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Problem in set theory
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin (1920) and published posthumously. It has
Suslin's_problem
Surname list
new Suslin sets Suslin operation Suslin's problem Suslin representation, a set of real numbers built up in a certain way Sergey Suslin (1944–1989), Soviet
Suslin
Russian mathematician
Institute. Suslin died of typhus in the 1919 Moscow epidemic following the Russian Civil War, at the age of 24. His name is associated to Suslin's problem, a
Mikhail_Suslin
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Mathematical tree
ℵ2-Suslin tree, is a longstanding open problem. Glossary of set theory Kurepa tree List of statements independent of ZFC List of unsolved problems in
Suslin_tree
American mathematician
having never published a proof of it, Tennenbaum had also studied Suslin's problem and computably enumerable sets. Over his academic career in the 1960s
Stanley_Tennenbaum
Intersection of Set Theory and General Topology
questions that can be solved using set-theoretic methods, for example, Suslin's problem. In the mathematical field of general topology, a Dowker space is a
Set-theoretic_topology
Proposition in mathematical logic
problems in set theory, and establishing its truth or falsehood was the first of Hilbert's 23 problems presented in 1900. The answer to this problem is
Continuum_hypothesis
German logician and mathematician (1871–1953)
theory. Proposed in 1931, Zermelo's navigation problem is a classic optimal control problem. The problem deals with a boat navigating on a body of water
Ernst_Zermelo
Commutative algebra theorem
The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between
Quillen–Suslin_theorem
Set with exactly one element
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Singleton_(mathematics)
Use of braces for specifying sets
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Set-builder_notation
Condition in order theory and topology
non-empty open subsets of X is countable. The name originates from Suslin's Problem. Every separable topological space has the ccc. Furthermore, a product
Countable_chain_condition
Paradox in set theory
Gottlob Frege of the paradox in Frege's 1879 Begriffsschrift and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's
Russell's_paradox
Pair of logical equivalences
are especially useful when simplifying logical expressions in proofs and problem solving. The laws are named after Augustus De Morgan (1806–1871), who introduced
De_Morgan's_laws
German mathematician (1831–1916)
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Richard_Dedekind
Mathematical concept
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Transfinite_induction
American mathematician (1934–2007)
locally compact group is the convolution of two such functions, solving a problem posed by Walter Rudin. In Cohen (1960), he made a significant breakthrough
Paul_Cohen
Set of the elements not in a given subset
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Complement_(set_theory)
Generalization of "n-th" to infinite cases
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Ordinal_number
The real numbers or their cardinality
axioms characterize the order type of the real number line. Aleph null Suslin's problem Transfinite number Weisstein, Eric W. "Continuum". mathworld.wolfram
Continuum_(set_theory)
Any one of the distinct objects that make up a set in set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Element_of_a_set
Branch of mathematics that studies sets
independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that
Set_theory
Axiom of set theory
} has a model. Unsolved problem in mathematics Does the partition principle imply the axiom of choice? More unsolved problems in mathematics There are
Axiom_of_choice
Elements in exactly one of two sets
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Symmetric_difference
Proof by Alan Turing
infallibly gives a correct "yes" or "no" answer to each instance of the problem. In Turing's own words: "what I shall prove is quite different from the
Turing's_proof
Mathematics textbook
axioms, and quickly develops combinatorial notions such as trees, Suslin's problem, the diamond principle, and Martin's axiom. It develops some basic
Set Theory: An Introduction to Independence Proofs
Set_Theory:_An_Introduction_to_Independence_Proofs
Concept in axiomatic set theory
Press, 0-674-55451-5 Toth, Gabor (2021-09-23). Elements of Mathematics: A Problem-Centered Approach to History and Foundations. Springer Nature. p. 32.
Axiom_schema_of_specification
Mathematician (1845–1918)
function by trigonometric series. Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred
Georg_Cantor
Ultrafilter lemma Tree (set theory) Tree (descriptive set theory) Suslin's problem Absorption law Prewellordering Stone duality Stone's representation
List_of_order_theory_topics
3-volume treatise on mathematics, 1910–1913
established for Principia's axioms of set theory. (See Hilbert's second problem.) Russell and Whitehead suspected that the system in PM is incomplete:
Principia_Mathematica
Axiom used in set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Axiom_of_extensionality
Standard system of axiomatic set theory
hypothesis and the negation of the Suslin hypothesis. Martin's axiom plus the negation of the continuum hypothesis implies the Suslin hypothesis. The constructible
Zermelo–Fraenkel_set_theory
Mathematical concept
Mathematical Reasoning, MacMillan D'Angelo; West (2000), Mathematical Thinking: Problem Solving and Proofs, Prentice Hall Cupillari, The Nuts and Bolts of Proofs
Equivalence_class
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Nested_set_collection
Finite ordered list of elements
preprint D'Angelo, John P.; West, Douglas B. (2000), Mathematical Thinking/Problem-Solving and Proofs (2nd ed.), Prentice-Hall, ISBN 978-0-13-014412-6 Keith
Tuple
Alternative to the standard Zermelo–Fraenkel set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
List of alternative set theories
List_of_alternative_set_theories
Mathematical logician and philosopher
Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: "Are the axioms of a formal system sufficient
Kurt_Gödel
Mathematical set formed from two given sets
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Cartesian_product
Set that is not a finite set
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Infinite_set
\aleph _{1}} -dense sets are order-isomorphic is independent of ZFC. Suslin's problem asks whether a specific short list of properties characterizes the
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Possible axiom for set theory
initial ordinals, and we have δ1 2n+2 = (δ1 2n+1)+, and for n < ω, the 2n-th Suslin cardinal is equal to δ1 2n−1. AD+ Axiom of real determinacy (ADR) Borel
Axiom_of_determinacy
Paradox in set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Burali-Forti_paradox
Infinite set that is not countable
_{1}} . In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that ℵ 1 = ℶ 1 {\displaystyle \aleph _{1}=\beth _{1}} is
Uncountable_set
Term in set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Almost
Czech mathematician
gave the first published proof of the consistency of the existence of a Suslin line. With Karel Prikry, he introduced the notion of precipitous ideal.
Thomas_Jech
Mathematical set containing no elements
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Empty_set
American philosopher and logician (1908–2000)
The problem of non-referring names is an old puzzle in philosophy, which Quine captured when he wrote, A curious thing about the ontological problem is
Willard_Van_Orman_Quine
Complement (set theory) Complete Boolean algebra Continuum (set theory) Suslin's problem Continuum hypothesis Countable set Descriptive set theory Analytic
List_of_set_theory_topics
Proof in set theory
For example, the conventional proof of the unsolvability of the halting problem is essentially a diagonal argument. Also, diagonalization was originally
Cantor's_diagonal_argument
English philosopher and logician (1872–1970)
Unwin 1921. The Analysis of Mind. London: George Allen & Unwin 1922. The Problem of China. London: George Allen & Unwin 1922. Free Thought and Official
Bertrand_Russell
Basic framework of mathematics
(multivariable polynomial equation) has a solution in integers. 1971: Suslin's problem is proven to be independent from ZFC. Starting in 1935, the Bourbaki
Foundations_of_mathematics
Axiom of Zermelo-Fraenkel set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Axiom_of_infinity
Sets whose elements have degrees of membership
"Named Sets in the Analysis of Uncertainty". Methodological and Theoretical Problems of Mathematics and Information Sciences. Kiev: 72–85. Cattaneo, Gianpiero;
Fuzzy_set
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Generic_filter
Set theory concept
1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7. von Neumann, John (1923). "Zur Einführung
Von_Neumann_universe
Branch of mathematics
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Order_theory
Concept in topology
space, a subset is a Suslin space if and only if it is a Suslin set (an image of the Suslin operation). The following are Suslin spaces: closed or open
Polish_space
Set theory axiom extension
determinacy. Axiom of projective determinacy Axiom of real determinacy Suslin's problem Topological game Larson, Paul B. (2023). Extensions of the Axiom of
AD+
Concept in axiomatic set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Axiom_of_power_set
One-to-one correspondence
Chapman & Hall/ CRC Press. D'Angelo; West (2000). Mathematical Thinking: Problem Solving and Proofs. Prentice Hall. Cupillari (1989). The Nuts and Bolts
Bijection
Order whose elements are all comparable
total partial order Ranking – Relationship between items in a set Suslin's problem – Problem in set theory Well-order – Class of mathematical orderings Let
Total_order
Collection of mathematical objects
characterized by the formula. There are several ways for avoiding the problem. One may prove that the formula defines a set; this is often almost immediate
Set_(mathematics)
Size of a possibly infinite set
Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), "Über das Problem der Wohlordnung", Math. Ann., Bd. 76 (4), Leipzig: B. G. Teubner: 438–443
Cardinal_number
Collection of sets in mathematics that can be defined based on a property of its members
Smullyan, Raymond M.; Fitting, Melvin (2010), Set Theory And The Continuum Problem, Dover Publications, ISBN 978-0-486-47484-7 Monk, Donald J. (1969), Introduction
Class_(set_theory)
Set of elements common to all of some sets
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Intersection_(set_theory)
Set whose elements all belong to another set
displaying short descriptions of redirect targets Subset sum problem – Decision problem in computer science Subsumptive containment – System of elements
Subset
Complement (set theory) Complete Boolean algebra Continuum (set theory) Suslin's problem Continuum hypothesis Countable set Descriptive set theory Analytic
List of mathematical logic topics
List_of_mathematical_logic_topics
In mathematics, operation on sets
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Disjoint_union
Every set is smaller than its power set
some natural number, say b {\displaystyle b} . However, this causes a problem. If b {\displaystyle b} is in B {\displaystyle B} , then b {\displaystyle
Cantor's_theorem
Mathematical set that can be enumerated
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Countable_set
Result in mathematics and set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Mostowski_collapse_lemma
German-Israeli mathematician and Zionist (1891–1965)
האוניברסיטה העברית 1943c. "Problems and Methods in Modern Mathematics – 1". In Scripta Mathematica IX (1). 1943d. "Problems and Methods in Modern Mathematics
Abraham_Fraenkel
Set of elements in any of some sets
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Union_(set_theory)
Pair of mathematical objects
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Ordered_pair
System of mathematical set theory
disposes of the Russell antinomy as far as we are concerned." This left the problem of "the domain B", which seems to refer to something. This led to the idea
Zermelo_set_theory
Diagram that shows all possible logical relations between a collection of sets
supplemental rules for the standard Venn diagram, in order to account for certain problem cases. For instance, regarding the issue of representing singular statements
Venn_diagram
Finite sets whose elements are all hereditarily finite sets
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Hereditarily_finite_set
Size of a set in mathematics
presented ten unsolved problems (of a total of 23, later published, now called Hilbert's problems). Of these, he placed "Cantor's problem" (now called the Continuum
Cardinality
Axiom of set theory proposed by Peter Aczel in 1988
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Aczel's_anti-foundation_axiom
Any collection of sets, or subsets of a set
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Family_of_sets
Template that specifies one or more axioms
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Axiom_schema
Axiom in set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Axiom_of_global_choice
Particular class of sets which can be described entirely in terms of simpler sets
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Constructible_universe
Set with algorithmic membership test
computable. The set of busy beaver champions is not computable. Hilbert's tenth problem is not computable. Both A, B are sets in this section. If A is computable
Computable_set
Type of cardinal number in mathematics
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Regular_cardinal
Mathematical concept for comparing objects
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Equivalence_relation
Mathematical set of all subsets of a set
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Power_set
Axiom in the mathematical field of set theory
are no Suslin lines). MA + ¬CH implies that there exists a Whitehead group that is not free; Shelah used this to show that the Whitehead problem is independent
Martin's_axiom
Family of subsets representing "large" sets
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Filter_on_a_set
American logician (1943–2011)
Hampshire. The mathematical context in which Baumgartner worked spans Suslin's problem, Ramsey theory, uncountable order types, disjoint refinements, almost
James_Earl_Baumgartner
Mathematical construction of a set with an equivalence relation
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Setoid
Possible axiom for set theory in mathematics
Diamondsuit Clubsuit Global square The existence of morasses The negation of the Suslin hypothesis The non-existence of 0# and as a consequence The non existence
Axiom_of_constructibility
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Gödel_logic
Concept in axiomatic set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Axiom_of_pairing
Axiom of set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Axiom_of_regularity
Concept in set theory
Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Paul Bernays Georg
Axiom_schema_of_replacement
Line formed by the real numbers
nonempty open intervals in R is countable. In order theory, the famous Suslin problem asks whether every linear continuum satisfying the countable chain condition
Number_line
Informal set theories
be the realm of axiomatic set theory or of axiomatic class theory. The problem, in this context, with informally formulated set theories, not derived
Naive_set_theory
Theorem in set theory
Klein; Walther von Dyck; David Hilbert; Otto Blumenthal (eds.), "Über das Problem der Wohlordnung", Mathematische Annalen (in German), 76 (4): 438–443, doi:10
Schröder–Bernstein_theorem
SUSLINS PROBLEM
SUSLINS PROBLEM
Surname or Lastname
English
English : perhaps a variant of Bullen or an altered form of Bullions, a variant of Bullion.
Boy/Male
Muslim
Slave of the one who gives life and sustains it
Surname or Lastname
English (Essex)
English (Essex) : variant of Sullen.
Surname or Lastname
English and Irish
English and Irish : variant of Mullins.
Surname or Lastname
English and Irish
English and Irish : occupational name from Old French molineux ‘miller’ (see Molyneux).William Mullins (d. 1621) was one of the Pilgrims who sailed on the Mayflower in 1620. He, his wife, and his son died during the first winter at Plymouth Colony, leaving behind his daughter Priscilla, who married John Alden, by whom she had eleven children.
Boy/Male
Arabic, Muslim
Slave of the One who Gives Life and Sustains it
Surname or Lastname
English (Essex)
English (Essex) : variant spelling of Sullens.
Girl/Female
Muslim/Islamic
Morning Star
Boy/Male
Indian
God of Shiva
Biblical
who nourishes, consumes, and sustains the whole
Boy/Male
Biblical
Who nourishes, consumes, and sustains the whole.
Surname or Lastname
English
English : variant of Pullen, with patronymic -s.
Surname or Lastname
English
English : variant of Hoskins.
Boy/Male
Arabic, Muslim
One who Gives Life and Sustains it
Boy/Male
Arabic, Muslim
Slave of the One who Gives Life and Sustains it
Girl/Female
Arabic
Princess of the Muslims; Maker of Dresses
Girl/Female
Arabic, Muslim
Morning Star
Girl/Female
Muslim
Morning star
Boy/Male
Muslim/Islamic
Slave of the one who gives life and sustains it
Surname or Lastname
English
English : variant of Mullins.
SUSLINS PROBLEM
SUSLINS PROBLEM
Boy/Male
Australian, Greek
Regal; Royal; Kingly
Boy/Male
Hindu
The Sun
Girl/Female
Muslim/Islamic
Active energetic
Girl/Female
Muslim/Islamic
Slave of the enlarger
Girl/Female
Arabic, Muslim
True Believer; Upright
Girl/Female
Indian, Punjabi, Sikh
Fish; Horoscope; Raashi
Boy/Male
Tamil
Pranand | பà¯à®°à®¾à®¨à®‚த
Happy life
Boy/Male
Australian, Chinese, French, German, Latin, Swiss
Blessed
Girl/Female
English
Faithful; Warm Hearted
Girl/Female
Tamil
Priyadutta | பà¯à®°à®¿à®¯à®¾à®¤à¯à®¤à¯à®¤à®¾
Earth
SUSLINS PROBLEM
SUSLINS PROBLEM
SUSLINS PROBLEM
SUSLINS PROBLEM
SUSLINS PROBLEM
n.
A thin cotton, white, dyed, or printed. The name is also applied to coarser and heavier cotton goods; as, shirting and sheeting muslins.
n.
Coarse plain India muslins.
a.
Wearing buskins.
n.
The suslik.
n.
One who sustains a great burden.
n.
See Suslik.
n.
A seaweed. See Baddrelocks.
n.
Muslin.
a.
Belonging to sutlers; engaged in the occupation of a sutler.
n.
A kind of muslin.
v. t.
To keep from falling; to bear; to uphold; to support; as, a foundation sustains the superstructure; a beast sustains a load; a rope sustains a weight.
pl.
of Sully
a.
Trodden by buskins; pertaining to tragedy.
n.
The suslik.
n.
A sour, morose fellow.
n.
One who, or that which, sustains.
n.
That which sullies or defiles.