In mathematical
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, an Aronszajn line (named after
Nachman Aronszajn
Nachman Aronszajn (26 July 1907 – 5 February 1980) was a Polish American mathematician. Aronszajn's main field of study was mathematical analysis, where he systematically developed the concept of reproducing kernel Hilbert space. He also con ...
) is a
linear ordering
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexiv ...
of cardinality
which contains no subset
order-isomorphic to
*
with the usual ordering
* the reverse of
* an
uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
subset of the
Real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s with the usual ordering.
Unlike
Suslin lines, the existence of Aronszajn lines is provable using the standard axioms of set theory. A linear ordering is an Aronszajn line if and only if it is the lexicographical ordering of some
Aronszajn tree In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal ''κ'', a ''κ''-Aronszajn tree is a tree of ...
.
References
Order theory
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