Kitcher (1984) believes that species are sets of organisms.
An individual with additional chromosome sets is called a polyploid.
We compared the older sets of our picks against the newer sets.
Sets range from camping sets for one, two, or four people to ski clothing sets for men, women, and even kids.
No matter how many sets have been formed, it is possible to form even more.
It implies the axiom of choice for countable sets of sets but is incompatible with the unrestricted axiom of choice.
Thus, the hierarchy of sets P((ωω)k) ∩ Lα(ℝ) (for α ∈ On) does indeed yield a transfinite extension of the projective sets.
Pure set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets.
This book introduces readers to the novel concept of spherical fuzzy sets, showing how these sets can be applied in practice to solve various decision-making problems.
We shall discuss three central regions of the hierarchy of definability—the Borel sets of reals, the projective sets of reals, and the sets of reals in L(ℝ).
The BBC’s Royal Charter sets out a clear framework for governing and regulating the BBC: a new Board that sets its strategy, runs its operations and is responsible for its output.
It is possible (but harder) to prove Replacement as well in the realm of well-founded sets (which can be the entire universe of sets if Foundation for classes is added as an axiom).
The axiom Δ̰11-determinacy (equivalently, Borel-determinacy) is a theorem of ZFC and lies at the heart of the structure theory of Borel sets (and, in fact, the Σ̰11-sets and Σ̰12 sets).
Type theories characteristically assign types to entities, distinguishing, for example, between numbers, sets of numbers, functions from numbers to sets of numbers, and sets of such functions.
Over the course of several weeks we paid attention to how comfortable the dining sets were for different purposes, as well as asked visitors to compare the comfort of the seats and the overall aesthetics of the sets.
These can be taken to be the axioms of Z (other than extensionality and choice, which are not needed): the sethood of pairs of sets, unions of sets, power sets of sets, and the existence of an infinite set are enough to give us the world of ZFC.
The category Set is a topos; so also are (i) the category Set(B) of Boolean-valued sets and mappings in any Boolean extension V(B) of the universe of sets; (ii) the category of sheaves of sets on a topological space; (iii) the category of all diagrams of maps of sets
Perhaps surprisingly, given that constructions like this provide complements for all sets, they give a central rôle to the concept of wellfounded set in that the wellfounded sets of the new model can be made to be an isomorphic copy of the wellfounded sets in the structure one starts with.
The topology on projective space appropriate for algebraic geometry is the Zariski topology, defined not by its open sets but rather by its closed sets, which are taken to be the algebraic sets, namely those sets constituting the common zeros of a set of homogeneous polynomials.
This is shown by demonstrating first that the discrete sets (and more particularly the (closed) sets of isolated points in the topology) satisfy an analogue of Replacement (a definable function (defined by a formula which need not be positive) with a discrete domain is a set), and so an analogue of separation, then by showing that well-founded sets are isolated in the topology and the class of well-founded sets is closed under the constructions of ZF.
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This is shown by demonstrating first that the discrete sets and more particularly the closed sets of isolated points in the topology satisfy an analogue of Replacement a definable function defined by a formula which need not be positive with a discrete domain is a set and so an analogue of separation then by showing that well-founded sets are isolated in the topology and the class of well-founded sets is closed under the constructions of ZF