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A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z 2). Such subsystems are essential to reverse mathematics , a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength.
The second order objects are arbitrary sets (or unary predicates) of finite binary strings. S2S has functions s→s0 and s→s1 on strings, and predicate s∈S (equivalently, S(s)) meaning string s belongs to set S. Some properties and conventions: By default, lowercase letters refer to first order objects, and uppercase to second order objects.
Second order approximation, an approximation that includes quadratic terms; Second-order arithmetic, an axiomatization allowing quantification of sets of numbers; Second-order differential equation, a differential equation in which the highest derivative is the second; Second-order logic, an extension of predicate logic
Monadic second-order logic comes in two variants. In the variant considered over structures such as graphs and in Courcelle's theorem, the formula may involve non-monadic predicates (in this case the binary edge predicate E ( x , y ) {\displaystyle E(x,y)} ), but quantification is restricted to be over monadic predicates only.
The system Hilbert asked for a completeness proof of is more like second-order arithmetic than first-order Peano arithmetic. As a nowadays common interpretation, a positive solution to Hilbert's second question would in particular provide a proof that Peano arithmetic is consistent. There are many known proofs that Peano arithmetic is ...
One millionth of a second. Symbol is μs millisecond: 10 −3 s: One thousandth of a second. Shortest time unit used on stopwatches. jiffy (electronics) ~ 10 −3 s: Used to measure the time between alternating power cycles. Also a casual term for a short period of time. centisecond: 10 −2 s: One hundredth of a second. decisecond: 10 −1 s ...
A vertex in a directed graph whose second neighborhood is at least as large as its first neighborhood is called a Seymour vertex. [5]In the second neighborhood conjecture, the condition that the graph have no two-edge cycles is necessary, for in graphs that have such cycles (for instance the complete oriented graph) all second neighborhoods may be empty or small.
In the monadic second-order logic of graphs, the variables represent objects of up to four types: vertices, edges, sets of vertices, and sets of edges. There are two main variations of monadic second-order graph logic: MSO 1 in which only vertex and vertex set variables are allowed, and MSO 2 in which all four types of variables are allowed ...