CH} $. This is, for example, the case with the Suslin hypothesis. Indeed, $ \mathsf{MA} + \neg \mathsf{CH} $ implies the Suslin hypothesis, while $ \diamondsuit_{\omega_{1}} $ implies the negation of the Suslin hypothesis.

The combination $ \mathsf{MA} + \neg \mathsf{CH} $ is compatible with $ \mathsf{ZFC} $ if $ \mathsf{ZF} $ is consistent.

References

[1] M. [M.Ya. Suslin] Souslin, “Problème 3”, Fundam. Math., 1 (1920), pp. 223.
[2] K.J. Devlin, H. Johnsbråten, “The Souslin problem”, Lect. notes in math., 405, Springer (1974).
[3] T.J. Jech, “Lectures in set theory: with particular emphasis on the method of forcing”, Lect. notes in math., 217, Springer (1971).
[4] J. Barwise (ed.), Handbook of mathematical logic, North-Holland (1977), Chapts. B4-B7.
[5] K.J. Devlin, “Aspects of constructibility”, Lect. notes in math., 354, Springer (1973).
[6] V.V. Fedorchuk, “Completely closed mappings and the compatibility of certain general topology theorems with the axioms of set theory”, Mat. Sb., 99: 1 (1976), pp. 3–33. (In Russian)
[7] D.A. Martin, R. Solovay, “Internal Cohen extensions”, Ann. Math. Logic, 2 (1970), pp. 143–178.
[8] V.I. Malykhin, “Topology and forcing”, Russian Math. Surveys, 38: 1 (1983), pp. 77–136; Uspekhi Mat. Nauk, 38: 1 (1983), pp. 69–118.

Comments

Jensen’s combinatorial principles $ \diamondsuit $ and $ \Box $ are called diamond and square respectively. The countable anti-chain condition is sometimes called the countable chain condition, and is then also abbreviated to c.c.c.

References

[a1] K.J. Devlin, “Constructibility”, Springer (1984).
[a2] D.H. Fremlin, “Consequences of Martin’s axiom”, Cambridge Univ. Press (1984).
[a3] T.J. Jech, “Multiple forcing”, Cambridge Univ. Press (1986).
[a4] T.J. Jech, “Set theory”, Acad. Press (1978). (Translated from German)
[a5] K. Kunen, “Set theory, an introduction to independence proofs”, North-Holland (1980).
How to Cite This Entry:
Suslin hypothesis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Suslin_hypothesis&oldid=39906
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article